Renormalization Theory for Hamiltonian Systemselib.suub.uni-bremen.de/diss/docs/E-Diss440_Pronine.pdf · Renormalization Theory for ... I have bene ted from useful discussions with

Renormalization Theoryfor

Hamiltonian Systems

Mikhail Pronine

Universitat Bremen

2002

Renormalization Theory

for

Hamiltonian Systems

Vom Fachbereich fur Physik und Elektrotechnikder Universitat Bremen

zur Erlangung des akademischen Grades

DOKTOR DER NATURWISSENSCHAFTEN (Dr. rer. nat.)

genehmigte Dissertation

von

Dipl.-Math. Mikhail Pronine

aus Simferopol

Referent: Professor Dr. P. RichterKorreferent: Professor Dr. H. Schwegler

Tag des Promotionskolloquiums: 16.12.2002

Preface

This thesis is devoted to the study of the breakup of invariant tori of irrational windingnumbers in Hamiltonian systems with two degrees of freedom. Due to topological reasons,the decay of invariant tori in such systems is closely related to the onset of widespreadchaos. To give an estimate of where in parameter space the breakup of an invariant torusoccurs, an approximate renormalization scheme is derived. The scheme is applied to anumber of systems (the paradigm Hamiltonian of Escande and Doveil, the Walker andFord model, a model of the ethane molecule, the double pendulum, the Baggott system,limacon billiards).

The work is organized as follows.Chapter 1 describes the behavior of a generic Hamiltonian system with more than

one degree of freedom. The emphasis is put on systems with two degrees of freedom.We introduce the main problem of the work, i.e., the problem of finding the thresholdto global chaos in terms of the breakup of the ”last” invariant KAM torus. There exista number of analytical and numerical methods to deal with the problem. We reviewthese methods in Chapter 2. Our version of the renormalization group approach to theproblem is discussed in Chapter 3. The method is applied to various systems in Chapter4. Chapter 5 summarizes the results of the work.

Appendix A is devoted to the normal form for Hamiltonian systems with two degreesof freedom. In Appendix B we discuss classical perturbation theory and its applicationto the normal form. Appendix C contains some useful formulae for the calculation ofderivatives of implicit functions. The realization of the RG approach to the study of thebreakup of invariant tori in the Maple computer algebra system is presented in AppendixD.

I would like to thank my scientific advisor Prof. Peter H. Richter for supervising thework. I have benefited from useful discussions with former and current members of thegroup Nichtlineare Dynamik. I particularly thank Dr. Holger Dullin, Jan Nagler, Dr.Hermann Pleteit, Dr. Holger Waalkens.

v

Contents

1 Introduction 1

2 Criteria for the Breakup of KAM Tori 7

2.1 Sup map analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 The method of overlapping resonances . . . . . . . . . . . . . . . . . . . . 11

2.3 Greene’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 The renormalization group approach . . . . . . . . . . . . . . . . . . . . . 16

2.5 Comparison between the different methods . . . . . . . . . . . . . . . . . . 18

3 Renormalization Theory 19

3.1 Normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 The renormalization operator . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 The renormalization map for the normal form . . . . . . . . . . . . . . . . 23

3.4 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Applications 31

4.1 Application to the paradigm Hamiltonian . . . . . . . . . . . . . . . . . . . 31

4.2 Application to the Walker and Ford model . . . . . . . . . . . . . . . . . . 36

4.3 Application to a model of the ethane molecule . . . . . . . . . . . . . . . . 41

4.4 Application to the double pendulum problem . . . . . . . . . . . . . . . . . 53

4.4.1 The Lagrange function . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4.2 Integrable cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4.3 The Hamilton function . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4.4 The integrable limit of high energies . . . . . . . . . . . . . . . . . 57

4.5 The Baggott H2O Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 65

4.6 Limacon billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Conclusions 89

A The Normal Form 91

B Classical Perturbation Theory 95

B.1 Application to the normal form . . . . . . . . . . . . . . . . . . . . . . . . 101

C Derivatives of Implicit Functions 105

vii

viii

D Maple Program 107D.1 Renormalization operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

D.1.1 Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107D.1.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108D.1.3 Numerical approximation . . . . . . . . . . . . . . . . . . . . . . . 109D.1.4 Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110D.1.5 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

D.2 The paradigm Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 112D.3 Application to a model of the ethane molecule . . . . . . . . . . . . . . . . 113D.4 Application to the Baggott Hamiltonian . . . . . . . . . . . . . . . . . . . 114

Chapter 1

Introduction

This thesis deals with the dynamic behavior of classical Hamiltonian systems with twodegrees of freedom. Such systems and their stability properties are of interest in diversefields (celestial mechanics [28, 45], plasma physics [12, 21], chemical physics [28, 24, 25]to name just a few). Hamiltonian systems can be divided into two classes, integrable andnon-integrable. Let us recall the notion of integrability which is of utmost importance inthe study of Hamiltonian systems. Consider a Hamiltonian system defined by a functionH(p, q, t) in the phase space T ∗Q(p, q). The function H is called the Hamiltonian of thesystem. The equations of motion are Hamilton’s equations

dp

dt= −∂H

∂q,

dq

dt=∂H

∂p. (1.1)

A function F in the phase space is said to be a constant of motion if along any trajectory itsvalue is constant. Constants of motion are also referred to as first integrals . In autonomoussystems (i.e., H is explicitly time independent, H = H(p, q)) the Hamiltonian H is aconstant of motion. In what follows we restrict ourselves to the case of autonomoussystems.

The Poisson bracket of two functions F = F (p, q, t) and G = G(p, q, t) is defined tobe

[F,G] =n∑i=1

(∂F

∂pi

∂G

∂qi− ∂F

∂qi

∂G

∂pi

). (1.2)

The time dependence of a function F = F (p, q, t) is given by

dF

dt= [H,F ] +

∂F

∂t. (1.3)

Assume that the function F does not depend on time explicitly. It follows from (1.3) thatF is a constant of motion if and only if [H,F ] = 0. Two functions F and G are said tobe in involution if their Poisson bracket [F,G] vanishes.

Two functions F (p, q) and G(p, q) are called (functionally) independent if their gradi-ents (∂F/∂p1, . . . , ∂F/∂qn) and (∂G/∂p1, . . . , ∂G/∂qn) are linearly independent for almostevery point (p, q).

We are ready now to introduce the notion of integrability for Hamiltonian systems. AHamiltonian system of n degrees of freedom is called integrable if there exist n independent

1

2 CHAPTER 1. INTRODUCTION

constants of motion F1, . . . , Fn which are in involution [3]. Integrable systems are alsoreferred to as completely integrable ones.

The geometric description of integrable systems is given by the Liouville-Arnold the-orem [3]. According to this result the phase space of an integrable Hamiltonian systemH0 of n degrees of freedom is foliated by the invariant sets

(p, q) ∈ T ∗Q : F1(p, q) = c1, . . . , Fn(p, q) = cn, (1.4)

with F1, . . . , Fn being constants of motion. Moreover, in generic situations the motionon these invariant sets is periodic or quasiperiodic. If the energy surface H0(p, q) = h0

is compact, then connected components of the invariant sets are just n-dimensional tori.Locally there exists a canonical transformation (p, q) → (I,θ) such that in the newcoordinates (I,θ) the Hamiltonian H0 does not explicitly depend on the angle variablesθ:

H0 = H0(I). (1.5)

The coordinates (I,θ) are referred to as action-angle coordinates . The equations of motionare readily solvable in the action-angle coordinates. Indeed, Hamilton’s equations are

dI

dt= −∂H0

∂θ= 0,

dθ

dt=∂H0

∂I. (1.6)

Thus, the actions I are constants of motion, and the time evolution of the angle variablesis periodic or quasiperiodic with constant frequencies ∂H0/∂I.

A generic Hamiltonian system with two or more degrees of freedom is non-integrable.In this case there is no simple geometric description of motion. Moreover, the dynamicbehavior of a generic Hamiltonian system is at least partially chaotic.

Recall some relevant definitions from the theory of dynamical systems [13]. Let M bean arbitrary set. Consider a one-parametric family of maps f t : M → M from M intoitself. The pair (M, f t) is called a dynamical system. The set M is referred to as thephase space of the dynamical system (M, f t). The family f t is said to be the dynamics .If the parameter t is continuous, then the dynamical system (M, f t) is called the flow .

In the case of Hamiltonian systems we define the corresponding flow in the followingway. Choose the phase space T ∗Q = (p, q) as the phase space of the flow. Given apoint (p0, q0) from T ∗Q, the map f t assigns to it the solution of Hamilton’s equationswith initial conditions (p0, q0) at time t = 0.

A subset M ′ of the phase space M is called an invariant set if f t(M ′) lies in M ′ forevery t. The notion of invariant set plays an important role in the study of dynamicalsystems. Indeed, having identified all invariant sets of a given dynamical system, we caninvestigate the behavior of the dynamics f t on the invariant sets independently from eachother.

Assume now that the phase space M of some dynamical system (M, f t) is a metricspace. For example, the phase space T ∗Q of a Hamiltonian system with the Euclidianmetric is a metric space.

The dynamical system (M, f t) is said to be topologically transitive if for any two opensets U and V from M there exists t > 0 such that the intersection f t(U)∩V is not empty.Note that a topologically transitive dynamical system has no non-trivial open invariant

3

sets. Remember that if the dynamics f t of a dynamical system has a dense orbit, thenthe system is topologically transitive.

We say that the dynamics f t of a dynamical system (M, f t) has sensitive dependence oninitial conditions if there exists δ > 0 such that, for any x from M and any neighborhoodO(x) of x, there exist y in O(x) and t > 0 such that |f t(x) − f t(y)| > δ. Recall that weassume M to be a metric space. Thus, the distance |f t(x)− f t(y)| is well-defined.

Finally, a dynamical system (M, f t) is said to be chaotic if (i) f t has sensitive depen-dence on initial conditions, (ii) f t is topologically transitive, and (iii) periodic points aredense in M , see [13].

Integrable systems and chaotic ones are exceptions among Hamiltonian systems withtwo or more degrees of freedom. A generic system contains both regular and chaoticregions in the phase space. Given an arbitrary Hamiltonian system H, the questionarises as to how the complicated interplay of its chaotic and regular motions can bedescribed. The usual approach based on perturbation theory consists of introducing theso-called perturbation parameter and finding an integrable limit H0 for the system H.The integrable limit corresponds to some value, say zero, of the perturbation parameter.As the perturbation parameter varies, we obtain a family of Hamiltonian systems. Thefamily includes the integrable system H0 and the initial system H. Using the knowledgeof the dynamics of the former system, we can now try to describe the dynamic behaviorof the latter system. For example, one can represent solutions to the initial problem asseries with respect to the perturbation parameter.

Formally, for every Hamiltonian systemH and every integrable Hamiltonian systemH0

the former system can be represented as perturbation to the latter one. Indeed, considerthe family H0 + ε(H −H0) of Hamiltonian systems. The initial system H corresponds tothe value one of the perturbation parameter ε. In the limit ε→ 0 we obtain the integrablesystem H0. If for the given H the integrable limit H0 is chosen in an arbitrary way, thenthe behavior of H can be very different from that of H0. In other words, there is almostno hope that the system H can be described on the basis of H0. Perturbation theoryworks well only provided that perturbation is small . The choice of an integrable limit fora given non-integrable system is thus by no means trivial. Sometimes, especially workingwith physical systems, a number of possible integrable limits arise in a natural way.

Let us consider the behavior of an integrable system under small perturbation. Werefer to such systems as near-integrable ones. In what follows we confine ourselves tothe case of Hamiltonian systems with two degrees of freedom. Moreover, assume thatthe energy surface H0 = h is compact so that according to the Liouville -Arnold theo-rem it is foliated by two-dimensional invariant tori. Introducing action-angle coordinates(I1, I2, θ1, θ2) for the integrable system H0, we can express the frequencies of a given in-variant torus as (ω1, ω2) = (∂H0/∂I1, ∂H0/∂I2). The ratio W = ω1/ω2 is said to be thewinding number of the torus. The winding number of a torus is also called the windingratio. We have to distinguish between tori of rational and of irrational winding numbers.We will refer to invariant tori of rational winding numbers as rational invariant tori , orresonant invariant tori . An invariant torus of irrational winding number is said to benon-resonant , or irrational .

The behavior of rational invariant tori under perturbation is described by the Poincare-Birkhoff theorem applied to the Poincare map which the continuous flow generates on


a suitable closed surface of section, see Chapter 2. Note that a rational invariant torusof winding number W = p/q is foliated by an infinite number of periodic orbits. ThePoincare -Birkhoff theorem states that only a finite number of them survive under per-turbation. Half of these periodic orbits are elliptic, and the other half are hyperbolic. Ina Poincare surface of section θ2 = 0 mod 2π, the rational torus itself gives rise to a chainof kq islands with k being 1, or 2, or 3, . . . . The islands are separated by chaotic bandswhich are located in a neighborhood of the stable and unstable manifolds, or separatrices ,of the hyperbolic periodic orbits.

The behavior of irrational invariant tori can be studied using the Kolmogorov-Arnold-Moser theorem [26, 2, 32]. In order to formulate this result we need a number of pre-liminary definitions. Consider a near-integrable Hamiltonian system with n degrees offreedom. We assume that the Hamiltonian is written in the form H = H0 + εH1, whereH0 is an integrable system, and ε is a perturbation parameter. Introducing action-anglevariables (I,θ) for the unperturbed part H0, we obtain the following expression for theHamiltonian:

H(I,θ) = H0(I) + εH1(I,θ). (1.7)

The system H0 is said to be nondegenerate if the condition

det∂2H0

∂I2 = det∂ω

∂I6= 0 (1.8)

is satisfied. We denote in (1.8) the frequencies of the unperturbed system by ω.The system H0 is said to be iso-energetically nondegenerate if one of the frequencies

does not vanish and the ratios of the remaining n − 1 frequencies to it are functionallyindependent on the energy surface H0 = h. Formally, the last condition can be writtenin the form

det

(∂2H0

∂I2 ωωt 0

)6= 0. (1.9)

The determinant in (1.9) represents the Jacobian of the mapping of the (n−1)-dimensionalsurface H0 = h into the (n−1)-dimensional projective space (∂H0/∂I1, . . . , ∂H0/∂In). Letus discuss this condition in the case of Hamiltonian systems with two degrees of freedom.The winding number W is given by

W =ω1

ω2

, ω1 =∂H0(I1, I2)

∂I1

, ω2 =∂H0(I1, I2)

∂I2

. (1.10)

Consider the three-dimensional space (I1, I2, h). The Hamiltonian H0(I1, I2) gives riseto a two-dimensional surface (I1, I2, H0(I1, I2)) in the space (I1, I2, h). The vector v1 =(ω1, ω2, 1) is orthogonal to this surface. The one-dimensional ”energy surface” H0 = h0

may be viewed as the intersection of the surface (I1, I2, H0(I1, I2)) with the plane h = h0

whose normal is given by the vector v2 = (0, 0, 1). Finally, the gradient of the functionW = W (I1, I2) reads

v3 =

(∂W

∂I1

,∂W

∂I2

, 1

)=

(1

ω22

(ω2∂ω1

∂I1

− ω1∂ω2

∂I1

),

1

ω22

(ω2∂ω1

∂I2

− ω1∂ω2

∂I2

), 1

). (1.11)

5

The linear independence of the vectors v1, v2, v3 ensures that the winding number Wvaries smoothly on every energy surface. It is easy to show that the condition for the vec-tors v1, v2, v3 to be linearly independent coincides with the isoenergetic non-degeneracycondition.

We are ready now to formulate the KAM theorem, see, for example, [4].Theorem. Consider a near-integrable Hamiltonian system which is nondegenerate

or iso-energetically nondegenerate. Then for a sufficiently small perturbation irrationalinvariant tori with Diophantine frequency vectors do not decay but are only deformed.

Recall that a vector (ω1, . . . , ωn) is called Diophantine if for every nonzero integer-valued vector (k1, . . . , kn) the following inequality is satisfied:

|k1ω1 + · · ·+ knωn| ≥K(ω)

‖k‖, (1.12)

with ‖k‖ = maxi|ki| being the norm of the vector (k1, . . . , kn) and K(ω) being a constant.We refer to the tori from the KAM theorem as KAM tori . Note further that given

a KAM torus, angle coordinates for this torus can be introduced, so that the motionon the torus is quasiperiodic with the same frequencies as for the motion on the initialunperturbed torus. If the isoenergetic non-degeneracy condition is fulfilled, then the KAMtorus lies on the energy surface H = h with the same value h of energy as the initialunperturbed torus.

The KAM theorem is of utmost theoretical importance. To name just a few conse-quences of this result, let us mention that the KAM theorem elucidates the problem ofthe small divisors in classical perturbation theory, clarifies the questions concerning theconvergence of series in perturbation theory, explains the stability of elliptic periodic or-bits in nonlinear Hamiltonian systems with two degrees of freedom. From the practicalpoint of view, the KAM theorem has a significant restriction. The theorem is valid onlyfor sufficiently small perturbation. Its original version was restricted to perturbationsof order ε ∼ 10−48. Though the subsequent versions have dramatically improved thisestimate, to find a realistic value of perturbation corresponding to the breakup of a givenKAM torus remains an open problem.

In the case of Hamiltonian systems with two degrees of freedom the KAM theoremhas an interesting geometric interpretation. Consider an unperturbed non-resonant toruswith Diophantine frequencies on the energy surface H0 = h. If the isoenergetic non-degeneracy condition is satisfied, then, in accordance with the KAM theorem, for suffi-ciently small perturbation there exists a KAM torus with the same winding number asthe initial one on the energy surface H = h. This energy surface can be viewed asa three-dimensional surface in the four-dimensional phase space. The KAM torus is ofdimension two. For topological reasons the energy surface H = h is divided by theKAM torus into two invariant three-dimensional regions (at least locally). The regionsare invariant sets of the Hamiltonian system in question, so that it is impossible for atrajectory starting in one region to reach the other.

Recall that according to the Poincare -Birkhoff theorem and related results resonantunperturbed tori give rise to chaotic regions in the phase space of the perturbed system.In the case of Hamiltonian systems with two degrees of freedom the chaotic regions cor-responding to different resonant tori are separated one from another by KAM tori. Thus,


KAM tori may be considered as barriers to widespread chaos.The following scenario for the transition from regular to chaotic motion can often be

observed. Consider a near-integrable Hamiltonian system with two degrees of freedom.Let us confine ourselves to a fixed value h of energy. For small perturbation the larger partof the energy surface consists of KAM tori. Resonant tori of the unperturbed system giverise to chaotic regions. These regions are relatively small for small perturbation. Differentchaotic regions are separated by KAM tori. As perturbation increases, the chaotic regionsbecome larger and larger. Simultaneously, more and more KAM tori are destroyed. Thechaotic regions corresponding to different resonant tori can merge provided that all theKAM tori between them broke up. In some systems for sufficiently large perturbation thechaotic regions form two invariant sets divided by a KAM torus. As perturbation increasesfurther, this KAM torus, which is called the last KAM torus , is also destroyed. The twochaotic regions merge into one, so that a transition to global chaos can be observed. Thescenario just described holds, e. g., in the case of a double pendulum, see [37].

We are ready now to formulate the main problem of this work. Consider a near-integrable Hamiltonian system H0 + εH1 with two degrees of freedom. Here H0 is anintegrable Hamiltonian system, ε is the perturbation parameter. Consider two resonanttori, or resonances, of the system H0 on the energy surface H0 = h. The tori give rise totwo chaotic regions on the energy surface H = h. For small values of the perturbationparameter ε these two chaotic regions are separated by KAM tori. As perturbationincreases, more and more of these KAM tori decay. At some value εcrit of the perturbationparameter ε the last KAM torus between the chaotic regions in question is destroyed.We refer to this value of perturbation as the critical value. The problem is as follows.Given two resonances on the energy surface H0 = h, find the critical value εcrit of theperturbation parameter.

The following empirical rule has been observed studying various systems. The windingnumber of the last KAM torus is a noble number . Recall the definition of a noble number.An arbitrary real number x can be represented in a unique way by a continued fraction

x = a0 +1

a1 + 1a2+...

(1.13)

with a0 being an integer, and ai, i > 0, positive integers, see [23, 33]. We will write thecontinued fraction expansion (1.13) for x in the form x = [a0, a1, . . . ]. Note that a rationalnumber has a finite continued fraction. The number x is said to be noble if its continuedfraction x = [a0, a1, . . . ] satisfies the property that ai = 1 if i > i0 for some i0.

In the next chapter we review methods for studying the breakup of KAM tori and thetransition to widespread chaos.

Chapter 2

Criteria for the Breakup of KAMTori

In this chapter we discuss a number of methods dealing with the study of the breakup ofKAM tori in Hamiltonian systems with two degrees of freedom.

The majority of the methods described below are concerned with two-dimensional area-preserving mappings. In order to make use of these approaches in the case of Hamiltoniansystems with two degrees of freedom we employ the technique of Poincare sections . Thistechnique reduces, at least locally, the study of these systems to that of two-dimensionalarea-preserving mappings. Consider a Hamiltonian system H with two degrees of freedom.Take the energy surface H = h corresponding to some value h of energy. Consider sometwo-dimensional surface S contained in the three-dimensional energy surface. Take somepoint p on the surface S. Let γ be the trajectory with initial condition p and p′ the nextintersection of γ with the surface S. The mapping P : S → S of the surface S into itselfdefined in this way is said to be the return map, or the Poincare map. The Poincare mapis a two-dimensional map. It is always possible to introduce some regular measure onS so that the return map becomes area-preserving with respect to this measure. Thus,we reduce the problem of the breakup of KAM tori in Hamiltonian systems with twodegrees of freedom to the problem of the breakup of KAM circles in two-dimensionalarea-preserving mappings.

2.1 Sup map analysis

This section is devoted to the so-called sup map analysis , see [20] and references therein.We describe the sup map analysis for the case of two-dimensional mappings.

Consider the famous standard map

In+1 = In +K sin θn, (2.1)

θn+1 = θn + In+1 (2.2)

with K being the perturbation parameter. The phase space is assumed to be the two-dimensional torus (I, θ)|0 ≤ I < 2π, 0 ≤ θ < 2π, . The standard map is also referred

7

8 CHAPTER 2. CRITERIA FOR THE BREAKUP OF KAM TORI

K=0.0000

0.00

1.04

2.09

3.14

4.18

5.23

6.28

0.00 1.04 2.09 3.14 4.18 5.23 6.28

I

θ

K=0.5000

0.00

1.04

2.09

3.14

4.18

5.23

6.28

0.00 1.04 2.09 3.14 4.18 5.23 6.28

I

θ

K=1.0000

0.00

1.04

2.09

3.14

4.18

5.23

6.28

0.00 1.04 2.09 3.14 4.18 5.23 6.28

I

θ

K=1.5000

0.00

1.04

2.09

3.14

4.18

5.23

6.28

0.00 1.04 2.09 3.14 4.18 5.23 6.28

I

θ

Figure 2.1: The phase portrait for the standard map with K = 0 (top left), K = 0.5 (topright), K = 1 (bottom left), K = 1.5 (bottom right).

2.1. SUP MAP ANALYSIS 9

to as the Chirikov-Taylor map. We present the phase portrait for the standard map inFigure 2.1.

Note that the value K = 0 corresponds to the integrable case. Indeed, if K = 0, thenthe standard map becomes

In+1 = In, (2.3)

θn+1 = θn + In+1, (2.4)

so that I is a constant of motion and I/2π the winding number. An analog of theKAM theorem for Hamiltonian maps, Moser’s twist theorem [32], guarantees the existenceof one-dimensional KAM tori (circles) for small values of perturbation K. With thehelp of Figure 2.1 one can easily identify two large chaotic regions in the vicinity of theresonances which correspond to the resonant circles I = 0 and I = π in the integrablelimit. As perturbation becomes larger, these two chaotic regions grow, and more and moreKAM tori are destroyed. The last KAM torus to be destroyed has the winding numberg = (

√5 − 1)/2, where g is the golden mean. Note that the last KAM torus does not

divide the phase space into two invariant sets. Indeed, because of the periodicity of theaction variable I it is possible for a trajectory starting in the bottom part of the phasespace to reach the upper one. Nevertheless, as long as the last KAM torus exists, thiskind of transition is the only possible one. The transition through the last KAM torus isnot possible.

If we omit the periodicity condition on the action variable I, then the phase spacebecomes a cylinder R × S1. In this case, the last KAM torus divides the phase spaceinto two different invariant sets. Actually, there are then an infinite number of last KAMtori. All of them are just copies of the last KAM torus in question. The torus becomes abarrier to widespread chaos. In the following we consider the standard map as lifted tothe cylinder.

As we have just mentioned, as long as the last KAM torus exists, a trajectory withinitial condition in the invariant set below the last KAM torus remains in the sameinvariant set. More precisely, let us fix some value of the perturbation parameter K andchoose a point (I0, θ0) as initial condition of some trajectory. Assume that the point(IKAM0 , θ0) belongs to the last KAM torus, and that the inequality I0 < IKAM0 holds.Then the trajectory with initial condition (I0, θ0) lies in the bottom invariant set, and theinequality

maxIn∣∣∣n = 0, 1, . . . < maxIKAMn

∣∣∣n = 0, 1, . . . (2.5)

is satisfied. Here In denotes the I-values of the images of (I0, θ0) under the standard map.Let Imax(I, θ, n) be the maximal value of I among the first n iterates of the orbit with

initial conditions (I, θ). Figure 2.2 shows Imax(I, θ, n) as a function of I for θ = 0 andn = 1000 in the case of the standard map. The following three structures are clearlyvisible: (i) monotonically increasing variations, (ii) noisy variations, and (iii) V-shapedstructures, see [20]. The monotonic variations correspond to the regions where manyKAM tori exist. The crossing into a chaotic region gives rise to a jump in Imax followedby a noisy variation. V-shaped structures are related to island chains.


0.00

2.00

4.00

6.00

8.00

10.0

0.00 1.05 2.10 3.15 4.20 5.25 6.30

K=0

sup

I

0.00

2.00

4.00

6.00

8.00

10.0

0.00 1.05 2.10 3.15 4.20 5.25 6.30

K=0.5

sup

I

0.00

2.00

4.00

6.00

8.00

10.0

0.00 1.05 2.10 3.15 4.20 5.25 6.30

K=1

sup

I

0.00

2.00

4.00

6.00

8.00

10.0

0.00 1.05 2.10 3.15 4.20 5.25 6.30

K=1.5

sup

I

Figure 2.2: The sup map for K = 0 (top left), K = 0.5 (top right), K = 1 (bottom left),K = 1.5 (bottom right).

2.2. THE METHOD OF OVERLAPPING RESONANCES 11

The sup map analysis can be applied to the standard map as follows. Consider theregion 0 < I < Imax = π. A number of resonances in this region can easily be identifiedwith the help of Figure 2.1. We study the onset of widespread chaos between the tworesonances corresponding to the island chains consisting of one and two islands respec-tively. First, we choose bracketing bounds for the critical value Kcrit. For example, wecan confine ourselves to the interval [0, 5]. For a fixed value of the perturbation parameterK we study the trajectories with initial conditions (Ij, θ0) where Ij = I0 + j∆. In ouractual computation we take θ0 to be π, I0 = 4π/5, the grid spacing ∆ = π/500, andj = 0, 1, . . . , 100. The initial value I0 is chosen in such a way that the point (I0, θ0) liesbelow the last KAM torus. For a given value of the perturbation parameter K, a trajec-tory with initial condition (Ij, θ0) is iterated n times. Denote the maximal and minimal

values of I among the first n iterates of (Ij, θ0) by I(j)max and I

(j)min respectively. Let I

(0,1,...,j)max

be the maximum of I(0)max, I

(1)max, . . . , I

(j)max. Consider the following conditions:

I(0,...,j−1)max < I(j)

max, (2.6)

I(0,...,j−2)max < I(j−1)

max , (2.7)

I(j−1)min < I

(j)min, (2.8)

I(j)max < Imax. (2.9)

If all these conditions are fulfilled for some j > 1, we conclude that for the given K thereexists a KAM torus between the resonances under study and the corresponding chaoticregions are localized. Otherwise, the onset of widespread chaos is expected for this valueof the perturbation parameter. Let us comment the conditions (2.6-2.9). Inequalities

(2.6-2.7) are fulfilled in regions where I(j)max varies monotonically with respect to j. Thus,

these conditions are a strong evidence that the trajectory lies either in a region dominatedby KAM tori or in a region corresponding to nested islands. If inequalities (2.6-2.7) aresatisfied, then inequality (2.8) guarantees that the trajectory does not lie in a regiondominated by nested islands. The last condition (2.9) implies that the trajectory liesoutside the top chaotic region.

In order to find the critical value of K on the given bracketing interval with prescribedaccuracy, we can now use for example the bisection method.

The results of the application of the sup map analysis to the study of the onset ofwidespread chaos for the standard map are presented in Figure 2.3 and in Table 2.1. Theyare to be compared with the estimate Kcrit = 0.9716 . . . delivered by Greene’s method,see below.

2.2 The method of overlapping resonances

The method of overlapping resonances is due to Chirikov [12].

We discuss the method in the case of non-autonomous Hamiltonian systems with onedegree of freedom. Consider the Hamiltonian

H(I, θ, t) = H0(I) +H1(I, θ, t) (2.10)


0.9000

0.9400

0.9800

1.0200

1.0600

1.1000

10000. 25000. 40000. 55000. 70000. 85000. 100000

K

n

Figure 2.3: The critical value of K for the onset of widespread chaos (numerical calcu-lation using sup map analysis). The accuracy of the bisection algorithm is 0.001.

n Kcrit

10000 1.00820000 1.00830000 1.00140000 0.98650000 0.98360000 0.98370000 0.98380000 0.97990000 0.979100000 0.979

Table 2.1: The critical value of K for the onset of widespread chaos against the numberof iterations per orbit (numerical calculation using sup map analysis). The accuracy ofthe bisection algorithm is 0.001.

2.2. THE METHOD OF OVERLAPPING RESONANCES 13

with H1(I, θ, t) being the perturbation term. Expand the perturbation part in a Fourierseries:

H1(I, θ, t) =∑m,n∈Z

Vmn(I) cos(mθ + nt+ γmn). (2.11)

As an example we consider the standard map. It can be represented as a non-autonomousHamiltonian system in the following way:

H =I2

2−K cos θ

∑n∈Z

δ(t− n). (2.12)

Figure 2.1 may now be viewed as the stroboscopic map of the system given by Hamiltonian(2.12). The corresponding perturbation part is

H1(I, θ, t) = −K cos θ(1 + 2∞∑m=1

cos(2πmt)). (2.13)

Our aim now is to study the onset of widespread chaos between the resonances (m1,m2) =(1, 0) and (p1, p2) = (1, 1). They correspond to the terms of the form V10 cos(θ) andV11 cos(θ+2πt) in the Fourier series expansion for the perturbation H1. Using the identity

2 cos θ cos(2πt) = cos(θ + 2πt) + cos(θ − 2πt), (2.14)

we obtain

H1 = −K

(cos θ + cos(θ + 2πt) + cos(θ − 2πt) + 2 cos θ

∞∑m=2

cos(2πmt)

). (2.15)

If we consider the standard map as a mapping of R1 × S1 into itself, the resonances inquestion correspond to the two islands centered at (I, θ) = (0, π) and (I, θ) = (2π, π). Anumerical study of the system shows that the last KAM torus between the resonances isof winding number g = (

√5− 1)/2 with g being the golden mean.

The idea of the method of overlapping resonances consists of the following. To studythe onset of widespread chaos between two given resonances we neglect all the other termsin the Fourier expansion of the perturbation part. We replace the initial system with asystem whose perturbation part is the sum of two resonance terms. We discuss now thesetwo remaining resonances independently. A Hamiltonian system of the form

H(I) = H0(I) + V cos(mθ + 2nπt) (2.16)

is integrable. Here m and n are assumed to be integers. However, compared with Figure2.1, K = 0, the stroboscopic plot becomes more involved. Actually, it looks qualitativelyjust like the phase portrait of a pendulum. The initial system can now be thought ofas perturbation to the integrable Hamiltonian given by Eq. (2.16). According to thePoincare-Birkhoff theorem the neighborhood of the separatrix gives rise to a chaotic bandin the vicinity of the resonance. Let Ir be the position of the resonance in question inthe case of the integrable system H0(I). The size of the resonance can be described by


the resonance half-width. For the Hamiltonian (2.16) it is defined to be the distancebetween Ir and the maximal value of I attained by points on the separatrix. Denote theresonance half-width by ∆I. Thus, if we consider the projection of the chaotic regionto the coordinate line I, we obtain the interval [Ir − ∆I, Ir + ∆I]. We now repeat ourconsiderations for the second resonance. As a result, we obtain an interval [I ′r −∆I ′, I ′r +∆I ′] related to the second resonance. Chirikov’s criterion in its basic form states thatwidespread chaos sets in if and only if the intervals [Ir−∆I, Ir+∆I] and [I ′r−∆I ′, I ′r+∆I ′]have a non-empty intersection.

Let us return to the standard map. Consider the first resonance independently of theothers. We have the Hamiltonian

H(I, θ, t) =I2

2−K cos θ. (2.17)

The frequency is given by ω = ∂H/∂I = I. The resonance corresponds to the frequencyzero. Thus, the resonance is centered at the point I = 0. Its half-width is ∆I = 2

√K.

For the second resonance we obtain

H(I, θ, t) =I2

2−K cos(θ + 2πt). (2.18)

The resonance corresponds to the value ω = 2π of frequency. Thus, the center of theresonance lies at the point I = 2π. The half-width is again given by ∆I = 2

√K.

The distance between the resonances is equal to 2π. The method of overlappingresonances states that the critical value Kcrit of the perturbation parameter is given by

2∆I = 4√Kcrit = 2π. (2.19)

Thus,

Kcrit =π2

4≈ 2.5 (2.20)

The estimate obtained is rather far from the value 0.97 which is thought to be the criticalvalue for the standard map. Nevertheless, Chirikov’s method provides us with an esti-mate of the right magnitude. Note also that there exist modifications of the method ofoverlapping resonances which deliver considerably better results.

2.3 Greene’s method

The method is introduced and applied to the standard map in [21]. The idea of thismethod is to relate the existence of a KAM torus to the stability properties of periodicorbits in the vicinity of the KAM torus.

First, let us introduce the notion of the residue of a periodic orbit.Let P be an area-preserving mapping of a two-dimensional region into itself. Assume

that a point (x0, y0) is a fixed point of the mapping P . Consider the linearization M ofthe mapping near the fixed point (x0, y0). It is given by the matrix

M =∂P (x, y)

∂(x, y)

∣∣∣(x,y)=(x0,y0)

=

(a bc d

). (2.21)

2.3. GREENE’S METHOD 15

The absolute value of the determinant of the matrix M equals one because the mappingP preserves area. Assume further that the mapping P is orientation-preserving. Thenthe determinant of M is equal to one. Consider the eigenvalues λ1, λ2 of the matrix M .They are given by solutions of the equation

λ2 − (a+ d)λ+ ad− bc = λ2 − tr(M)λ+ det(M) = λ2 − tr(M)λ+ 1 = 0. (2.22)

Explicitly,

λ1,2 =trM ±

√tr2M − 4

2. (2.23)

The quantity

R =1

4(2− trM) (2.24)

is called the residue of the fixed point (x0, y0). In terms of the residue the eigenvaluesbecome

λ1,2 = 1− 2R± 2√R(R− 1). (2.25)

A fixed point (x0, y0) of a two-dimensional area-preserving mapping is called elliptic ifall the eigenvalues of its linearization operator are complex of magnitude unity. It is easyto see that in the case of orientation-preserving mappings the residue of an elliptic fixedpoint satisfies the inequality

0 < R < 1. (2.26)

A fixed point (x0, y0) of a two-dimensional area-preserving mapping is said to be hyperbolicif the eigenvalues of its linearization operator are real and different from ±1. For ahyperbolic fixed point we have in the case of an orientation-preserving mapping

R < 0 or R > 1. (2.27)

Consider a KAM torus with winding number W = [a0, a1, . . . ]. It is known that thebest rational approximants Pn/Qn to W are given by the truncated continued fractionsPn/Qn = [a0, a1, . . . , an], see [23, 33]. According to the Poincare-Birkhoff theorem theresonant torus with winding number Pn/Qn gives rise to two periodic orbits. One of themis hyperbolic, the other elliptic. Consider these two orbits. Let R+

Pn/Qndenote the residue

of the elliptic periodic orbit, and R−Pn/Qn the residue of the hyperbolic periodic orbit.

We are interested in the behavior of the quantities R+Pn/Qn

and R−Pn/Qn as n increases.Based on numerical investigations of the standard map, Greene formulated the followingcriterion. Given a two-dimensional area-preserving map, there is a smooth invariant KAMcurve of winding number W = [a0, a1, . . . ] if and only if the sequence R±Pn/Qn with Pn/Qn

being [a0, a1, . . . , an] converges to zero.Greene’s method seems to be very effective for the study of the breakup of KAM tori

or KAM curves. In the case of the standard map, the method predicts the critical valueKcrit to be 0.971635406 . . . .

MacKay proposed another method for studying the breakup of noble KAM tori usingthe notion of residue, see [29, 30] for details.

Paul and Richter have applied Greene’s method and the Mackay residue criterion tothe double pendulum, see [35].


2.4 The renormalization group approach

The idea of the renormalization group (RG) approach to the study of the breakup ofKAM tori in Hamiltonian systems with two degrees of freedom was developed in differentversions by MacKay [29, 30] and Escande [16]. In our presentation we follow the approachof Escande.

Consider the following functional space H. Let H be a Hamiltonian defining aHamiltonian system with two degrees of freedom. A point in H is given by a setH,m1,m2, p1, p2,W, h with m1, m2, p1, p2 being integers, and W and h being realnumbers. We interprete the point H,m1,m2, p1, p2,W, h as a Hamiltonian system withtwo fixed resonances defined by the pairs (m1,m2) and (p1, p2). The value of energy isgiven by h. The number W corresponds to the winding number of a KAM torus on theenergy surface H = h. Assume further that we have found a mapping R : H → H of thespace H into itself with the following property. For every point H,m1,m2, p1, p2,W, hin H there exists a KAM torus of winding number W located between the resonances(m1,m2) and (p1, p2) on the energy surface H = h if and only if there exists a KAMtorus of winding number W ′ located between the resonances (m′1,m

′2) and (p′1, p

′2) on

the energy surface H ′ = h′ where the point H ′,m′1,m′2, p′1, p′2,W ′, h′ is the image ofthe point H,m1,m2, p1, p2,W, h under the mapping R. Suppose that the mapping Rhas two invariant stable surfaces in the functional space H corresponding to the cases ofintegrable and strongly chaotic systems respectively. Assume that almost every orbit ofthe mapping R converges to one of these surfaces.

Having found a mapping R with the property just described, we could study thebreakup of KAM tori as follows. Given a system H, two resonances (m1,m2) and(p1, p2), an irrational number W , and a value of energy h, consider the orbit of thepoint H,m1,m2, p1, p2,W, h ∈ H under the mapping R. If the orbit converges to theinvariant surface corresponding to integrable systems, then the KAM torus of windingnumber W exists on the energy surface H = h. Indeed, the property of the system H tocontain the KAM torus of winding number W on the energy surface H = h remains invari-ant under the mapping R. After a sufficient number of iterations, the orbit of the pointH,m1,m2, p1, p2,W, h is located in the neighborhood of an integrable system where theKAM theorem is valid. Thus, for all the points along the orbit there exists a KAM torusof corresponding winding number. In particular, the system H itself contains the KAMtorus of winding number W on the energy surface H = h.

If the orbit of a point H,m1,m2, p1, p2,W, h ∈ H is attracted by the invariant stablesurface corresponding to chaotic systems, then it is reasonable to assume that the KAMtorus is destroyed.

Unfortunately, till now a mapping R on H with the properties introduced above hasnot been discovered. It is also not clear whether such a mapping can exist.

The usual way to avoid this difficulty is to replace the infinite dimensional space H bysome finite dimensional subspace N of H. Given a point H,m1,m2, p1, p2,W, h fromH, we consider the projection of the point into the subspace N . We refer to the resultingHamiltonian system as the normal form of H,m1,m2, p1, p2,W, h. In some cases theinitial system is already written as a normal form so that the process of projection intoN can be omitted.

2.4. THE RENORMALIZATION GROUP APPROACH 17

A number of various normal forms have been suggested by different authors. Escande[16] discusses the so-called paradigm Hamiltonian

H(v, x, t) =v2

2−M cosx− P cos k(x− t). (2.28)

The system is a non-autonomous Hamiltonian system with one degree of freedom. It caneasily be rewritten as an autonomous Hamiltonian system with two degrees of freedom.The parameters of the Hamiltonian are the magnitudes M , P , and the wave number k.Thus, normal forms make up a finite dimensional space N .

Paul [34] chooses the function

H(I1, I2, θ1, θ2) = WI2 +1

2(I2

1 +2βI1I2 +γI22 )−M cos(θ1 +θ2)−P cos(k1θ1 +k2θ2) (2.29)

as the normal form for a Hamiltonian system with two degrees of freedom.Chandre et al. [11] consider the family

H = ωI1 − I2 +1

2(I1 + kI2)2 −M cos θ1 − P cos θ2. (2.30)

We choose as the normal form the Hamiltonian

H = ωI1 + I2 + aI21 + 2bI1I2 + cI2

2 −M cos θ1 − P cos θ2 (2.31)

with a2 + b2 + c2 = 1.The next step of the RG analysis is to introduce the renormalization map on the

space N . Two approaches have been used for this purpose. Chandre et al. [11, 8]make use of Lie transformations. Escande et al. [16, 19, 18, 17], and Paul [34] employthe technique of generating functions in the framework of classical perturbation theory.We also use the latter approach in our work. Qualitatively the renormalization mapcan be described as follows. Given a point from N , transform its Hamiltonian in sucha way that the transformed Hamilton function contains no term linear in perturbationparameters. The resulting Hamiltonian does not belong to N anymore. Generally, it hasan infinite number of resonances. We define the projection of the transformed Hamiltonianinto N in the following way. The two initial resonances may be viewed as the firstrational approximants to the irrational winding number in question. Out of an infinitenumber of resonances presented in the new Hamiltonian, we take into account only two ofthem. These two resonances correspond to the next rational approximants of the windingnumber. In this way we obtain a near-integrable system whose Hamiltonian contains onlytwo resonances. Rescaling the angle variables we come to resonances of the same form asinitial ones. Using Taylor expansion about the KAM torus in question, the integrable partof this new Hamiltonian can be approximated by the integrable part of the correspondingnormal form. The magnitudes of the resonances have also to be approximated. Thesesimplifications lead to a new Hamilton function in normal form. The renormalization mapassigns this resulting system to the initial point from N .

The qualitative behavior of the renormalization map is thought to be rather simple.If we restrict ourselves to the two-dimensional plane given by the magnitudes of the


resonances, then there are two stable and one unstable fixed points. The stable fixedpoints correspond to the stable surfaces discussed above. Again, the existence of the KAMtorus is related to the behavior of iterations of the normal norm under the renormalizationmap just described.

2.5 Comparison between the different methods

We end this chapter with some remarks concerning the methods discussed above.We described four methods for the study of the onset of widespread chaos in Hamil-

tonian systems with two degrees of freedom. Two of them, namely, the sup map analysisand Greene’s methods, are entirely numerical. The other two, the method of overlappingresonances and the RG approach, are of analytical nature.

The sup map analysis is a numerical method, easy to implement and intuitively ap-pealing. However, it has a number of disadvantages. First, the method is rather time-consuming. Even in the case of the standard map one has to carry out many iterationsin order to obtain a realistic estimate for the critical value of the perturbation parameter.Second, one needs a good localization of the two main resonances for the method to workproperly.

Greene’s method is also a numerical one. Usually, the use of the method or its mod-ifications leads to very reliable estimates. The implementation of the method becomesessentially easier if the system in question possesses symmetries as it is the case for thestandard map.

Chirikov’s method is an analytical, clear, and easy to apply approach. However, theestimates which can be obtained in this way are usually not very precise. The methoddoes not work well if resonances of very different magnitudes are studied. It does notmake use of the KAM theory.

Another analytical method is based on the RG approach. Like Greene’s method itemploys the qualitative picture of the onset of widespread chaos provided by the KAMtheory. Like in Chirikov’s method one must carry out a number of essential approxima-tions to the initial Hamilton function. It is by no means obvious whether the study of thetransition to global chaos in the simplified system leads qualitatively to the same resultsas that of the initial system. A number of difficult technical problems must be overcomein order to apply the RG approach to a given system. One of them is to introduce action-angle variables for the unperturbed system and express the perturbation part in terms ofthese variables.

Nevertheless, from a theorist’s point of view, the RG approach seems to be the mostjustified method for the study of the onset of widespread chaos. We describe our versionof the renormalization theory in Chapter 3.

Chapter 3

Renormalization Theory

In this section we introduce a new version of the renormalization operator for Hamiltoniansystems with two degrees of freedom. This version is essentially based on the renormal-ization operator of Escande [16], [31], [34]. To rescale the Hamiltonian we also use theclassical perturbation theory approach. We choose, however, a different normal form forthe Hamilton function. Our choice is like in [11].

We start with a compact integrable Hamiltonian system with two degrees of freedom.According to the Liouville-Arnold theorem [3], one can introduce local action-angles vari-ables (I,θ) = (I1, I2, θ1, θ2). In these variables the Hamiltonian H0 is independent ofangles. So we have

H0(I,θ) = H0(I). (3.1)

Let us now consider some perturbation of the Hamiltonian system H0 which depends ona small parameter ε. The perturbed Hamiltonian is then

H(I,θ) = H0(I) + εH1(I,θ). (3.2)

We expand the non-integrable part H1 in a Fourier series,

H(I,θ) = H0(I) + ε∑

n=(n1,n2)

Vn(I) cos(n · θ + γn). (3.3)

3.1 Normal form

Let us consider a KAM torus Tε with frequencies ω = (ω1, ω2), ω1 = ∂H0/∂I1, ω2 =∂H0/∂I2 on the energy surface H = h. Assume that only two resonances are relevant forthe torus Tε. Let m = (m1,m2) and p = (p1, p2) be these resonances in the sense that thefrequencies (ωm

1 , ωm2 ) and (ωp

1 , ωp2 ) of the corresponding resonant tori of the unperturbed

system satisfy the relations

m1ωm1 +m2ω

m2 = 0, (3.4)

p1ωp1 + p2ω

p2 = 0. (3.5)

As shown in Appendix A, it is possible to approximate the given Hamilton function Hwith the following normal form

H(I,θ) = ωI1 + I2 + aI21 + 2bI1I2 + cI2

2 +M cos θ1 + P cos θ2 (3.6)

19

20 CHAPTER 3. RENORMALIZATION THEORY

with 0 < ω < 1, 0 < a, a2 + b2 + c2 = 1. The coefficients ω, a, b, c, M , P depend on theinitial Hamilton function, the value of energy h, the torus, and the resonances m, p, seeAppendix A for details. Note that

ω =m1ω1 +m2ω2

p1ω1 + p2ω2

, (3.7)

and the torus after normalization lies on the energy surface H = 0. In the integrable case,the torus is located at the origin (I1, I2) = (0, 0).

3.2 The renormalization operator

Assume that the Hamiltonian system in question has the normal form given by (3.6).Let us discuss changes in qualitative behavior of the system as the parameters M and

P vary. If the values of the amplitudes M and P are zero, then one has an integrablecase. Figure 3.1 shows a Poincare section θ1 = θ2, θ1 < θ2, h = 0, for the normal formwith ω = g, a = (g + 1)/2, b = −1/2, c = g/2, M = 0, P = 0, where g = (

√5 − 1)/2 is

the golden mean. In a general case, for non-zero values of M and P , the system becomesnon-integrable. Figure 3.2 shows a Poincare section θ1 = θ2, θ1 < θ2, h = 0, for the normalform with ω = g, a = (g + 1)/2, b = −1/2, c = g/2, M = 0.06, P = 0.06. Obviously, thetwo resonances M cos θ1, P cos θ2 give rise to chaotic regions.

h =0.00000 M=0.0000

-0.60

-0.25

0.10

0.00 1.57 3.14 4.71 6.28

I2

θ2

Figure 3.1: A Poincare section θ1 − θ2 = 0, θ1 − θ2 < 0, h = 0, M = 0, P = 0 for thenormal form.

Let us discuss Figure 3.2 in more detail. A number of nested islands can be seen in thePoincare section. Consider the large region containing one island structure in the bottompart of Figure 3.2. The numerical investigation of the trajectories from this region showsthat it corresponds to the resonances ω1 : ω2 = 0 : 1. In other words, the correspondingtorus in the unperturbed Hamilton system is of winding number W = ω1/ω2 = 0. Another

3.2. THE RENORMALIZATION OPERATOR 21

h =0.00000 M=0.0600

-0.60

0.06

0.00 1.57 3.14 4.71 6.28

I2

θ2

Figure 3.2: A Poincare section θ1 − θ2 = 0, θ1 − θ2 < 0, h = 0, M = 0.06, P = 0.06 forthe normal form in projection to the (I2,θ2) plane.

h =0.00000 M=0.0600

-0.10

0.06

0.00 1.57 3.14 4.71 6.28

I2

θ2

Figure 3.3: Blowup of the top part of Figure 3.2.


h =0.00000 M=0.0600

-0.05

0.04

0.00 1.57 3.14 4.71 6.28

I2

θ2

Figure 3.4: Further blowup of the top part of Figure 3.2.

elliptic region containing one island structure is located in the top part of Figure 3.2. Itcorresponds to the resonance ω1 : ω2 = 2 : 3 = 2/3. The third one island structureis located between the resonance 2/3 and the large chaotic region corresponding to theresonance 0/1. This structure is related to the resonance ω1 : ω2 = 1 : 2 = 1/2. Considerthe blowup of Figure 3.2, see Figure 3.3. The resonances 2/3 and 1/2 are seen in moredetail. Two further resonant regions can be identified. Both lie between the resonances2/3 and 1/2. They correspond to the resonances ω1 : ω2 = 3 : 5 = 3/5 and ω1 : ω2 =5 : 8 = 5/8. The former consists of two islands, the latter of three. An invariant linecorresponding to a KAM torus can clearly be seen in Figure 3.3. The line lies between theresonances 3/5 and 5/8. A further blowup of Figure 3.2 is given by Figure 3.4. The lastKAM torus is thought to be of winding ratio ω = g. The rationals w2 = 1/2, w3 = 2/3,w4 = 3/5, w5 = 5/8 corresponding to the resonances discussed are rational approximantsof the noble number g given by its continued fraction representation.

Figures 3.2-3.4 show that the chaotic regions originating from the resonances given byrationals wn, n = 2, 3, 4, 5, become smaller as n is increased. One of the main assumptionsof the RG approach in the study of KAM tori is that the existence of a KAM torus isconnected to the perturbation properties of the resonances in the vicinity of the torus. Aswe see from Figures 3.2-3.4, the resonances defined by the best rational approximants tothe winding number of the KAM torus can and usually do lead to the strongest chaoticregions. Note that the best rational approximants for a given irrational number can beobtained using its continued fraction representation, see [23, 33] for details. In what followsonly these resonances will be taken into account. The first two rational approximants ofthe winding number are included in the normal form. The resonances defined by the otherrational approximants will be used in the next steps of the renormalization map.

Using the classical methods of perturbation theory, one can introduce a canonicaltransformation such that the Hamilton function in the new variables contains no pertur-bation terms linear in M or P . In other words, we get rid of the two resonances presented

3.3. THE RENORMALIZATION MAP FOR THE NORMAL FORM 23

in the normal form. The next two rational approximants of the winding number are cho-sen to define the two-resonance approximation of the new Hamiltonian, see Section 3.3and Appendix B.1 for details. One can then approximate the new Hamiltonian with thenormal form

H ′(I1, I2, θ1, θ2) = ω′I1 + I2 + a′I21 + 2b′I1I2 + c′I2

2

+ M ′ cos θ1 + P ′ cos θ2. (3.8)

In this way we define a map R : N → N from the parameter space to itself. The map Ris called the renormalization map. Note that normal forms with M = P = 0 correspondto integrable cases.

We use the following criterion for the existence of the KAM torus. The KAM torusexists for the value ε if and only if

limn→∞

R(n)(ω, a, b, c,M, P ) = (ω∞, a∞, b∞, c∞, 0, 0), (3.9)

for some parameters ω∞, a∞, b∞, c∞. Here ω, a, b, c, M , P are the parameters given bythe normal form of the Hamiltonian H = H0 + εH1.

In general, the implementation of the approach just described is not easy. First, for agiven Hamiltonian system one has to find a suitable presentation in terms of the pertur-bation of some integrable case. Second, action-angle coordinates have to be determinedwhich may be a formidable task in practice. The Hamiltonian must then be rewrittenin these coordinates. Third, we have to identify KAM tori that are particularly stableagainst the perturbation. Having fixed one of these tori, one has to determine its mostimportant resonances. At this point, we are ready to write down the normal form. Thenormal form depends on the energy level, the KAM torus, and the value of the pertur-bation parameter. One of the possible strategies is now as follows: keep the energy leveland the winding number ω of the KAM torus fixed; change the perturbation parameter;obtain a family of normal forms. Using the criterion just described, one can determinethe critical value for the perturbation parameter. Having calculated the critical values fordifferent KAM tori, we choose their largest value as the relevant value for the transitionto global chaos at the given energy level.

In the following, we consider the renormalization operator in more detail. The meth-ods of classical perturbation theory enable us to get rid of the two given resonances inlinear order. But the corresponding canonical transformation gives rise to infinitely manyresonance terms that are of the second or larger power with respect to the perturbation.This necessitates a procedure for choosing the two relevant resonances out of these in-finitely many ones. One of the possible solutions to this problem is to use the continuedfraction representation of the torus’ winding number.

3.3 The renormalization map for the normal form

We discuss now the renormalization map in more detail. Consider the normal form in theslightly more general version

H(I1, I2, θ1, θ2) = ω1I1 + ω2I2 + aI21 + 2bI1I2 + cI2

2 +M cos θ1 + P cos θ2. (3.10)


Apply the canonical transformation (I1, I2, θ1, θ2)→ (J1, J2, φ1, φ2) defined by the gener-ating function

F (J1, J2, θ1, θ2) = θ1J1 + θ2J2 −M sin θ1

w1(J1, J2)− P sin θ2

w2(J1, J2), (3.11)

where

w1(J1, J2) =∂H0(J1, J2)

∂J1

= ω1 + 2aJ1 + 2bJ2, (3.12)

w2(J1, J2) =∂H0(J1, J2)

∂J2

= ω2 + 2bJ1 + 2cJ2 (3.13)

are the frequencies of the unperturbed part

H0(I1, I2) = ω1I1 + ω2I2 + aI21 + 2bI1I2 + cI2

2 . (3.14)

The generating function F (J1, J2, θ1, θ2) is a function of the old angles θ1, θ2 and newmomenta J1, J2. The new angles φ1, φ2 and old momenta I1, I2 are represented in thefollowing manner. The new angle φ1 has the form

φ1 =∂F

∂J1

= θ1 + (−M sin θ1)(−1)∂w1

∂J1

w21(J1, J2)

+ (−P sin θ2)(−1)∂w2

∂J1

w22(J1, J2)

, (3.15)

= θ1 +2aM sin θ1

w21(J1, J2)

+2bP sin θ2

w22(J1, J2)

. (3.16)

Similarly, the expression for φ2 can be obtained. Thus, we have

φ1 = θ1 +2aM sin θ1

w21(J1, J2)

+2bP sin θ2

w22(J1, J2)

, (3.17)

φ2 = θ2 +2bM sin θ1

w21(J1, J2)

+2cP sin θ2

w22(J1, J2)

. (3.18)

The old momenta are given by

I1 =∂F

∂θ1

= J1 −M cos θ1

w1(J1, J2), (3.19)

I2 =∂F

∂θ2

= J2 −P cos θ2

w2(J1, J2). (3.20)

Let us express the Hamilton function H in the new coordinates. Assuming M and P tobe small, we obtain for the Taylor expansion near the point I = J the following result:

H(I1, I2, θ1, θ2) = H(I1(J ,θ), I2(J ,θ), θ1, θ2)

= H0(J1, J2) +

+w1(J1, J2)

(− M cos θ1

w1(J1, J2)

)+ w2(J1, J2)

(− P cos θ2

w2(J1, J2)

)+

+M cos θ1 + P cos θ2 +O(M2 + P 2)

= H0(J1, J2) +O(M2 + P 2), (3.21)

3.3. THE RENORMALIZATION MAP FOR THE NORMAL FORM 25

where O(M2 + P 2) denotes terms of the second and higher order in M and P . Thus, theHamilton function H expressed in the new coordinates has no terms linear in M and P .Let us consider the Taylor expansion up to the quadratic terms in M and P .

H(J1, J2, θ1, θ2) = H0(J1, J2) +

+1

2

(−M cos θ1

w1

,−P cos θ2

w2

)∂2H0

∂J1∂J2

( −M cos θ1w1

−P cos θ2w2

)+

+O((M2 + P 2)3/2), (3.22)

where O((M2 + P 2)3/2) denotes terms of the third and higher order in M and P . TheHessian ∂2H0/∂J1∂J2 is given by

∂2H0

∂J1∂J2

= 2

(a bb c

). (3.23)

Thus,

H(J1, J2, θ1, θ2) = H0(J1, J2) +

(−M cos θ1

w1

,−P cos θ2

w2

)(a bb c

)( −M cos θ1w1

−P cos θ2w2

)+

+O((M2 + P 2)3/2). (3.24)

From now on, we neglect terms of magnitude (M2 + P 2)3/2 and higher:

H(J1, J2, θ1, θ2) = H0(J1, J2) +aM2 cos2 θ1

w21

+2bMP cos θ1 cos θ2

w1w2

+cP 2 cos2 θ2

w22

. (3.25)

Using the trigonometric formulae

cos2 θi =1 + cos 2θi

2, (3.26)

cos θ1 cos θ2 =1

2(cos(θ1 + θ2) + cos(θ1 − θ2)), (3.27)

we obtain

H(J ,θ) = H0(J) +aM2

2w21

+cP 2

2w22

+aM2

2w21

cos 2θ1 +cP 2

2w22

cos 2θ2 +

+bMP

w1w2

(cos(θ1 + θ2) + cos(θ1 − θ2)) . (3.28)

Note that in order to express the Hamiltonian in the new variables J1, J2, φ1, φ2, we needto represent the old angles θ1, θ2 as functions of J1, J2, φ1, φ2 and replace θ1, θ2 in Eq.(3.28) by these expressions. We will not need the explicit expression for H as a functionof J1, J2, φ1, φ2. From the point of view of the renormalization map, it is enough torepresent the integrable part as a function of J1, J2 and to find the amplitudes of themain resonances. See Appendix B for details. As the new integrable part, we choose

H0(J1, J2) = H0(J1, J2) +aM2

2w21

+cP 2

2w22

. (3.29)

Appendix D contains the realization of the renormalization map for the normal formin the Maple computer algebra system.


3.4 Fixed points

This part is devoted to the study of the stable fixed point (M,P ) = (0, 0) of the renor-malization map.

As mentioned in Chapter 2, it is expected that the renormalization map has thefollowing qualitative behavior in projection to the (M,P ) plane, see Figure 3.5. Thereexist two stable fixed points. One of them is the point (M,P ) = (0, 0). Note that insome sufficiently small neighborhood of this point the KAM theorem is valid. Thus, forany point x with (M,P ) sufficiently small the existence of the last KAM torus can beobtained as a consequence of the KAM theorem. Another stable fixed point in the (M,P )plane is the point (M,P ) = (∞,∞). This point is thought to correspond to the strongperturbation of the initial integrable part. Thus, the onset of widespread chaos for pointsin a sufficiently small neighborhood of the fixed point (M,P ) = (∞,∞) is expected. Thetwo basins of attraction corresponding to the two fixed points are believed to be separatedby a one-dimensional critical surface.

KAM

P

M

Figure 3.5: The renormalization map in the (M,P ) plane.

Note that the fixed point (M,P ) = (0, 0) is super-stable. It means that the corre-sponding eigenvalues are zero. Indeed, let x = (ω1, ω2, a, b, c,M, P ) be a point in theparameter space. Assume that one of its last components, say M , equals zero. Let usconsider the image of the Hamilton function H defined by x under the renormalization

3.4. FIXED POINTS 27

map. We refer to Section 3.3 and Appendix B for notation. In the coordinates J, θ thefunction H reads

H(J,θ) = H0(J) +cP 2

2w22

+cP 2

2w22

cos 2θ2. (3.30)

The normal form for this Hamiltonian defines the image x′ = (ω′1, ω′2, a′, b′, c′,M ′, P ′) of

the point x under the renormalization map. Consider the linearization R of the renor-malization map near the fixed point (M,P ) = (0, 0). Then the new amplitudes are(

M ′

P ′

)= R

(MP

)+O(M2 + P 2) = R

(0P

)+O(M2 + P 2). (3.31)

In other words, M ′ and P ′ are proportional to P unless R = 0. However, from Eq. (3.30)it can easily be seen that M ′ and P ′ must be proportional to P 2. Thus, the linearizedpart R of the renormalization map is identically zero. In particular, the eigenvalues ofthe fixed point (M,P ) = (0, 0) are zero, i.e., the fixed point is super-stable.

Let (ω1, ω2) be as usual (ω, 1). Consider the representation of ω as a continued fraction:

ω = [a0, a1, a2, a3, . . . ]. (3.32)

Assume without restriction of generality that a0 is equal to zero. Note that the firstrational approximants to ω are given by

a0 = 0, a0 +1

a1

=1

a1

, a0 +1

a1 + 1a2

=a2

a1a2 + 1, a0 +

1

a1 + 1a2+ 1

a3

=a2a3 + 1

a1a2a3 + a1 + a3

.

(3.33)We are interested in the last two of these approximants. Note that they correspond tothe resonances (a1a2 + 1,−a2) and (a1a2a3 + a1 + a3,−a2a3 − 1). These two resonancesrepresent a reasonable choice for the main resonances corresponding to the KAM toruswith the winding ratio ω. Thus, we use them for the two-resonance approximation of theperturbation part in our version of the renormalization theory.

Let us now discuss the fixed point (M,P ) = (0, 0) in more detail. First of all,note that the other parameters in the normal form (the other coordinates of the point(ω1, ω2, a, b, c, 0, 0)) can change under the iterations of the renormalization map.

We restrict ourselves to the case ω = ω1 = g, where g = (√

5 − 1)/2 is the goldenmean. The normal form defined by x reads

H = gI1 + I2 + aI21 + 2bI1I2 + cI2

2 . (3.34)

We study the orbit of the point x = (g, 1, a, b, c, 0, 0) under the renormalization map.First, we apply perturbation theory to the normal form H as described in Appendix B.In the new coordinates J1, J2, w1, w2 the Hamiltonian H preserves its form

H = gJ1 + J2 + a′J21 + 2b′J1J2 + c′J2

2 , (3.35)

but the main resonances are given by

(m1,m2) = (a1a2 + 1,−a2) = (2,−1), (3.36)

(p1, p2) = (a1a2a3 + a1 + a3,−a2a3 − 1) = (3,−2). (3.37)


Second, we bring the new Hamiltonian to the normal form, see Appendix B for details.We transform the resonances (m1,m2) = (2,−1) and (p1, p2) = (3,−2) to (1, 0) and (0, 1).To do so, we use the linear canonical transformation defined by the matrix

R =

(a1a2 + 1,−a2 −a2

a1a2a3 + a1 + a3 −a2a3 − 1

)=

(2 −13 −2

). (3.38)

As shown in Appendix A, the coefficients (ω, 1) = (g, 1) of the linear part of the normalform H are transformed in the following way:(

ω′1ω′2

)= R

(ω1

ω2

)= R

(g1

)=

(2g − 13g − 2

)(3.39)

with (ω′1, ω′1) being the linear coefficients in the new coordinates. Using the fact that the

golden mean g satisfies the equation g2 + g − 1, one obtains for the winding ratio in thenew coordinates

W ′ =ω′1ω′2

=2g − 1

3g − 2= −g − 1. (3.40)

To bring the linear part of the normal form to ωJ1 + J2 with 0 < ω < 1, we use thefollowing two linear canonical transformations. First, we interchange the action coordi-nates. Second, we change the sign of the first action coordinate. Finally, we multiply theresulting Hamiltonian by 1/(2g − 1). It is easy to see that the frequency ω becomes

ω = −ω′2

ω′1=

1

g + 1= g. (3.41)

As shown in Appendix A, up to multiplication by a constant we obtain the followingexpression for the quadratic part of the normal form(

a′ b′

b′ c′

)= QtP tR

(a bb c

)RtPQ, (3.42)

where the matrices

P =

(0 11 0

), Q =

(−1 0

0 1

)(3.43)

corresponds to the two linear canonical transformations in question.It is easy to check that the transformation can be written as a′

b′

c′

= S

abc

=

9 −12 4−6 7 −2

4 −4 1

abc

. (3.44)

The eigenvalues of the matrix S are −1, 9+4√

5, 9−4√

5. The largest eigenvalue 9+4√

5corresponds to the stable fixed point in the subspace (a, b, c) of the parameter space. Thecorresponding eigenvector of length one is given by ((g + 1)/2,−1/2, g/2).

We obtain the following result. The point(g, 1,

g + 1

2,−1

2,g

2, 0, 0

)(3.45)

3.4. FIXED POINTS 29

–5

–4

–3

–2

–1

0

1

I2

–5 –4 –3 –2 –1 0 1 2I1

Figure 3.6: The energy surface (h = 0) for the trivial fixed point of the renormalizationoperator.

is a stable fixed point with respect to the coordinates a, b, c, M , P of the renormalizationmap. Figure 3.6 shows the energy surface h = 0 of the corresponding normal form.

Figure 3.7 shows the results of the numerical investigations of the renormalization mapin projection to the (M,P ) plane. Consider a point

x =

(g, 1,

g + 1

2,−1

2,g

2,M, P

)(3.46)

from the parameter space N . Choose some small neighborhood M2 + P 2 < δ2 of theorigin. We take δ = 0.01 in our actual computation. We consider three iterates x1, x2, x3

of the point x = x0. Let n be the minimal integer such that the projection of xn to the(M,P ) plane lies in the neighborhood M2 +P 2 < δ. If the projections of all three iteratesare outside this region take n to be 4. The regions in the (M,P ) plane corresponding todifferent values of n are visualized in Figure 3.7. Three regions can easily be identified.The largest region corresponds to n = 4, that is, to the points in N where the onsetof widespread chaos is expected. The region in the neighborhood of the origin containsthe points with n = 1 or n = 0. The third region is located between the first two andcorresponds to the initial conditions leading to n = 2. The region n = 3 can hardly beseen. The chaotic region and the region n = 2 are separated by a critical surface.


0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.1 0.2 0.3 0.4 0.5

P

M

Figure 3.7: The renormalization map in the (M,P ) plane.

Chapter 4

Applications

In this chapter the RG approach to the breakup of KAM tori is applied to a number ofHamiltonian systems.

Section 4.1 deals with the paradigm Hamiltonian of Escande and Doveil. In Section4.2 we discuss the Walker and Ford model. Section 4.3 is devoted to the study of asimple model of the ethane molecule. Section 4.4 is concerned with the double pendulumproblem. The Baggott H2O Hamiltonian is studied in Section 4.5. Our last example isthe family of limacon billiards, see Section 4.6.

Given a Hamiltonian system with two degrees of freedom, the following steps must beexecuted in order to make use of the renormalization theory. First, using the Poincaresection technique we visualize the onset of chaos in the system. Usually, many differentresonances can be identified. We fix two of them and discuss the onset of widespread chaosbetween these two resonances. Then we express the Hamilton function as a near-integrableHamiltonian system. Note that to choose an integrable limiting case which is suitable forthe study of the breakup of KAM tori between the resonances is not always an easy task.Having determined such a limiting case, introduce action-angle variables, and expressthe Hamiltonian in these new coordinates. Using the two-resonance approximation, thecentered-resonance approximation, the approximation of the integrable part by its Taylorexpansion to the second order, we find the parameters of the normal form corresponding tothe pair of resonances, the value of energy, and the winding ratio of the KAM torus whichis thought to be the last KAM torus between the two resonances in question. In particular,we need the expression of the Hamilton function in the action-angle coordinates, and thefirst and second derivatives of the Hamilton function with respect to the action variables.Usually, the corresponding formulae are easily written in implicit form. Appendix Cdescribes how one finds the quantities in question in the case when the energy is given byan implicit function of the action variables. To calculate the magnitudes of the resonanceswe need formulae for the angle variables.

4.1 Application to the paradigm Hamiltonian

This section is devoted to the study of the so-called paradigm Hamiltonian of Escandeand Doveil , see [16, 28, 36].

31

32 CHAPTER 4. APPLICATIONS

The paradigm Hamiltonian reads

H(p, q, t) =p2

2−M cos q − P cos k(q − t) (4.1)

with k being a rational number. The system is a non-autonomous Hamiltonian systemwith one degree of freedom. Let us introduce the parameter

S = 2√M + 2

√P . (4.2)

We will refer to S as the stochasticity parameter [16].In what follows we restrict ourselves to the case k = 1. A number of strobe plots

is presented in Figure 4.1. One can see that system becomes more and more chaotic asthe stochasticity parameter increases. Two large chaotic regions corresponding to theresonances (1, 0) and (1,−1) can easily be identified. For small values of the stochasticityparameter these regions are separated by KAM tori. The onset of widespread chaos canbe observed at the value S ≈ 0.7, see [16, 36].

The Chirikov overlapping criterion can be easily applied to the problem in question.The half-width of the resonance (1, 0) is given by 2

√M , and the half-width of the reso-

nance (1,−1) by 2√P . The sum of the half-widths is precisely the stochasticity parameter

S. The distance between the two resonances is equal to one. Thus, the Chirikov crite-rion gives the estimate S = 1 for the value of the stochasticity parameter at which thelast KAM torus between the resonances (1, 0) and (1,−1) is destroyed. Obviously, thisestimate is rather far from the actual critical value of the stochasticity parameter. Theestimates obtained by the RG approach give much more satisfactory results, see [16, 36].

In the following we apply our version of the renormalization operator to the paradigmHamiltonian.

The paradigm Hamiltonian can be represented as an autonomous Hamiltonian systemwith two degrees of freedom in the following way

H(p1, p2, θ1, θ2) =p2

1

2−M cos θ1 − P cos k(θ1 − θ2) + p2. (4.3)

The system may be viewed as perturbation to the integrable system governed by theHamiltonian

H0(p1, p2) =p2

1

2+ p2. (4.4)

Note that the coordinates (p1, p2, θ2, θ2) = (I1, I2, θ2, θ2) are action-angle coordinates forthe Hamiltonian system H0.

We study the onset of widespread chaos between the resonances (1, 0) and (1,−1). Asthe last KAM torus between the resonances, we choose the KAM torus with the windingratio W = g with g = (

√5 − 1)/2 being the golden mean, see [16, 28, 36]. Figure 4.2

shows the energy surface h = 1 and the positions of the resonances and the last KAMtorus.

The renormalization theory described in Chapter 3 can easily be applied to theparadigm Hamiltonian. Indeed, up to rotation in angle coordinates, the paradigm Hamil-tonian is already written in the normal form. We consider the case M = P = S2/16. The

4.1. APPLICATION TO THE PARADIGM HAMILTONIAN 33

quantity S is considered to be the perturbation parameter. The Hamiltonian reads

H(p1, p2, θ1, θ2) =p2

1

2− S2

16cos θ1 −

S2

16cos k(θ1 − θ2) + p2. (4.5)

We present the application of the RG approach to the paradigm Hamiltonian in Ap-pendix D.

Let us make some remarks. Due to the fact that the paradigm Hamiltonian is a non-autonomous system with one degree of freedom the threshold to the onset of widespreadchaos does not depend on the energy value h. Formally, it can be shown as follows. Theposition of the last KAM torus on the energy surface H0 = h is given by a solution of thesystem of equations

p21

2+ p2 = h, (4.6)

∂H0

∂p1

− g∂H0

∂p2

= 0. (4.7)

The solution reads

(pT1 , pT2 ) = (g, h− g2

2). (4.8)

The Taylor expansion to the second order at the point (pT1 , pT2 ) is given by

H = h+ g(p1 − g) + (p2 − h+g2

2) +

1

2(p1 − g)2 −M cos θ1 − P cos k(θ1 − θ2). (4.9)

By shifting (p1 − g, p2 − h+ g2

2)→ (I ′1, I

′2) we get

H = gI ′1 + I ′2 +1

2I ′21 −M cos θ1 − P cos k(θ1 − θ2). (4.10)

The normal form is obtained from the function H by a rotation transforming the res-onances (1, 0) and (k,−k) to (1, 0) and (0, 1) and subsequent rescaling. Note that thefunction H does not depend on h anymore. Thus, the normal form will also be indepen-dent of h. Consequently, the threshold to widespread chaos is constant as a function ofh.

Our result is presented in Figure 4.3. The estimate S ≈ 0.69 ± 0.01 for the criticalvalue of the stochasticity parameter S is in good agreement with Figure 4.1.


h =1.00000 S=0.0000

-0.2

1.2

0.0 6.3

p

q

h =1.00000 S=0.5000

-0.2

1.2

0.0 6.3

p

q

h =1.00000 S=0.7000

-0.2

1.2

0.0 6.3

p

q

h =1.00000 S=0.7250

-0.2

1.2

0.0 6.3

p

q

h =1.00000 S=0.7500

-0.2

1.2

0.0 6.3

p

q

h =1.00000 S=1.0000

-0.2

1.2

0.0 6.3

p

q

Figure 4.1: Stroboscopic plots q vs. p for the paradigm Hamiltonian with k = 1, M =P = S2/16, S = 0 (top left), S = 0.5 (top right), S = 0.7 (middle left), S = 0.72 (middleright), S = 0.75 (bottom left), S = 1 (bottom right).

4.1. APPLICATION TO THE PARADIGM HAMILTONIAN 35

–3

–2

–1

0

1

2

I2

–2 –1 0 1 2I1

Figure 4.2: The energy surface (h0 = 1) in action-angle variables. The positions of theKAM torus with winding ratio g = (

√5−1)/2 and the main resonances (1, 0) and (1,−1)

are indicated by arrows.

0.000

0.250

0.500

0.750

1.000

1.000 1.250 1.500 1.750 2.000

S

h

Figure 4.3: The critical value of ε for the KAM torus with winding ratio g = (√

5− 1)/2.The accuracy of the bisection method is 0.01.


4.2 Application to the Walker and Ford model

Consider the system governed by the Hamilton function

H(I1, I2, θ1, θ2) = H0(I1, I2) + αI1I2 cos(2θ1 − 2θ2) + βI1I322 cos(2θ1 − 3θ2), (4.11)

H0(I1, I2) = I1 + I2 − I21 − 3I1I2 + I2

2 . (4.12)

This system is called the Walker and Ford model and represents a model for the systemof two coupled nonlinear oscillators [43].

The Hamiltonian system given by Hamiltonian (4.11) may be viewed as perturbationto the integrable system governed by the Hamiltonian H0.

Note that a near-integrable system with a single perturbing term is always integrable,see, for example, [41]. Only two perturbing terms are enough to make an integrable systemnon-integrable. The Walker and Ford model is one of the first examples demonstratingthis effect.

The coordinates (I1, I2, θ1, θ2) are action-angle coordinates for the unperturbed system.The perturbation contains only two resonances. These two features of the system make itrelatively easy to apply the RG methods to this problem [19, 31, 34]. In order to obtaina perturbed system that depends only on one parameter ε, we choose α = β = ε. TheHamilton function reads

H(I1, I2, θ1, θ2) = H0(I1, I2) + ε(I1I2 cos(2θ1 − 2θ2) + I1I

322 cos(2θ1 − 3θ2)

),(4.13)

H0(I1, I2) = I1 + I2 − I21 − 3I1I2 + I2

2 . (4.14)

We investigate the Walker and Ford model with the help of the RG approach. SomePoincare sections are shown in Figures 4.4 and 4.5 to illustrate the behavior for differentvalues of the perturbation parameter ε.

Two large chaotic regions are clearly visible in the Poincare sections. One of themcorresponds to the resonance (m1,m2) = (2,−2), and the other to the resonance (p1, p2) =(2,−3). To study the onset of widespread chaos between these two chaotic regions, weinvestigate the breakup of the KAM torus of winding ratio

W =(p2 −m2)g − p2

(m1 − p1)g + p1

=3− g

2, (4.15)

where g = (√

5− 1)/2 is the golden mean.The resonant tori of resonances (m1,m2) = (2,−2) and (p1, p2) = (2,−3) are also

studied in [43, 31, 34]. However, these works refer to the resonant tori which are differentfrom the tori related to the chaotic regions presented in Figures 4.4 and 4.5. The res-onances from [43, 31, 34] are shown in Figure 4.7. The resonances we are interested inare presented in Figure 4.6. As can be seen, Figures 4.7 and 4.6 correspond to differentbranches of the energy surface h = 0.2. Figure 4.8 shows both parts of the energy surface.

The application of the RG method leads to the results presented in Figure 4.9 andin Table 4.1. In their h-dependence, the results are in reasonable agreement with thePoincare sections presented in Figures 4.4 and 4.5. However, the estimates given by therenormalization theory are considerably larger than the values of perturbation at whichthe onset of widespread chaos can be observed using the Poincare sections.

4.2. APPLICATION TO THE WALKER AND FORD MODEL 37

h =0.18000 ε =0.0000

0.05

0.06

0.08

0.10

0.11

0.13

0.15

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

I2

θ2

h =0.18000 ε =0.0025

0.05

0.06

0.08

0.10

0.11

0.13

0.15

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

I2

θ2

h =0.18000 ε =0.0050

0.05

0.06

0.08

0.10

0.11

0.13

0.15

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

I2

θ2

Figure 4.4: Poincare sections θ1 = 0, θ1 < 0, h = 0.18, ε = 0 (top), ε = 0.0025 (middle),ε = 0.005 (bottom), for the Walker and Ford model in projection to the (I2, θ2) plane.


h =0.22000 ε =0.0000

0.05

0.06

0.08

0.10

0.11

0.13

0.15

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

I2

θ2

h =0.22000 ε =0.0010

0.05

0.06

0.08

0.10

0.11

0.13

0.15

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

I2

θ2

h =0.22000 ε =0.0025

0.05

0.06

0.08

0.10

0.11

0.13

0.15

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

I2

θ2

Figure 4.5: Poincare sections θ1 = 0, θ1 < 0, h = 0.22, ε = 0 (top), ε = 0.001 (middle),ε = 0.0025 (bottom), for the Walker and Ford model in projection to the (I2, θ2) plane.

4.2. APPLICATION TO THE WALKER AND FORD MODEL 39

–0.2

–0.1

0

0.1

0.2

0.3

I2

0.2 0.4 0.6 0.8 1I1

Figure 4.6: The energy surface (h0 = 0.2) in action-angle variables. The positions of theKAM torus with winding ratio (3− g)/2 and the main resonances (2,−2) and (2,−3) areindicated by arrows.

–0.2

–0.1

0

0.1

0.2

0.3

0.4

I2

0 0.1 0.2 0.3 0.4 0.5 0.6I1

Figure 4.7: The energy surface (h0 = 0.2) in action-angle variables. The positions of theKAM torus with winding ratio (3− g)/2 and the main resonances (2,−2) and (2,−3) areindicated by arrows.

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

I2

0 0.1 0.2 0.3 0.4 0.5 0.6I1

Figure 4.8: The energy surface (h0 = 0.2) in action-angle variables.


0.000

0.003

0.007

0.011

0.015

0.180 0.190 0.200 0.210 0.220

ε

h

Figure 4.9: The critical value of ε for the KAM torus (3− g)/2.

h εcrit0.180 0.01430.182 0.01390.184 0.01350.186 0.01310.188 0.01270.190 0.01230.192 0.01190.194 0.01150.196 0.01110.198 0.01070.200 0.01040.202 0.01000.204 0.00920.206 0.00880.208 0.00840.210 0.00760.212 0.00720.214 0.00640.216 0.00610.218 0.00530.220 0.0049

Table 4.1: The critical value of ε for the KAM torus of winding number (3 − g)/2. Theaccuracy of the bisection method is 0.0001.

4.3. APPLICATION TO A MODEL OF THE ETHANE MOLECULE 41

4.3 Application to a model of the ethane molecule

In this section we apply our variant of the renormalization theory to the following me-chanical model of the ethane molecule, see [14]. A molecule of ethane consists of sixatoms of hydrogen and two atoms of carbon, see Figure 4.10. Let us assume that onecarbon atom and three hydrogen atoms are situated at the corners of a rigid pyramid,and that the molecule consists of two such pyramids connected via the C-C bond alongthe z-axis. Further we assume that rotations of the pyramids and an oscillation of thecarbon bond are the only possible degrees of freedom of the system. In what follows werestrict ourselves to the motion of one of the two pyramids with respect to the other. Therelative position of the pyramid is given by the angle ϕ. The system obtained in this wayis a system with two degrees of freedom z, ϕ.

H

H

H HH

H

CC

Figure 4.10: A model of the ethane molecule and its degrees of freedom: rotation aboutthe horizontal axis (ϕ) and oscillations along this axis (z).

Let mH be the mass of an atom of hydrogen, mC the mass of an atom of carbon, andm = m/2 = 3mH +mC the mass of the pyramid. The moment of inertia of the pyramidabout the z-axis is Θ = 2Θ = 3mHr

2, where r is the distance between the carbon atomand z-axis.

We assume the carbon bond to be a Hookian spring with spring constant k = k/4 andequilibrium displacement 2l.

The kinetic energy reads

K =m

2z2

1 +m

2z2

2 +Θ

2ϕ2

1 +Θ

2ϕ2

2. (4.16)

The position of the center of mass is

Z =z1 + z2

2. (4.17)

Let z be the distance between the first pyramid and the center of mass. Then,

z1 = Z + z, z2 = Z − z. (4.18)

Let the angles ϕ1, ϕ2 describe the positions of the first and second pyramids with respectto the z-axis. The relative position of the first pyramid with respect to the second is givenby the angle ϕ = ϕ1 − ϕ2. Let the values ϕ = 0, ϕ = 2π/3, ϕ = −2π/3 correspond tothe case when the hydrogen atoms of the pyramids are located symmetrically, that is, the


projections of the pyramids to the plane perpendicular to the axis z coincide. Introducethe angle Φ = (ϕ1 + ϕ2)/2. The kinetic energy can now be written as

K = mZ2 + mz2 + ΘΦ2 +Θ

4ϕ2. (4.19)

We assume further that the motion is hindered by a steric potential with the strength∆. The potential energy is the sum of the Hookian and steric potentials:

U =k

2(2z − 2l)2 +

∆

z(1 + cos 3ϕ) =

k

2(z − l)2 +

∆

z(1 + cos 3ϕ). (4.20)

We assume that the initial condition in projection to the configuration space is given by(z, ϕ) with z being positive. The factor 1/z by the second term prevents the pyramidfrom oscillations in the region z < 0 where the second pyramid is located.

We take the total impulse P = 2mZ and the total angular momentum LΦ = 2ΘΦ tobe zero. The kinetic energy becomes

K = mz2 +Θ

4ϕ2 =

m

2z2 +

Θ

2ϕ2. (4.21)

The Hamiltonian function reads

H =p2

2m+L2

2Θ+k

2(z − l)2 +

∆

z(1 + cos 3ϕ), (4.22)

where p and L are the conjugate to z and ϕ momenta.Let E be some value of energy, not necessarily E = h. Introduce dimensionless

variables by scaling energy with E, time with√E/Θ, angular momentum L with 1/

√EΘ,

momentum p with 1/√Em, and z with l. We obtain

H =p2

2+L2

2+k

2(z − 1)2 +

∆

z(1 + cos 3ϕ) (4.23)

with

k =kl2

E, ∆ =

∆

El. (4.24)

Thus, the rescaled system depends on the two parameters k, ∆ which in turn dependon the value E according to (4.24). Choosing the value E appropriately, we can alwaysassume that one of these parameters takes on a fixed value. In this way, we will fix theparameter k = 3 and consider the parameter ∆ to be the only parameter of the system.

In what follows we will omit tildes over rescaled variables and parameters. The Hamil-tonian reads

H(p, L, z, ϕ) =p2

2+L2

2+k

2(z − 1)2 +

∆

z(1 + cos 3ϕ). (4.25)

The Hamiltonian system given by Hamiltonian (4.25) is not integrable. Figures 4.11and 4.12 show Poincare sections z = 1, z > 0 for different values of energy and parameters


k and ∆ in projection to the (L, ϕ) plane. As can be seen, the system is regular for ∆ = 0and becomes more and more chaotic as the value of ∆ increases.

We express the Hamiltonian H as the sum H = H0 +H1 of the two following functions

H0 =p2

2+L2

2+k

2(z − 1)2, (4.26)

H1 =∆

z(1 + cos 3ϕ). (4.27)

Note that H0 does not explicitly depend on ϕ and thus defines an integrable Hamiltoniansystem. The first integrals of this system are the energy H0 and the angular momentumL. The function H1 can be thought of as a perturbation to the system H0. Note thatthe representation of the initial system as a near-integrable system with perturbationparameter ∆ seems to be reasonable from the point of view of numerical simulations, seeFigures 4.11 and 4.12.

Let us discuss Poincare sections 4.11 and 4.12 in more detail. If the value of ∆equals zero, then the system is integrable with the angular momentum L being a constantof motion along with the Hamilton function itself. According to the Liouville -Arnoldtheorem the energy surface H0 = h is foliated by two-dimensional tori. The projections ofthese tori to the (L, φ) plane are straight lines L = const. As the perturbation parameterincreases, the system becomes more and more chaotic. A number of chains of islands canbe identified in the Poincare sections. They correspond to resonances in the unperturbedsystem. In the vicinity of these chains of islands chaotic regions are clearly visible. Thetwo large chains both consisting of three islands can be seen in the top and bottom partsof the Poincare sections. They correspond to the resonances (0,−3) and (1,−3). Thelarger the perturbation parameter, the larger the chaotic regions in the vicinity of theseresonances become. For some critical value of ∆ all the KAM tori between these chaoticregions are destroyed. In the following we study the onset of widespread chaos betweenthe resonances (0,−3) and (1,−3) using the RG approach.

First, we introduce the action-angle coordinates I1, I2, θ1, θ2 for the integrable systemgiven by the Hamiltonian H0. As fundamental paths we choose the curves γz : dz = 0 andγϕ : dϕ = 0. The actions I1 and I2 of a torus with H0 = h, L = l are

I1 =1

2π

∮γz

pdq, (4.28)

I2 =1

2π

∮γϕ

pdq. (4.29)


0.80

-0 .20.00 6.30

h=1.0000 ∆=0.0000

L

ϕ

0.80

-0 .20.00 6.30

h=1.0000 ∆=0.0050

L

ϕ

0.80

-0 .20.00 6.30

h=1.0000 ∆=0.0100

L

ϕ

0.80

-0 .20.00 6.30

h=1.0000 ∆=0.0125

L

ϕ

Figure 4.11: Poincare sections z = 1, z > 0, h = 1, k = 3, ∆ = 0 (top left), ∆ = 0.005(top right), ∆ = 0.01 (bottom left), ∆ = 0.0125 (bottom right), for the model of ethanemolecule in projection to the (L, ϕ) plane.


0.80

-0 .20.00 6.30

h=1.2000 ∆=0.0000

L

ϕ

0.80

-0 .20.00 6.30

h=1.2000 ∆=0.0050

L

ϕ

0.80

-0 .20.00 6.30

h=1.2000 ∆=0.0080

L

ϕ

0.80

-0 .20.00 6.30

h=1.2000 ∆=0.0100

L

ϕ

Figure 4.12: Poincare sections z = 1, z > 0, h = 1.2, k = 3, ∆ = 0 (top left), ∆ = 0.005(top right), ∆ = 0.008 (bottom left), ∆ = 0.01 (bottom right), for the model of ethanemolecule in projection to the (L, ϕ) plane.


We obtain

I1 =1

2π

∮γz

pdq =1

2π

∮γz

(pdz + Ldϕ) =1

2π

∮γz

pdz

=2

π

∫ zmax

1

√2h− l2 − k(z − 1)2dz

=2

π

∫ zmax−1

0

√2h− l2 − kz2dz

=2

π

∫ zmax−1

0

√2h− l2

√√√√1−

(z

√k

2h− l2

)2

dz, (4.30)

where zmax = ((2h− l2)/k)1/2 + 1. Introducing the new variable

y = z

√k

2h− l2, (4.31)

we find

I1 =2

π

∫ 1

0

(2h− l2)√k

√1− y2dy

=2

π

(2h− l2)√k

∫ π2

0

cos2 tdt

=2

π

(2h− l2)√k

π

4=

2h− l2

2√k. (4.32)

The action I2 is given by

I2 =1

2π

∮γϕ

pdq =1

2π

∮γϕ

Ldϕ = l. (4.33)

The corresponding angles θ1, θ2 can be found with the help of the generating functionF (z, ϕ, I1, I2) defined by

F (z, ϕ, I1, I2) =

∫ (z,ϕ)

(z0,ϕ0)

pdq. (4.34)

The angles θ1, θ2 are given by

θ1 =∂F

∂I1

, θ2 =∂F

∂I2

. (4.35)


Let the initial point (z0, ϕ0) be (1, 0). We have

θ1 =∂

∂I1

(

∫ z

1

pdz +

∫ ϕ

0

Ldϕ) =∂

∂I1

∫ z

1

pdz

=∂

∂I1

∫ z

1

√2h− l2 − k(z − 1)2dz

=∂

∂I1

∫ z−1

0

√2h− l2 − kz2dz

=∂

∂I1

∫ z−1

0

4√k

√2I1 −

√kz2dz

=

∫ z−1

0

4√k(2I1 −

√kz2)−

12 dz

=

∫ z−1

0

4√k√

2I1

1−

(z 4√k√

2I1

)2− 1

2

dz

=4√k√

2I1

∫ 4√k

(z−1)√2I1

0

(1− y2)−12 dy

√2I1

4√k, (4.36)

where y = 4√kz/(2I1)

12 . We obtain

θ1 =

∫ arcsin

(4√k(z−1)√

2I1

)0

dt = arcsin

(4√k(z − 1)√

2I1

). (4.37)

The angle θ2 becomes

θ2 =∂

∂I2

(

∫ z

1

pdz +

∫ ϕ

0

Ldϕ) =∂

∂I2

∫ z

1

pdz + ϕ. (4.38)

Note that the momentum p is given by

p =√

2h− I22 − k(z − 1)2. (4.39)

The quantity h is assumed to be a function of the action variables I1, I2. We have foundabove that this function is given implicitly by the relation

I1 =2h− I2

2

2√k

. (4.40)

Thus, the partial derivative of the momentum p with respect to I2 can be expressed as

∂

∂I2

p(I1, I2) =∂

∂I2

√2h(I1, I2)− I2

2 − k(z − 1)2 =∂p(h, I2)

∂I2

+∂p(h, I2)

∂h

∂h

∂I2

. (4.41)

The frequency ∂h/∂I2 can be found with the help of Eq. (4.40). The result is

∂h

∂I2

= I2. (4.42)


Note that∂p(h, I2)

∂I2

= −I2

p, (4.43)

∂p(h, I2)

∂h=

1

p. (4.44)

Hence,∂

∂I2

p(I1, I2) = −I2

p+

1

pI2 = 0. (4.45)

We obtainθ2 = ϕ. (4.46)

Let us summarize our calculations. The old coordinates can be expressed in the newones as follows.

z = 1 +

√2I1

ωsin θ1, (4.47)

p =√

2I1ω cos θ1, (4.48)

ϕ = θ2, (4.49)

L = I2, (4.50)

where ω =√k. Note that (I1, θ1) are just action-angle variables for the harmonic oscilla-

tor, see for example [28].The Hamiltonian for the initial system in action-angle variables reads

H = I1ω +I2

2

2+

∆

1 +√

2I1ω

sin θ1

(1 + cos 3θ2). (4.51)

We write the Hamiltonian function in the following form:

H = H0 + ∆H1, (4.52)

where

H0 = I1ω +I2

2

2, (4.53)

H1 =(1 + cos 3θ2)

1 +√

2I1ω

sin θ1

. (4.54)

The energy surface h0 = 1 for the integrable part H0 is shown in Figure 4.13.The resonances qM = (0,−3) and qP = (1,−3) correspond to the resonant tori of

winding ratios WM = 1/0 and WP = 3/1, respectively. If we change the order of thefundamental paths, that is, replace the winding ratio W = ω1/ω2 by W ′ = ω2/ω1, thenthe winding numbers of the resonances become W ′

1 = W ′M = 0 and W ′

2 = W ′P = 1/3,

. The two subsequent chains of islands are found to be the resonant tori with winding


–4

–2

0

2

4

I2

–5 –4 –3 –2 –1 0 1 2I1

The energy surface in the action-angle variables

Figure 4.13: The energy surface h0 = 1 in action-angle variables. The positions of theKAM torus with winding ratio 3/g, and the main resonances (0,−3) and (1,−3) areindicated by arrows.

ratios W ′3 = 1/6 and W ′

4 = 2/9. Taking into account the 2π/3 periodicity of the angleϕ, we interpret these winding ratios as sub-harmonics of the ratios 0, 1, 1/2, 2/3 whichare the first rational approximants to the golden mean g = (

√5 − 1)/2. We conclude

that the resonant tori of winding ratios W ′1, W ′

2, W ′3, W ′

4 correspond to the rationalapproximants of the winding ratio W ′ = g/3. Returning to the initial choice of the orderof the fundamental paths, we obtain the winding ratio W = 3/g. It is the breakup of theKAM torus of winding ratio W = 3/g that we study below.

Figure 4.13 shows the positions of the KAM torus with winding ratio 3/g, and themain resonances (0,−3) and (1,−3) on the energy surface h0 = 1.

The results of the application of the RG approach to the study of the onset ofwidespread chaos between the resonances (0,−3) and (1,−3) are presented in Figure4.14 and in Table 4.2. The minimum step width in ∆ is chosen to be 0.0001. See Ap-pendix D for details. For comparison, we use the results obtained using the sup mapanalysis. As initial conditions we consider 100 points on the line ϕ = π between L = 0.0and L = 0.7. Figures 4.15, 4.16 show the numerical results of the sup map analysis withthe number of iterations for each orbit being 100 and 1000 respectively. The step width is0.001. Figure 4.17 shows the numerical results of the sup map analysis with the numberof iterations for each orbit being 1000 and the step width being 0.0001. See also Table4.3.

We conclude the discussion of the model with the following remark. The results of theRG approach are qualitatively in good agreement with the results of the numerical studyusing the sup map analysis of the Poincare sections. Quantitatively, the agreement withnumerical results presented in Table 4.3 is within 12.7% for h = 1.0 and 22% for h = 1.2.


0.007

0.008

0.010

0.011

0.013

1.000 1.050 1.100 1.150 1.200

∆

h

Figure 4.14: The critical value of ∆ for the KAM torus of winding number 3/g.

h ∆crit

1.00 0.01241.01 0.01211.02 0.01211.03 0.01181.04 0.01161.05 0.01141.06 0.01121.07 0.01101.08 0.01091.09 0.01071.10 0.01051.11 0.01041.12 0.01021.13 0.01001.14 0.00991.15 0.00961.16 0.00951.17 0.00931.18 0.00921.19 0.00901.20 0.0088

Table 4.2: The critical value of ∆ for the KAM torus of winding number 3/g. The stepwidth is 0.0001.


0.008

0.009

0.011

0.013

0.015

1.000 1.050 1.100 1.150 1.200

∆

hFigure 4.15: The critical value of ∆ for the onset of widespread chaos between the reso-nances (0,−3) and (1,−3) (numerical calculation using sup map analysis). The numberof iterations for each orbit is 100, the step width is 0.001.

0.006

0.008

0.010

0.012

0.015

1.000 1.050 1.100 1.150 1.200

∆

hFigure 4.16: The same as Figure 4.15 with the number of iterations for each orbit being1000.


h ∆crit

1.00 0.01101.01 0.01101.02 0.01081.03 0.00991.04 0.00961.05 0.00951.06 0.00911.07 0.00891.08 0.00891.09 0.00901.10 0.00911.11 0.00901.12 0.00861.13 0.00821.14 0.00781.15 0.00771.16 0.00771.17 0.00751.18 0.00741.19 0.00791.20 0.0072

Table 4.3: The critical value of ∆ for the onset of widespread chaos between the reso-nances (0,−3) and (1,−3) (numerical calculation using sup map analysis). The numberof iterations for each orbit is 1000, the step width is 0.0001.

0.006

0.007

0.009

0.010

0.012

1.000 1.050 1.100 1.150 1.200

∆

hFigure 4.17: The same as Figure 4.15 with the number of iterations for each orbit being1000 and the step width being 0.0001.

4.4. APPLICATION TO THE DOUBLE PENDULUM PROBLEM 53

4.4 Application to the double pendulum problem

This section deals with the double pendulum, see [27, 37, 34]. The system has served asa key example of non-integrable, chaotic motion [37, 38].

4.4.1 The Lagrange function

S

s

1

1

2

φ

φ

1

2

S2

A

Aa

2

y

xs1

Figure 4.18: The double pendulum model

Let us introduce the following notations, see Figure 4.18. Let Si, mi, and Ai be thecenter of mass, the mass, and the pivot of the i-th pendulum, i = 1, 2. Let a denote thedistance between the pivots A1 and A2. Let si be the distance between the pivot Ai andthe center of mass Si. Let ΘS

i denote the moment of inertia of the i-th pendulum aboutthe center of mass. We introduce the following coordinate system. The x axis shows inthe horizontal direction from left to right, the y axis shows in the vertical direction fromtop to bottom. Finally, we choose the pivot A1 of the first pendulum as the origin of ourcoordinate system. As generalized coordinates we choose angles φ1 and φ2 that the innerand outer pendulum make up with the y-axis.

The kinetic energy of the first pendulum is

T1 =m1

2(x2

1 + y21) +

1

2ΘS

1 φ21, (4.55)

where x1, y1 are the coordinates of the center of the mass of the first pendulum. Thekinetic energy of the second pendulum is

T2 =m2

2(x2

2 + y22) +

1

2ΘS

2 φ22. (4.56)


We have

x1 = s1 sinφ1,

y1 = s1 cosφ1,

x1 = s1 cosφ1φ1,

y1 = −s1 sinφ1φ1,

x2 = a sinφ1 + s2 sinφ2,

y2 = a cosφ1 + s2 cosφ2,

x2 = a cosφ1φ1 + s2 cosφ2φ2,

y2 = −a sinφ1φ1 − s2 sinφ2φ2.

(4.57)

Substituting Equations (4.57) into the expressions for the kinetic energies, we obtain

T1 =1

2(ΘS

1 +m1s21)φ2

1, (4.58)

T2 =m2

2(a2φ2

1 + s22φ

22 + 2s2a cos(φ2 − φ1)φ1φ2) +

1

2ΘS

2 φ22. (4.59)

The total kinetic energy is the sum of the kinetic energies of the pendulums.

T = T1 + T2

=1

2(ΘS

1 +m1s21 +m2a

2)φ21 +

+ m2s2a cos(φ2 − φ1)φ1φ2 +1

2(ΘS

2 +m2s22)φ2

2. (4.60)

Let us introduce the quantities

m11 = ΘS1 +m1s

21 +m2a

2, (4.61)

m12 = m21 = m2s2a, (4.62)

m22 = ΘS2 +m2s

22. (4.63)

Then

T =1

2m11φ

21 +m12 cos(φ2 − φ1)φ1φ2 +

1

2m22φ

22. (4.64)

The potential energy of the first pendulum is given by

U1 = −gm1y1 = −gm1s1 cosφ1. (4.65)

The potential energy of the second pendulum is

U2 = −gm2y2 = −gm2s2 cosφ2 − gm2a cosφ1. (4.66)

Using the quantities

k11 = g(m1s1 +m2a), (4.67)

k22 = gm2s2, (4.68)


we can write the total potential energy in the form

U = U1 + U2 = −k11 cosφ1 − k22 cosφ2. (4.69)

The Lagrange function is

L = T − U

=1

2m11φ

21 +m12 cos(φ2 − φ1)φ1φ2 +

1

2m22φ

22 +

+ k11 cosφ1 + k22 cosφ2. (4.70)

Adding an arbitrary constant to the Lagrange function does not change the equations ofmotion. So, we can replace the Lagrange function (4.70) by

L = T − U

=1

2m11φ

21 +m12 cos(φ2 − φ1)φ1φ2 +

1

2m22φ

22 +

+ k11(1− cosφ1) + k22(1− cosφ2). (4.71)

4.4.2 Integrable cases

Generally, the system governed by the Lagrangian (4.71) is not integrable. However, anumber of integrable cases can be identified.

First, if the force of gravity vanishes, then the Lagrange function depends only on thedifference φ2 − φ1, and the quantity

∂L

∂φ1

+∂L

∂φ2

(4.72)

becomes a constant of motion. Note that this limiting case may be viewed as a limit ofhigh energies, see below.

Second, if the pivots A1 and A2 of the two pendulums lie at the same point, that is,a = 0, then m12 = 0, and the pendulums become uncoupled.

Third, if the pivot A2 of the second pendulum coincides with its center of mass S2,then s2 = 0, m12 = 0, k22 = 0, and the angle φ2 becomes cyclic.

Finally, if the energy is small, that is,

h min(k11, k22), (4.73)

then we have the case of small oscillations, see [22].

4.4.3 The Hamilton function

Fix some energy value E. Measuring energies in E, time in√m22/E, momenta in m22,

we obtain the Lagrange function in the form

L =1

2m11

(dφ1

dt

)2

+ m12 cos(φ2 − φ1)dφ1

dt

dφ2

dt+

1

2

(dφ2

dt

)2

+ k11(1− cosφ1) + k22(1− cosφ2). (4.74)


with L, mij, t, kij being given by

L = EL, (4.75)

mij = m22mij = (ΘS2 +m2s

22)mij, (4.76)

t =

√m22

Et, (4.77)

kij = Ekij. (4.78)

Introduce the new parameters δ, γ, α, ε which are defined by

α = m11 =m11

m22

=Θ1S +m1s

21 +m2a

2

ΘS2 +m2s2

2

, (4.79)

ε = m22 =m12

m22

=m2s2a

ΘS2 +m2s2

2

, (4.80)

δ = k22 =k22

E=gm2s2

E, (4.81)

γ =1

δk11 =

m1s1g +m2ag

gm2s2

. (4.82)

Note that α > ε2.The Lagrange function reads

L =1

2αφ2

1 + ε cos(φ2 − φ1)φ1φ2 +1

2φ2

2 −

− δγ(1− cosφ1)− δ(1− cosφ2). (4.83)

Here we have omitted tildes for brevity.We are particularly interested in the case of the mathematical pendulum. In this case

ΘSi = 0, s1 = a, s2 = a, m1, m2. For the parameters α, ε, γ this yields

α =m11

m22

=ΘS

1 +m1s21 +m2a

2

ΘS2 +m2s2

2

=2ma2

ma2= 2, (4.84)

ε =m12

m22

=m2s2a

ΘS2 +m2s2

2

=ma2

ma2= 1, (4.85)

γ =k11

k22

=m1s1g +m2ag

m2s2g=

2ma

ma= 2. (4.86)

To derive the Hamiltonian for the system, we find first the generalized momenta that aregiven by(

p1

p2

)=

(∂L/∂φ1

∂L/∂φ2

)=

(α ε cos(φ2 − φ1)

ε cos(φ2 − φ1) 1

)(φ1

φ2

). (4.87)

The generalized velocities as functions of generalized momenta are(φ1

φ2

)=

1

α− ε2 cos2(φ2 − φ1)

(1 −ε cos(φ2 − φ1)

−ε cos(φ2 − φ1) α

)(p1

p2

). (4.88)


The Hamilton function reads

H = (p1, p2)

(φ1

φ2

)− L

=1

2(α− ε2 cos2(φ2 − φ1))(p1, p2)

(1 −ε cos(φ2 − φ1)

−ε cos(φ2 − φ1) α

)(p1

p2

)+

+ γδ(1− cosφ1) + δ(1− cosφ2)

=1

1− (ε2/α) cos2(φ2 − φ1)

(p2

1

2α− ε

αcos(φ2 − φ1)p1p2 +

1

2p2

2

)+

+ γδ(1− cosφ1) + δ(1− cosφ2). (4.89)

4.4.4 The integrable limit of high energies

We discuss now the integrable limit of high energies.We choose now the new angle coordinates ϕ1 = φ1, ϕ2 = φ2−φ1. We use the generating

functionF2(φ, L) = φ1L1 + (φ2 − φ1)L2. (4.90)

The old momenta and the new angles are given by formulae

p1 =∂F2

∂φ1

= L1 − L2, (4.91)

p2 =∂F2

∂φ2

= L2, (4.92)

ϕ1 =∂F2

∂L1

= φ1, (4.93)

ϕ2 =∂F2

∂L2

= φ2 − φ1. (4.94)

The Hamiltonian in the new coordinates is

H =1

1− (ε2/α) cos2 ϕ2

((L1 − L2)2

2α− ε

αcosϕ2(L1 − L2)L2 +

1

2L2

2

)+

+ γδ(1− cosϕ1) + δ(1− cos(ϕ1 + ϕ2))

=1

α(1− (ε2/α) cos2 ϕ2)

(1

2L2

1 − (1 + ε cosϕ2)L1L2 +1

2(1 + α + 2ε cosϕ2)L2

2

)+

+ γδ(1− cosϕ1) + δ(1− cos(ϕ1 + ϕ2)). (4.95)

Recall that for the mathematical double pendulum α = 2, ε = 1, γ = 2. If we take thevalue of E to be gm2s2, then δ = 1. In this case we have

H =1

2− cos2 ϕ2

(1

2L2

1 − (1 + cosϕ2)L1L2 +1

2(3 + 2 cosϕ2)L2

2

)+

+ 2(1− cosϕ1) + (1− cos(ϕ1 + ϕ2)). (4.96)

It is the Hamiltonian considered in [38].


We are interested in the limit of high energies E. In this case the parameter δ tendsto zero. The remaining part of the Hamiltonian is then independent of the angle ϕ1 andthus integrable. The terms in H that are linear in δ are considered to be a perturbationto this integrable part. More formally, one has

H(L1, L2, ϕ1, ϕ2) = H0(L1, L2, ϕ2) + δH1(L1, L2, ϕ1, ϕ2), (4.97)

where

H0(L1, L2, ϕ2) =12L2

1 − (1 + ε cosϕ2)L1L2 + 12(1 + α + 2ε cosϕ2)L2

2

α(1− (ε2/α) cos2 ϕ2), (4.98)

and

H1(L1, L2, ϕ1, ϕ2) = γ(1− cosϕ1) + (1− cos(ϕ1 + ϕ2)). (4.99)

The constants of motion for the Hamiltonian H0 are the total angular momentum L1 andthe Hamiltonian itself.

Figure 4.19 shows a number of Poincare sections for different values of the perturbationparameter δ. Two large chaotic regions can be identified. It is not difficult to check thatthese regions correspond to the resonances (1, 0) and (1, 1). The chaotic region originatedfrom the resonance (1, 0) lies in the central part of the Poincare section. The resonance(1, 1) leads to the smaller chaotic region in the lower part of the Poincare section, seeFigure 4.19. As the perturbation parameter δ increases, the chaotic regions becomelarger. At the value δ ≈ 0.1 we can observe the onset of widespread chaos betweenthe resonances (1, 0) and (1, 1). The last KAM torus is believed to be of winding ratioW = −g2 = −1/(2 + g) ≈ −0.381966 with g = (

√5 − 1)/2 being the golden mean, see

[37, 38].From the point of view of the renormalization theory, the study of the onset of

widespread chaos between the resonances (m1,m2) = (1, 0) and (p1, p2) = (1, 1) is dif-ficult. The reason is as follows. The two resonances lie inside the same branch of theenergy surface in action space, see Figures 4.20 and 4.21. There exists another chaoticregion playing an important role in the onset of global chaos. This region corresponds tothe resonance (p′1, p

′2) = (1, 1). The latter lies outside the branch of the energy surface

containing the resonances (m1,m2) and (p1, p2). For small values of δ the chaotic regionof the resonance (p′1, p

′2) is separated from the resonances (m1,m2) and (p1, p2). In Figure

4.19 it is located below the resonance (p1, p2) = (1, 1). As the perturbation parameterincreases, we observe the onset of widespread chaos between the resonances (p′1, p

′2) and

(p1, p2). It occurs before the transition to global chaos due to the resonances (m1,m2) and(p1, p2). Hence, in order to describe the onset of global chaos properly, we need to takeinto account the influence of the resonance (p′1, p

′2). Unfortunately, there is no consistent

procedure to deal with this problem in the framework of the RG approach.In what follows we neglect the impact of the resonance (p′1, p

′2) and discuss the KAM

torus with the winding number −g2. Its breakup is considered to be the threshold toglobal chaos.

To study the structure of the energy surface for the unperturbed system, let us in-troduce action-angle variables for the system with the Hamiltonian H0. As fundamental


h =50.0000 µ =0.7071

-15.0

-10.0

-5.00

0.00

5.00

10.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

p1

φ1

h =20.0000 µ =0.7071

-10.0

-6.00

-2.00

2.00

6.00

10.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

p1

φ1

h =10.3000 µ =0.7071

-10.0

-6.00

-2.00

2.00

6.00

10.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

p1

φ1

h =9.00000 µ =0.7071

-10.0

-6.00

-2.00

2.00

6.00

10.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

p1

φ1

Figure 4.19: Poincare sections φ2 − φ1 = 0, φ2 − φ1 < 0, 1/δ = 50 (top, left), 1/δ = 20(top, right), 1/δ = 10.3 (bottom, left), 1/δ = 9 (bottom, right), for the mathematicaldouble pendulum α = 2, ε = 1, γ = 2, in projection to the (p1, φ1) plane.


paths we choose ϕ1- and ϕ2-cycles. We have

I1 =1

2π

∮γϕ1

pdq =1

2π

∮γϕ1

L1dϕ1 = L1, (4.100)

I2 =1

2π

∮γϕ2

pdq =1

2π

∮γϕ2

L2dϕ2. (4.101)

Let h be the value of energy. The generalized momentum L2 can be expressed by

L2 =(1 + ε cosϕ2)L1 ±

√(α− ε2 cos2 ϕ2)(2h(1 + α + 2ε cosϕ2)− L2

1)

1 + α + 2ε cosϕ2

. (4.102)

Define the quantityLsep1 =

√2h(1 + α− 2ε). (4.103)

It can be shown [37] that the ϕ2-motion is rotation for |L1| < Lsep1 , and libration for|L1| > Lsep1 . It follows that for |L1| < Lsep1 we have

I2 =1

2π

∫ 2π

0

(1 + ε cosϕ2)L1 ±√

(α− ε2 cos2 ϕ2)(2h(1 + α + 2ε cosϕ2)− L21)

1 + α + 2ε cosϕ2

dϕ2,

(4.104)and for |L1| > Lsep1

I2 =1

π

∫ ϕmax2

0

(1 + ε cosϕ2)L1 +√

(α− ε2 cos2 ϕ2)(2h(1 + α + 2ε cosϕ2)− L21)

1 + α + 2ε cosϕ2

dϕ2

− 1

π

∫ ϕmax2

0

(1 + ε cosϕ2)L1 −√

(α− ε2 cos2 ϕ2)(2h(1 + α + 2ε cosϕ2)− L21)

1 + α + 2ε cosϕ2

dϕ2

=1

π

∫ ϕmax2

0

2√

(α− ε2 cos2 ϕ2)(2h(1 + α + 2ε cosϕ2)− L21)

1 + α + 2ε cosϕ2

dϕ2, (4.105)

where ϕmax2 = arccos((L2

1 − 2h(1 + α))/(4hε)).Figure 4.20 shows the energy surface in action-angle variables for the Hamiltonian H0

with the parameters α = 2, ε = 1, γ = 2.We are interested in the resonances (1, 0) and (1, 1). These resonances are located in

the region where the ϕ2-motion is rotation. Figure 4.21 shows the corresponding branchof the energy surface (h0 = 1) in action-angle variables and the positions of the KAMtorus with winding ratio −1/(2 + g) and the main resonances (1, 0) and (1, 1).

The transformation to action-angle coordinates (I1, I2, θ1, θ2) can be obtained with thehelp of generating function

F (ϕ1, ϕ2, I1, I2) =

∫ (ϕ1,ϕ2)

(ϕ01,ϕ

02)

L1(ϕ1, ϕ2, I1, I2)dϕ1 +

∫ (ϕ1,ϕ2)

(ϕ01,ϕ

02)

L2(ϕ1, ϕ2, I1, I2)dϕ2. (4.106)

If we choose the point (0, 0) as the starting point (ϕ01, ϕ

02), then

F (ϕ1, ϕ2, I1, I2) = I1ϕ1 +

∫ ϕ2

0

L2(ϕ2, I1, I2)dϕ2. (4.107)


Energy surface in action space

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

I 1I 2

Figure 4.20: Energy surface in action space for the high energy integrable limiting caseof the double pendulum for h = 1

The new angles θ1, θ2 and the old momenta L1, L2 read

θ1 =∂F (ϕ1, ϕ2, I1, I2)

∂I1

, θ2 =∂F (ϕ1, ϕ2, I1, I2)

∂I2

, (4.108)

L1 =∂F (ϕ1, ϕ2, I1, I2)

∂ϕ1

, L2 =∂F (ϕ1, ϕ2, I1, I2)

∂ϕ2

. (4.109)

We have

θ1 =∂F

∂I1

= ϕ1 +

∫ ϕ2

0

∂L2

∂I1

dϕ2. (4.110)

Here the generalized momentum L2 is assumed to be a function of the variables ϕ2, I1,I2. Eq. (4.102) yields the expression of the momentum L2 as a function of ϕ2, I1, h.To obtain from this expression the partial derivative ∂L2(ϕ2, I1, I2)/∂I1 one can use therelation

∂L2(ϕ2, I1, I2)

∂I1

=∂L2(ϕ2, I1, h)

∂h

∂h

∂I1

+∂L2(ϕ2, I1, h)

∂I1

. (4.111)

We have

θ1 = ϕ1 +

∫ ϕ2

0

(∂L2

∂h

∂h

∂I1

+∂L2

∂I1

)dϕ2. (4.112)

Here the function L2 is assumed to be a function of ϕ2, I1, and h.Further,

θ2 =∂F

∂I2

=

∫ ϕ2

0

∂L2(ϕ2, I1, I2)

∂I2

dϕ2 =

∫ ϕ2

0

∂L2(ϕ2, I1, h)

∂h

∂h

∂I2

dϕ2. (4.113)

Thus, in order to find the angles θ1 and θ2 we have to calculate the partial derivatives∂h/∂I1 and ∂h/∂I2. In the case of rotational motion in ϕ2, Eq. (4.104) yields

I2 =1

2π

∫ 2π

0

(1 + ε cosϕ2)I1 ±√

(α− ε2 cos2 ϕ2)(2h(1 + α + 2ε cosϕ2)− I21 )

1 + α + 2ε cosϕ2

dϕ2.

(4.114)


Eq. (4.114) gives an implicit representation of the energy h as a function of the actionsI1, I2. The calculation of the partial derivatives of h with respect to the actions I1, I2 inthis situation is discussed in Appendix C. We obtain

∂h

∂I1

= −∂I2(I1, h)/∂I1

∂I2(I1, h)/∂h, (4.115)

∂h

∂I2

=1

∂I2(I1, h)/∂h. (4.116)

The winding ratio W = (∂h/∂I1)/(∂h/∂I2) is

W = −∂I2(I1, h)/∂I1

∂I2(I1, h)/∂h· ∂I2(I1, h)

∂h= −∂I2(I1, h)

∂I1

. (4.117)

The partial derivatives of I2 with respect to I1 and h are given by formulae

∂I2(I1, h)

∂I1

=1

2π

∫ 2π

0

∂L2

∂I1

dϕ2, (4.118)

∂I2(I1, h)

∂h=

1

2π

∫ 2π

0

∂L2

∂hdϕ2. (4.119)

The frequencies ∂h/∂I1 and ∂h/∂I2 can be written as

∂h

∂I1

= −∂I2/∂I1

∂I2/∂h= −

12π

∮∂L2

∂I1dϕ2

12π

∮∂L2

∂hdϕ2

= −∮

∂L2

∂I1dϕ2∮

∂L2

∂hdϕ2

(4.120)

∂h

∂I2

=

(∂I2

∂h

)−1

=

(1

2π

∮∂L2

∂hdϕ2

)−1

. (4.121)

Finally, we find for the winding ratio

W = −∂I2

∂I1

= − 1

2π

∮∂L2

∂I1

dϕ2. (4.122)

To make use of Eqs. (4.120–4.122), we have to calculate the partial derivatives of thegeneralized momentum L2 with respect to h and I1. They are given by

∂L2

∂h=

1

1 + α + 2ε cosϕ2

·(±1

2

)2(1 + α + 2ε cosϕ2)(α− ε2 cos2 ϕ2)√

(α− ε2 cos2 ϕ2)(2h(1 + α + 2ε cosϕ2)− I21 )

= ±

√α− ε2 cos2 ϕ2

2h(1 + α + 2ε cosϕ2)− I21

, (4.123)

∂L2

∂I1

=1

1 + α + 2ε cosϕ2

((1 + ε cosϕ2)±

√α− ε2 cos2 ϕ2 · (−2)I1

2√

2h(1 + α + 2ε cosϕ2)− I21

)

=1

1 + α + 2ε cosϕ2

((1 + ε cosϕ2)∓ I1

√α− ε2 cos2 ϕ2

2h(1 + α + 2ε cosϕ2)− I21

).(4.124)


Figure 4.22 shows the winding ratio W as a function of I2 for the value of energy h0 = 1.The expressions for the second derivatives of the energy h with respect to the actions

I1, I2 can be derived using the formulae from Appendix C.We obtain the following formula for the Jacobian |∂(θ1, θ2)/∂(ϕ1, ϕ2)|:∣∣∣ ∂(θ1, θ2)

∂(ϕ1, ϕ2)

∣∣∣ = det

(1 ∂θ1

∂ϕ2

0 ∂L2

∂h· ∂h∂I2

)=∂L2

∂h· ∂h∂I2

. (4.125)

The amplitudes M and P of the resonances (m1,m2) and (p1, p2) can be calculated asfollows:

M =√M2

cos +M2sin, P =

√P 2

cos + P 2sin, (4.126)

where

Mcos =1

2π2

∫ 2π

0

∫ 2π

0

δH1(θ1, θ2, I1, I2) cos(m1θ1 +m2θ2)dθ1dθ2, (4.127)

Msin =1

2π2

∫ 2π

0

∫ 2π

0

δH1(θ1, θ2, I1, I2) sin(m1θ1 +m2θ2)dθ1dθ2. (4.128)

The quantities I1 and I2 correspond to the position of the resonance (m1,m2) on theenergy surface H0 = h of the integrable part of the system. Similar formulae hold for Psin

and Pcos.The formula for Mcos can be rewritten as

Mcos =1

2π2

∫ 2π

0

∫ 2π

0

δH1(ϕ1, ϕ2, L1, L2) cos(m1θ1(ϕ1, ϕ2, I1, I2) +m2θ2(ϕ1, ϕ2, I1, I2))

×∣∣∣ ∂(θ1, θ2)

∂(ϕ1, ϕ2)

∣∣∣dϕ1dϕ2, (4.129)

where the Jacobian ∂(θ1, θ2)/∂(ϕ1, ϕ2) is given by Eq. (4.125). Note that the quantitiesL1 and L2 are given by L1 = I1 and Eq. (4.102).

Having calculated the first and second derivatives of the energy h with respect to theactions I1, I2, and the amplitudes of the resonances (m1,m2), (p1, p2), we are ready toderive the normal form for the Hamilton function H on the given energy surface H = h.Take δ0 = 0.5. We have obtained the following values for the coefficients of the non-scalednormal form

ω1 = 1.10914, ω2 = −0.685458, (4.130)

a = 0.584595, b = 0.0684753, c = 0.387442, (4.131)

M = 0.376518, P = 0.21139. (4.132)

The perturbation part is linear in δ. Thus, to calculate the normal form for a value δ, wereplace the values of the coefficients M and P in (4.130) by the values δM/δ0 and δP/δ0.

The RG analysis leads to the estimate δ ≈ 0.55 for the threshold δcrit. The result isessentially larger than the numerical value δcrit ≈ 0.1.

We conclude that the decay of the KAM torus with winding ratio −g2 in the caseof the double pendulum cannot satisfactorily be described by the application of the RG


approach to the onset of widespread chaos between the resonances (1, 1) and (1, 0). Theinfluence of resonances outside the branch of the energy surface in question on the KAMtorus turns out to be rather strong. In particular, the resonance (p′1, p

′2) = (1, 1) cannot

be neglected.

-1.00

-0.25

0.50

1.25

2.00

-1.75 -1.00 -0.25 0.50 1.25

I2

I1

Figure 4.21: The energy surface (h0 = 1) in action-angle variables. The positions of theKAM torus with winding ratio −1/(2 + g) and the main resonances (1, 0) and (1, 1) areindicated by arrows.

-5.00

-3.33

-1.66

0.00

1.66

3.33

5.00

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

W

I1

Figure 4.22: The winding ratio W as a function of I2 for the value of energy h0 = 1

4.5. THE BAGGOTT H2O HAMILTONIAN 65

4.5 The Baggott H2O Hamiltonian

We discuss here the so-called Baggott Hamiltonian and apply the RG approach to studythe breakup of KAM tori in this system. The Hamilton function is given by

H = H0 +H1:1 +H2:2 +H2:11 +H2:1

2 , (4.133)

H0 = Ωs(I1 + I2) + ΩbI3 + αs(I21 + I2

2 ) + αbI23 + εssI1I2 + εsbI3(I1 + I2),

H1:1 = (β12 + λ′(I1 + I2) + λ′′I3)(I1I2)1/2 cos(θ1 − θ2),

H2:2 = β22I1I2 cos 2(θ1 − θ2),

H2:11 = βsb(I1I

23 )1/2 cos(θ1 − 2θ3),

H2:12 = βsb(I2I

23 )1/2 cos(θ2 − 2θ3), (4.134)

where

Ωs = 3885.57,

Ωb = 1651.72,

αs = −81.99,

αb = −18.91,

εss = −12.17,

εsb = −19.12,

β12 = −112.96,

λ′ = 6.04,

λ′′ = −0.16,

β22 = −1.82,

βsb = 18.79.

(4.135)

The values in (4.135) for the parameters of the Baggott Hamiltonian are given in cm−1.These values are fitted according to the experimental spectra. The Hamiltonian systemrepresents a classical version of the vibrational spectroscopic Hamiltonian for the watermolecule. The model was introduced by Baggott [5]. The Baggott Hamiltonian in form(4.133) is due to [25]. The canonical action-angle variables (I1, θ1, I2, θ2, Ib, θb) correspondto the two local mode stretches and the bend mode. The system has three degrees offreedom. It possesses a constant of motion

P = 2(I1 + I2) + I3. (4.136)

This quantity corresponds to the quantum Polyad number. Due to the existence of aconstant of motion, we are able to reduce our three degree of freedom to a system withtwo degrees of freedom. We follow the approach in [42]. The reduction can be done byintroducing new action-angle variables (N1, N2, N3, ψ1, ψ2, ψ3) such that N3 = P . We fixnow the value of P to be 34.5 and consider the resulting Hamiltonian system with twodegrees of freedom, see [42] for details. The Hamiltonian reads

H = α1N1 + α2N2 + α3 + β3N21 + β4N1N2 + β5N

22

+β1

√N1 +N2(−2N2 + P ) cosψ2

−β2(N21 +N1N2) cos 2ψ1, (4.137)


where

P = 34.5,

α1 = −112.96− 0.16P = −118.48,

α2 = 525.65 + 56.44P = 2472.83,

α3 = 1651.72P − 18.91P 2 = 34476.71,

β1 = 26.5731,

β2 = −76.8150,

β3 = 73.1750,

β4 = 79.5350,

β5 = −78.7125.

(4.138)

Again, the values in (4.138) for the parameters are given in cm−1.Figure 4.23 shows Poincare sections of the Baggott system for some values of energy.

On the Poincare sections for the system one can distinguish two large chaotic regions. Weare interested only in one of them. Namely, the chaotic region in question is located inthe upper part of the Poincare sections, see Figure 4.23. In the region the system is inthe oscillatory phase with respect to ψ1 and in the rotational phase with respect to ψ2.Several resonances can be identified by studying numerically the dynamics in this part ofthe phase space. Some of them are also visible on the Poincare sections. Figures 4.24 and4.25 show the Poincare sections for the values of energy h = 35000 and h = 40000 withvarying values of β1. Compared with Figure 4.23 the chaotic region we are interested inis shown in more detail. One can recognize the chains of islands corresponding to theresonances (1,−1), (3,−2), (2, 1). Indeed, the large island structure in the bottom partof the Poincare section corresponds to the resonance (1,−1), see for example Figure 4.24,the case β1 = 15. The resonance (3,−2) gives rise to the two island chain in the middle.The resonance (2,−1) leads to the one island structure in the top part of the figure. Westudy the breakup of KAM tori located between the resonances (1,−1) and (2,−1).

We apply the renormalization theory to the Baggott system in the following way.First, we express the initial Hamiltonian as perturbation to an integrable system. Let usconsider two functions

H0 = α1N1 + α2N2 + α3 + β3N21 + β4N1N2 + β5N

22

−β2(N21 +N1N2) cos 2ψ1, (4.139)

and

H1 = β1

√N1 +N2(−2N2 + P ) cosψ2. (4.140)

The Baggott Hamiltonian H is just the sum of H0 and H1. The function H0 definesan integrable Hamiltonian system. Indeed, the angle ψ2 is cyclic and thus, the functionF = N2 is a constant of motion. The function H1 is assumed to be perturbation to theintegrable system H0.

To make use of the renormalization theory, we introduce action-angle coordinates(I1, I2, θ1, θ2) for the system given by the Hamiltonian H0. As fundamental cycles we


25.0

5.00-3.1 3.14

h=46500. P=34.500

N2

ψ2

25.0

5.00-3.1 3.14

h=48200. P=34.500

N2

ψ2

25.0

5.00-3.1 3.14

h=50600. P=34.500

N2

ψ2

25.0

5.00-3.1 3.14

h=52000. P=34.500

N2

ψ2

25.0

5.00-3.1 3.14

h=53100. P=34.500

N2

ψ2

25.0

5.00-3.1 3.14

h=53600. P=34.500

N2

ψ2

Figure 4.23: Poincare sections ψ1 = 0, ψ1 > 0, P = 34.5, h = 46500cm−1 (top, left),h = 48200cm−1 (top, right), h = 50600cm−1 (middle, left), h = 52000cm−1 (middle,right), h = 53100cm−1 (bottom, left), h = 53600cm−1 (bottom, right), for the BaggottHamiltonian in projection to the (N2, ψ2) plane.


h =35000.0 P=34.500

22.0

24.1

26.3

28.5

30.6

32.8

35.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

N 2

ψ 2

h =35000.0 P=34.500

22.0

24.1

26.3

28.5

30.6

32.8

35.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

N 2

ψ 2

h =35000.0 P=34.500

22.0

24.1

26.3

28.5

30.6

32.8

35.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

N 2

ψ 2

h =35000.0 P=34.500

22.0

24.1

26.3

28.5

30.6

32.8

35.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

N 2

ψ 2

h =35000.0 P=34.500

22.0

24.1

26.3

28.5

30.6

32.8

35.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

N 2

ψ 2

h =35000.0 P=34.500

22.0

24.1

26.3

28.5

30.6

32.8

35.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

N 2

ψ 2

Figure 4.24: Poincare sections ψ1 = 0, ψ1 > 0, P = 34.5, h = 35000cm−1 β1 = 0 (top,left), β1 = 15 (top, right), β1 = 26.5730 (middle, left), β1 = 35 (middle, right), β1 = 40(bottom, left), β1 = 50 (bottom, right), for the Baggott Hamiltonian in projection to the(N2, ψ2) plane.


h =40000.0 P=34.500

22.0

24.1

26.3

28.5

30.6

32.8

35.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

N 2

ψ 2

h =40000.0 P=34.500

22.0

24.1

26.3

28.5

30.6

32.8

35.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

N 2

ψ 2

h =40000.0 P=34.500

22.0

24.1

26.3

28.5

30.6

32.8

35.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

N 2

ψ 2

h =40000.0 P=34.500

22.0

24.1

26.3

28.5

30.6

32.8

35.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

N 2

ψ 2

h =40000.0 P=34.500

22.0

24.1

26.3

28.5

30.6

32.8

35.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

N 2

ψ 2

h =40000.0 P=34.500

22.0

24.1

26.3

28.5

30.6

32.8

35.0

-3.14 -2.09 -1.04 0.00 1.04 2.09 3.14

N 2

ψ 2

Figure 4.25: Poincare sections ψ1 = 0, ψ1 > 0, P = 34.5, h = 40000cm−1 β1 = 0 (top,left), β1 = 15 (top, right), β1 = 26.5730 (middle, left), β1 = 35 (middle, right), β1 = 40(bottom, left), β1 = 50 (bottom, right), for the Baggott Hamiltonian in projection to the(N2, ψ2) plane.


choose the ψ1- and ψ2-cycles. We will be interested in the case when the system is in theoscillational phase with respect to ψ1 and in the rotational phase with respect to ψ2. Letus find the actions I1, I2 for the system. The action I2 is given by

I2 =1

2π

∫γ2

pdq =1

2π

∫ 2π

0

N2dψ2 = N2. (4.141)

To calculate the action

I1 =1

2π

∫γ1

pdq =1

2π

∫N1dψ1, (4.142)

we express the generalized momentum N1 as a function of energy h, the generalizedmomentum N1, and the angle ψ2. We have

h = α1N1 + α2N2 + α3 + β3N21 + β4N1N2 + β5N

22 −

−β2(N21 +N1N2) cos 2ψ1. (4.143)

Thus, the momentum N1 can be found as a solution of the quadratic equation

AN21 +BN1 + C = 0, (4.144)

where

A = β3 − β2 cos 2ψ1 (4.145)

B = α1 + β4N2 − β2N2 cos 2ψ1 (4.146)

C = α2N2 + α3 + β5N22 − h. (4.147)

Hence, the momentum N1 reads

N1 =−B ±

√D

2A, (4.148)

where D = B2 − 4AC is the determinant of Eq. (4.144).Recall that the motion in ψ1 is assumed to be oscillatory. The minimal and maximal

values of the angle ψ1 for the given values of the constants of motion h and N2 can beobtained using the condition

D = 0. (4.149)

Choose the orientation of the fundamental path dψ2 = 0 as it is shown in Figure 4.26.We have

I1 =1

2π

∫N1dψ1 =

1

2π

∫ ψmax1

ψmin1

−B −√D

2Adψ1 +

1

2π

∫ ψmin1

ψmax1

−B +√D

2Adψ1

=1

2π

∫ ψmax1

ψmin1

−√D

Adψ1. (4.150)

Eq. (4.150) gives the implicit representation I1 = I1(h, I2) of the energy in action space.


−πψ1

N 1

π

Figure 4.26: The projection of the fundamental path dψ2 = 0 to the (N1, ψ1) plane

20.0

22.5

25.0

27.5

30.0

-10.0 -7.50 -5.00 -2.50 0.00

I2

I1

Figure 4.27: Energy surface in action space for the integrable part of the Baggott Hamil-tonian, h = 42000cm−1. The positions of the KAM torus with winding ratio 1/(2 − g)and the main resonances (1,−1) and (2,−1) are indicated by arrows.


Figure 4.27 shows the energy surface of the system given by the Hamiltonian H0 inaction space for the energy h = 42000cm−1.

To determine the magnitudes of the resonances we are interested in, let us find theangle variables θ1 and θ2. To transform from the initial coordinates N1, N2, ψ1, ψ2 toaction-angle variables we use the following generating function, see [1] for details.

F (ψ, I) =

∫ ψ

ψ0

pdq =

∫ ψ

ψ0

Ndψ

=

∫ ψ1

ψ01

N∣∣∣ψ2=ψ0

2

dψ +

∫ ψ2

ψ02

N∣∣∣ψ1=ψ0

1

dψ

=

∫ ψ1

ψ01

N1

∣∣∣ψ2=ψ0

2

dψ1 +

∫ ψ2

ψ02

N2

∣∣∣ψ1=ψ0

1

dψ2

=

∫ ψ1

ψ01

N1(h,N2, ψ1)dψ1 +N2(ψ2 − ψ02)

=

∫ ψ1

ψ01

N1(h,N2, ψ1)dψ1 + I2(ψ2 − ψ02). (4.151)

Figure 4.28 shows the integration path for the generating function F (ψ, I). We have

N =∂F

∂ψ, (4.152)

θ =∂F

∂I. (4.153)

The angles θ1 and θ2 become

θ1 =∂F

∂I1

=∂

∂I1

∫ ψ1

ψ01

N1(h, I2, ψ1)dψ1

=

∫ ψ1

ψ01

∂N1(h, I2, ψ1)

∂h

∂h

∂I1

dψ1, (4.154)

θ2 =∂F

∂I2

=∂

∂I2

∫ ψ1

ψ01

N1(h, I2, ψ1)dψ1 + ψ2 − ψ02

=

∫ ψ1

ψ01

(∂N1(h, I2, ψ1)

∂h

∂h

∂I2

+∂N1(h, I2, ψ1)

∂I2

)dψ1 + ψ2 − ψ0

2. (4.155)

Let us consider the formulae derived for the angles θ1 and θ2 in more detail. Figure4.26 shows the projection of the fundamental path dψ2 = 0 with the chosen orientationto the (N1, ψ1) plane. We have to differentiate between two cases. First, the projection(N1, ψ1) resides on the bottom part γbottom of the curve dψ2 = 0 in Figure 4.26. In this


case we obtain

θ1 =

∫ ψ1

0

∂N1

∂h

∂h

∂I1

dψ1,

=1

∂I1/∂h

∫ ψ1

0

∂N1

∂hdψ1,

=1

∂I1/∂h

∫ ψ1

0

∂

∂h

(−B −

√D

2A

)dψ1, (4.156)

where we have set ψ01 = 0 and used the fact that ∂h/∂I1 = 1/(∂I1/∂h). Note that the

coefficients A and B do not explicitly depend on h, so that ∂A/∂h = ∂B/∂h = 0. Theexpression for the angle θ1 reads

θ1 =1

∂I1/∂h

∫ ψ1

0

∂

∂h

(−√D

2A

)dψ1. (4.157)

If the projection (N1, ψ1) lies on the top part γtop of the curve, we find

θ1 =1

∂I1/∂h

(∫ ψmax1

0

∂

∂h

(−B −

√D

2A

)dψ1 +

∫ ψ1

ψmax1

∂

∂h

(−B +

√D

2A

)dψ1

)

=1

∂I1/∂h

(∫ ψmax1

0

∂

∂h

(−B −

√D

2A

)dψ1 −

∫ ψmax1

0

∂

∂h

(−B +

√D

2A

)dψ1+

+

∫ ψ1

0

∂

∂h

(−B +

√D

2A

)dψ1

)

=1

∂I1/∂h

∂

∂h(πI1) +

1

∂I1/∂h

∫ ψ1

0

∂

∂h

(−B +

√D

2A

)dψ1

= π +1

∂I1/∂h

∫ ψ1

0

∂

∂h

(√D

2A

)dψ1 (4.158)

Consider the angle θ2. Let ψ02 be equal to zero. Then Eq. (4.155) takes the form

θ2 =

∫ ψ1

0

(∂N1(h, I2, ψ1)

∂h

∂h

∂I2

+∂N1(h, I2, ψ1)

∂I2

)dψ1 + ψ2. (4.159)

We discuss again two cases. If (N1, ψ1) ∈ γbottom, we obtain

θ2 = ψ2 +∂h

∂I2

∫ ψ1

0

∂

∂h

(−B −

√D

2A

)dψ1 +

∫ ψ1

0

∂

∂I2

(−B −

√D

2A

)dψ1. (4.160)

Note again that ∂A/∂h = ∂B/∂h = 0. We have

θ2 = ψ2 +∂h

∂I2

∫ ψ1

0

∂

∂h

(−√D

2A

)dψ1 +

∫ ψ1

0

∂

∂I2

(−B −

√D

2A

)dψ1. (4.161)


Recall that the derivative ∂h/∂I2 can be expressed in the following way:

∂h

∂I2

= −∂I1/∂I2

∂I1/∂h. (4.162)

If (N1, ψ1) ∈ γtop, we have

θ2 = ψ2 +

∫ ψmax1

0

(∂h

∂I2

∂

∂h

(−B −

√D

2A

)+

∂

∂I2

(−B −

√D

2A

))dψ1 +

+

∫ ψ1

ψmax1

(∂h

∂I2

∂

∂h

(−B +

√D

2A

)+

∂

∂I2

(−B +

√D

2A

))dψ1

= ψ2 +

∫ ψmax1

0

(∂h

∂I2

∂

∂h

(−B −

√D

2A

)+

∂

∂I2

(−B −

√D

2A

))dψ1 +

+

∫ 0

ψmax1

(∂h

∂I2

∂

∂h

(−B +

√D

2A

)+

∂

∂I2

(−B +

√D

2A

))dψ1 +

+

∫ ψ1

0

(∂h

∂I2

∂

∂h

(−B +

√D

2A

)+

∂

∂I2

(−B +

√D

2A

))dψ1

= ψ2 +∂

∂I2

(2πI1)+

∫ ψ1

0

(∂h

∂I2

∂

∂h

(−B +

√D

2A

)+∂

∂I2

(−B +

√D

2A

))dψ1.(4.163)

For the sake of convenience we collect here the formulae for the angle variables.

θ1 =

1

∂I1/∂h

∫ ψ1

0∂∂h

(−√D

2A

)dψ1, (N1, ψ1) ∈ γbottom

π + 1∂I1/∂h

∫ ψ1

0∂∂h

(√D

2A

)dψ1, (N1, ψ1) ∈ γtop

(4.164)

θ2 =

ψ2 + ∂h∂I2

∫ ψ1

0∂∂h

(−√D

2A

)dψ1 +

∫ ψ1

0∂∂I2

(−B−

√D

2A

)dψ1, (N1, ψ1) ∈ γbottom

ψ2 + π ∂I1∂I2

+∫ ψ1

0

(∂h∂I2

∂∂h

(−B+

√D

2A

)+ ∂

∂I2

(−B+

√D

2A

))dψ1, (N1, ψ1) ∈ γtop.

(4.165)It is easy to calculate the Jacobian of the transformation (ψ1, ψ2)→ (θ1, θ2). We have

∂θ1

∂ψ1

=

1

∂I1/∂h∂∂h

(−√D

2A

), (N1, ψ1) ∈ γbottom

1∂I1/∂h

∂∂h

(√D

2A

), (N1, ψ1) ∈ γtop

(4.166)

∂θ1

∂ψ2

= 0, (4.167)

∂θ2

∂ψ1

=

∂h∂I2

∂∂h

(−√D

2A

)+ ∂

∂I2

(−B−

√D

2A

), (N1, ψ1) ∈ γbottom

∂h∂I2

∂∂h

(√D

2A

)+ ∂

∂I2

(−B+

√D

2A

), (N1, ψ1) ∈ γtop

(4.168)


∂θ2

∂ψ2

= 1. (4.169)

The absolute value of the determinant of the Jacobi matrix ∂(θ1, θ2)/∂(ψ1, ψ2) is given by

J =

∣∣∣∣∣ 1

∂I1/∂h

∂

∂h

(√D

2A

)∣∣∣∣∣ . (4.170)

We calculate the frequencies ∂h/∂I1 and ∂h/∂I2 in order to get a closed expressionfor angle coordinates. We have

∂h

∂I1

=

(∂I1(h, I2)

∂h

)−1

, (4.171)

∂h

∂I2

= −∂I1(h, I2)/∂I2

∂I1(h, I2)/∂h. (4.172)

(ψ , ψ )

(ψ , ψ )

1 2 1 2

1 2

(ψ , ψ )000

Figure 4.28: The path for the generating function F (ψ, I)

We study of the onset of widespread chaos between the resonances (m1,m2) = (2,−1)and (p1, p2) = (1,−1).

Consider the breakup of the KAM torus of winding ratio W = g. Note that therational numbers 1 and 1/2 corresponding to the pair of resonances (m1,m2) = (2,−1)and (p1, p2) = (1,−1) are the first two rational approximants of the irrational numberW = g.

The numerical calculation of the normal form is as follows. Given an energy level h, wefind the positions of the KAM torus and resonances on the energy surface H0(I1, I2) = hwith the help of the implicit representation (4.150) for the energy surface in action space.Let (IT1 , I

T2 ) be the position of the KAM torus. To find the coefficients of the linear and

quadratic terms of the Taylor expansion of H0(I1, I2) at the point (IT1 , IT2 ), we employ the

formulae from Appendix C. To calculate numerically the magnitudes of the resonances mand p, we proceed in the same way as in the case of the double pendulum, see formulae(4.126-4.129). Consider for example the resonance m. The perturbation part (4.140) isa function of N1, N2 = I2, and ψ2. According to the centered-resonance approximation,


-2.00

-1.60

-1.20

-0.80

-0.40

0.00

21.0 22.0 23.0 24.0 25.0 26.0 27.0

W

I2

Figure 4.29: The winding ratioW as a function of I2 for the value of energy h = 42000cm−1

we substitute the second component Im2 of the position of the resonance on the energy

surface H0 = h for N2. The quantity N1 is replaced by its expression as a function of h,I2 = N2, and ψ1 given by Eq. (4.143). The resulting representation of the perturbationpart H1 is a function of the angles ψ1, ψ2. The magnitude M is given by

M =√M2

cos +M2sin, (4.173)

Mcos =1

2π2

∫ 2π

0

∫ 2π

0

β1(−2I2 + P )√N1(h, I2, ψ1) + I2 cosψ2 ×

× cos(m1θ1(ψ1, ψ2, h, I2) +m2θ2(ψ1, ψ2, h, I2))Jdψ1dψ2, (4.174)

Msin =1

2π2

∫ 2π

0

∫ 2π

0

β1(−2I2 + P )√N1(h, I2, ψ1) + I2 cosψ2 ×

× sin(m1θ1(ψ1, ψ2, h, I2) +m2θ2(ψ1, ψ2, h, I2))Jdψ1dψ2. (4.175)

The Jacobian J is given by Eq. (4.170). The angle variables θ1, θ2 are expressed asfunctions of ψ1, ψ2, h, I2 in Eqs. (4.164-4.165). Note that the magnitudes depend linearlyon the perturbation parameter β1. Thus, for a given value of energy, we need to calculatethe integrals (4.174-4.175) only one time for some fixed value of β0

1 . We took β01 = 1 in

our actual computation. Denote the corresponding value of the magnitude by M0. Theparameter M in the normal form for some new value β1 of the perturbation parameter isgiven by M = β1M

0/β01 . The same remains true for the magnitude P .

Having determined the parameters of the normal form, we are ready to apply therenormalization theory to the Baggott Hamiltonian, see Appendix D for details.

Figure 4.30 presents the results of the application of the RG approach to the breakupof the torus of winding ratio W = g.

Take the KAM of winding ratio W = 1/(2− g). Its continued fraction representationis

W =1

2− g= [0, 1, 2, (1)] (4.176)


with (1) being an infinite sequence of ones. Thus, the irrational number W is a nobleirrational. Figure 4.27 shows the positions of the torus W = 1/(2 − g) and resonances(m1,m2) = (2,−1) and (p1, p2) = (1,−1) on the energy surface h = 42000cm−1. Theapplication of the RG approach to the study of the breakup of the torus W = 1/(2− g)gives the results presented in Figure 4.31. The comparison with Figure 4.30 shows thatfrom the point of view of the RG approach this torus is essentially more stable than thatof winding ratio W = g.

The numerical results of the sup map analysis with number of iteration for each orbitbeing 1000 and 10000 respectively are shown Figures 4.32, 4.33. The step width in β1 is0.1.

Let us compare the results obtained using the RG approach (W = 1/(2 − g)) withthe results of the sup-map analysis. Denote the estimate for the critical value of β1 asa function of h by βRG1 (h) in the case of the RG approach and βsup1 (h) in the case ofthe sup-map analysis. Note that the critical values βsup1 (h) are larger than βRG1 (h). Thedifference is up to 30% in β1. Taking into account the fact that the critical values βsup1 (h)get smaller as the number of iterations for each orbit is increased, we can say that theRG approach leads to rather realistic estimates.

10.00

11.00

12.00

13.00

14.00

15.00

16.00

17.00

18.00

19.00

20.00

35000 36000 37000 38000 39000 40000

β1

h

Figure 4.30: The critical value of β1 for the KAM torus with winding ratio g = (√

5−1)/2.


20.00

22.00

24.00

26.00

28.00

30.00

32.00

34.00

36.00

38.00

40.00

35000 36000 37000 38000 39000 40000

β1

h

Figure 4.31: The critical value of β1 for the KAM torus with winding ratio 1/(2 − g),where g = (

√5− 1)/2.

20.00

23.00

26.00

29.00

32.00

35.00

38.00

41.00

44.00

47.00

50.00

35000 36000 37000 38000 39000 40000

β1

h

Figure 4.32: The critical value of β1 for the KAM torus with winding ratio 1/(2 − g),where g = (

√5 − 1)/2 (numerical calculation using sup-map analysis). The number of

iterations for each orbit is 1000.


30.00

30.50

31.00

31.50

32.00

32.50

33.00

33.50

34.00

34.50

35.00

35000 35500 36000 36500 37000 37500

β1

h

Figure 4.33: The same as Figure 4.32 with the number of iterations for each orbit being10000.


4.6 Limacon billiards

This section is devoted to the study of the so-called limacon billiards [40, 6]. The billiardboundary for the limacon billiard is given by

ρ(ϕ) = 1 + ε cosϕ, ϕ ∈ [−π, π], (4.177)

where ρ and ϕ are polar coordinates, ε ∈ [0, 1] a parameter, see Figure 4.34.The system’s phase space is a four-dimensional manifold

M4 =

(x, y, px, py)∣∣∣(x, y) ∈ Bε

, (4.178)

whereBε =

(x, y) = (r cosϕ, r sinϕ)

∣∣∣r ≤ 1 + ε cosϕ

(4.179)

is the set of accessible points for the given value ε. The Hamiltonian is given by

H =1

2(p2x + p2

y) =1

2

(p2r +

p2ϕ

r2

). (4.180)

It is sufficient to study the system’s behavior for some fixed value of energy h becausethe energy may be scaled away. The motion takes place on the three-dimensional energysurface

E3 =

(x, y, px, py)∣∣∣(x, y) ∈ Bε, p

2x + p2

y = 2. (4.181)

We use Poincare surfaces of section technique in order to study the dynamics of thelimacon billiards. To do that, we consider orbits just after the reflection at the boundaryof the limacon billiard. More formally, as surface of section we choose the surface

S2 =

(x, y, px, py)∣∣∣(x, y) ∈ ∂Bε, p⊥ > 0, p2

x + p2y = 2

⊂ E3, (4.182)

where ∂Bε is the boundary of the limacon billiard, p⊥ = pxnx + pyny is the momentum’sprojection onto the normal direction (nx, ny) to ∂Bε at point (x, y). We consider theprojection of the surface S2 onto the cylinder

Pε =

(p‖, ϕ)∣∣∣p‖ ∈ [−1, 1], ϕ ∈ [0, 2π]

. (4.183)

Note that this choice differs from the usual representation in terms of Birkhoff’s coordi-nates , see [7]; the angle ϕ is not the arc length along the boundary. The Poincare mapP : Pε → Pε for different values of the perturbation parameter is shown in Figure 4.37.The main features of a typical Poincare surface of section for the limacon billiard areKAM-Tori, chains of islands, corresponding to elliptic orbits , and chaotic regions.

Consider Figure 4.37 in more detail. For smaller values of the perturbation parameterε, that is, ε = 0.1, ε = 0.2, two large chaotic regions can be identified in the upper half ofthe Poincare plots. One of them is located exactly in the center of the Poincare plots. Thecorresponding island chain consists of two islands. The other chaotic region is located inthe top part of the Poincare plots. The corresponding island chain contains three ellipticislands. The chaotic regions correspond to the resonances (2,−1) and (3,−1).

4.6. LIMACON BILLIARDS 81

2.00

-2.0-2.0 2.00

ε = 0

y

x

2.00

-2.0-2.0 2.00

ε = 2ε−1

y

x

2.00

-2.0-2.0 2.00

ε = 4ε−1

y

x

2.00

-2.0-2.0 2.00

ε = 6ε−1

y

x

2.00

-2.0-2.0 2.00

ε = 8ε−1

y

x

2.00

-2.0-2.0 2.00

ε = 1

y

x

Figure 4.34: The family of limacon billiards for ε = 0, ε = 0.2, ε = 0.4, ε = 0.6, ε = 0.8,ε = 1.


Our aim is to study the onset of widespread chaos between the resonances (m1,m2) =(3,−1) and (p1, p2) = (2,−1). As the last KAM torus we choose the torus of windingratio

W =(p2 −m2)g − p2

(m1 − p1)g + p1

=1

g + 2= 1− g (4.184)

with g = (√

5− 1)/2 being the golden mean, see Appendix A. Note that the first rationalapproximants to the irrational number W given by the continued fraction expansion are

P1

Q1

=1

2,

P2

Q2

=1

3. (4.185)

They are the winding numbers corresponding to the resonances (m1,m2) = (3,−1) and(p1, p2) = (2,−1).

In what follows we study the breakup of the KAM torus with winding ratio 1/(g+ 2).Note that the case ε = 0 is integrable. The corresponding billiard is the circle billiard.

The system possesses two independent constants of motion. These are the Hamiltonianitself and the modulus of the angular momentum squared, p2

ϕ.In order to apply the renormalization theory to the limacon billiards we proceed as

follows. First, we introduce action-angle coordinates for the integrable case ε = 0. Second,we find the position of the last KAM torus on the energy surface in action space andreplace the unperturbed Hamiltonian by its Taylor expansion at the position of the lastKAM torus to the second order. We represent the Hamilton function as a near-integrablesystem in the following way:

H = H0 +H1. (4.186)

We make use of the two-resonance approximation of the perturbation part. We approxi-mate the amplitudes of the two resonances using the centered-resonance approximation.

It is necessary for us to introduce action-angle coordinates for the integrable case ε = 0.A pair (h, lϕ) of the values of the constants of motion defines a two-dimensional torus T 2

h,lϕ

in phase space. As fundamental cycles around these tori we choose γϕ = (r, ϕ, pr, pϕ) ∈T 2h,lϕ|dr = 0, γr = (r, ϕ, pr, pϕ) ∈ T 2

h,lϕ|dϕ = 0. We choose the orientation on the path

γr as it is shown in Figure 4.35.

r

p r

Figure 4.35: The orientation of the path γr.

The action coordinates Ir and Iϕ are given by

Ir =1

2π

∮γr

pdq =1

2π

∮γr

prdr, (4.187)


Iϕ =1

2π

∮γϕ

pdq =1

2π

∮γr

pϕdϕ = lϕ. (4.188)

We obtain

Ir =1

2π

∮γr

prdr =2

2π

∫ rmax

rmin

√2(h−

l2ϕ2r2

)dr

=

√2h

π

∫ rmax

rmin

√1−

l2ϕ2hr2

dr =

√2h

π

∫ arccos lϕ/√

2h

0

sin2 t

cos2 tdt

lϕ√2h

=lϕπ

∫ arccos lϕ/√

2h

0

sin td( 1

cos t

)=

lϕπ

( sin t

cos t

∣∣∣arccos lϕ/√

2h

0−∫ arccos lϕ/

√2h

0

d sin t

cos t

)=

lϕπ

( sinφ

lϕ/√

2h− φ) =

√2h

π(sinφ− φ cosφ), (4.189)

where φ is defined by

cosφ =lϕ√2h. (4.190)

The energy surface in action space is shown in Figure 4.38, see also [39, 44].Using action variables

Iϕ = lϕ, Ir =

√2h

π(sinφ− φ cosφ), (4.191)

we can express the winding ratio W for the torus T 2h,lϕ

as

W = −∂Ir(Iϕ, h)

∂Iϕ. (4.192)

We obtain

W = − ∂Ir∂Iϕ

= −∂Ir∂lϕ

= −√

2h

π(cosφφ′ − φ′ cosφ+ φ sinφφ′)

= −√

2h

πφ sinφφ′ = −

√2h

πφd cosφ = −

√2h

πφd cosφ

= −√

2h

πφ

dlϕ√2h

=φ

π=

arccos (lϕ/√

2h)

π(4.193)

with φ′ being ∂φ/∂lϕ. The winding ratio as a function of I1 is shown in Figure 4.39, seealso [39, 44].

To find the coefficients of the quadratic terms in the normal form, we need the secondderivatives of the energy with respect to actions. They can be found with the help of the


formulae from Appendix C using the second derivatives of Ir(h, Iϕ) with respect to h andIϕ. They are given by

∂2Ir∂I2

ϕ

=1

π√

2h− I2ϕ

, (4.194)

∂2Ir∂Iϕ∂h

= − Iϕ

2hπ√

2h− I2ϕ

, (4.195)

∂2Ir∂h2

=I2ϕ − h

2h2π√

2h− I2ϕ

. (4.196)

To find the angle coordinates θ1, θ2, we make use of the generating function

F (ϕ, r, Iϕ, Ir) = Iϕϕ+

∫ r

1

prdr. (4.197)

The angle coordinates are given by the derivatives of the generating function F withrespect to the action Iϕ, Ir. We obtain

θ1 =∂F

∂Iϕ= ϕ+

∂

∂Iϕ

∫ r

1

pr(r, Iϕ, Ir)dr, (4.198)

θ2 =∂F

∂Ir=

∂

∂Ir

∫ r

1

pr(r, Iϕ, Ir)dr. (4.199)

Note that

∂pr(r, Iϕ, Ir)

∂Iϕ=

∂pr(r, Iϕ, h)

∂Iϕ+∂pr(r, Iϕ, h)

∂h

∂h

∂Iϕ, (4.200)

∂pr(r, Iϕ, Ir)

∂Ir=

∂pr(r, Iϕ, h)

∂h

∂h

∂Ir. (4.201)

We obtain

θ1 =ϕ+arccos

(Iϕ

r√

2h

)−arccos

(Iϕ√2h

)+

(√2h− I2

ϕ −√

2hr2 − I2ϕ

2h

)∂h

∂Iϕ,(4.202)

θ2 =

(√2h− I2

ϕ −√

2hr2 − I2ϕ

2h

)∂h

∂Ir, (4.203)

if the point (pr, r) is located on the bottom part of the integration path, see Figure 4.35,and

θ1 =ϕ−arccos

(Iϕ

r√

2h

)−arccos

(Iϕ√2h

)+

(√2h− I2

ϕ +√

2hr2 − I2ϕ

2h

)∂h

∂Iϕ,(4.204)

θ2 =

(√2h− I2

ϕ +√

2hr2 − I2ϕ

2h

)∂h

∂Ir, (4.205)

if the point (pr, r) is on the top part of the integration path.


The determinant of the Jacobi matrix for the transformation (ϕ, r)→ (θ1, θ2) reads

J = − ∂h∂Ir

r√2hr2 − I2

ϕ

, (4.206)

if the point (pr, r) is on the bottom part of the integration path, and

J =∂h

∂Ir

r√2hr2 − I2

ϕ

, (4.207)

if the point (pr, r) is on the top part of the integration path.To represent the Hamiltonian as a near-integrable system

H = H0 +H1, (4.208)

note that the boundary r = 1 + ε cosϕ can be obtained as the image of the unit circlein the complex z = x + iy plane under the quadratic conformal map w = u + iv =z+εz2/2, see [40, 6]. The solutions of the equations of motion of the limacon billiards aregeodesics of the Riemannian metric ds2 = du2 + dv2. At the boundary the trajectoriessatisfy the reflection law. Consider the coordinate transformation (u, v)→ (x, y). In thecoordinates (x, y) the trajectories of the system are geodesics of the Riemannian metricds2 = du(x, y)2 + dv(x, y)2. We have

ds2 = ((1 + εx)2 + ε2y2)(dx2 + dy2). (4.209)

Due to the fact that the transformation is conformal, the trajectories again satisfies thereflection law at the boundary which is the unit circle. Introduce polar coordinates (r, ϕ)in the (x, y) plane. It is not difficult to check that the Hamiltonian in polar coordinatesreads

H =1

2

p2r +

p2ϕ

r2

1 + 2εr cosϕ+ ε2r2= H0 +H1, (4.210)

with

H0 =1

2(p2r +

p2ϕ

r2), (4.211)

H1 = −(2εr cosϕ+ ε2r2)(p2

r +p2ϕ

r2 )

2(1 + 2εr cosϕ+ ε2r2). (4.212)

The integrable part H0 corresponds to the circle billiard. Using the two-resonance andcentered-resonance approximations, we can now calculate the coefficients M and P in thenormal form. Note that the coefficients M and P depend non-linearly on the perturbationvalue ε.

We have calculated the coefficients of the normal form numerically for different valuesof ε. Using the equation for the energy surface in action space (4.191), the formula for thewinding ratio (4.193), the formulae for the Hesse matrix (4.194- 4.196), the formulae forthe angle variables (4.202-4.205), and the formula for the Jacobian (4.206), we proceededin the similar way as in the cases of the double pendulum and the Baggott Hamiltonian.


The parameters ω1, ω2, a, b, c of the non-scaled normal form in the case h = 1 are foundto be

ω1 = −1.12603, ω2 = 0.694589, (4.213)

a = −1.55053, b = −1.54643, c = −1.35033. (4.214)

The magnitudes M , P as functions of the perturbation parameter ε are shown in Figure4.36.

0.0

1.2

2.4

3.6

4.8

6.0

0.0 0.2 0.4 0.6 0.8 1.0

M

ε

0.0

1.2

2.4

3.6

4.8

6.0

0.0 0.2 0.4 0.6 0.8 1.0

P

ε

Figure 4.36: The parameters M , P of the normal form as functions of the perturbationparameter.

The estimate for the threshold to widespread chaos between the resonances (2,−1)and (3,−1) is found to be εRGcrit ≈ 0.47. As can be seen from Figure 4.37, the actual valueof the threshold is approximately given by εcrit = 0.3. Thus, the RG approach leads toan essentially larger value of the threshold in comparison with the actual one.


1.00

-1.00.00 6.28

ε=0.0000

p

ϕ

1.00

-1.00.00 6.28

ε=0.1000

p

ϕ

1.00

-1.00.00 6.28

ε=0.2000

p

ϕ

1.00

-1.00.00 6.28

ε=0.3000

p

ϕ

1.00

-1.00.00 6.28

ε=0.4000

p

ϕ

1.00

-1.00.00 6.28

ε=0.5000

p

ϕ

Figure 4.37: Poincare sections for the limacon billiards for ε = 0, ε = 0.1, ε = 0.2, ε = 0.3,ε = 0.4, ε = 0.5.


0.45

0.00-1.4 1.40

h = 1

I 2

I 1

Figure 4.38: Energy surface of a circle billiard, h = 1.

0.50

0.020.00 1.41

h = 1

W

I 1

Figure 4.39: Winding ratio of a circle billiard, h = 1.

Energy surface

0

0.2

0.4

0.6

0.8

1

1.2

1.4

I2

0 0.2 0.4 0.6 0.8 1 1.2 1.4I1

Figure 4.40: The energy surface of a circle billiard, h = 1, in action-angle coordinates andthe positions of the KAM torus with winding ratio 1/(g + 2), and the main resonances(2,−1) and (3,−1).

Chapter 5

Conclusions

In this work we studied the onset of widespread chaos in Hamiltonian systems with twodegrees of freedom using the renormalization group approach.

In the framework of the renormalization theory for Hamiltonian systems with twodegrees of freedom KAM tori are viewed as barriers to the onset of widespread chaosbetween a pair of resonances. Thus the transition to widespread chaos coincides withthe breakup of the last KAM torus. It is not easy to determine which of the KAM toriturns out to be the last one for a given Hamiltonian system, pair of resonances and valueof energy. A KAM torus of noble winding ratio represents a reasonable choice. Thequestion of the existence of a KAM torus with a given winding ratio can be studied usingperturbation theory for near-integrable Hamiltonian systems. The application of the RGapproach to a given Hamiltonian systems can be divided into two steps. First, for agiven pair of resonances, value of energy and winding ratio of the prospective last KAMtorus we approximate the Hamilton function by its normal form. The resulting normalform defines a starting point for the renormalization map in the finite-dimensional spaceof normal forms. Second, using perturbation theory we get rid of the resonances in thenormal form to the first order in perturbation parameter. The resulting Hamiltonian isagain approximated by its normal form. The renormalization map can then be applied tothis new point in the space of normal forms and the procedure can be iterated. It turnsout that the breakup of the KAM torus in question is highly connected with the behaviorof the corresponding orbit in the parameter space of normal forms.

The normal form and the renormalization map for the chosen normal form can beintroduced in different ways. Our version of the renormalization theory is based on [16, 34].The version works reliably in the case where the last KAM torus has a winding ratio whichis given by the golden mean.

We applied the renormalization group approach to a number of Hamiltonian systems.We discussed the paradigm Hamiltonian of Escande and Doveil, the Walker and Fordmodel, a model of the ethane molecule, the double pendulum, the Baggott system, limaconbilliards. The results are compared with numerical studies using the Poincare surface ofsection technique.

To apply the renormalization theory to a given system we must represent it as a near-integrable Hamiltonian system. Sometimes, to identify a suitable integrable limiting caseturns out to be a non-trivial problem. For example, in the case of the double pendulum

89

90 CHAPTER 5. CONCLUSIONS

we discussed the integrable limit of high energies. As the study with the help of Poincaresection technique shows, from the point of view of the KAM theory this limiting caseseems to be a reasonable choice for the study of the onset of global chaos. However, whenaction-angle variables for this choice of the unperturbed Hamiltonian are introduced,relevant resonances turn out to lie on different sheets of the energy surface in actionspace. Apparently this is a severe difficulty for the RG approach and seems to make ituseless. It would be very exciting to generalize the RG approach to handle such systemsproperly.

One of the technical obstacles for the application of the RG approach to a given Hamil-tonian system is to introduce action-angle variables for the corresponding unperturbedsystem and to express the perturbation part in these new coordinates. For example, inthe case of the double pendulum the action-angle variables for the integrable case studiedare given by hyperelliptic integrals [15]. We had to use numerical methods in order tocalculate the magnitudes of the resonances and other parameters of the normal form insuch cases.

In this thesis we discussed the RG approach to the breakup of the KAM tori inHamiltonian systems with two degrees of freedom. New developments, see [9, 10], provideus with the generalization of the renormalization theory to the case of Hamiltonian systemswith three degrees of freedom.

A number of simplifications are employed in order to apply the RG approach to a givensystem. In particular, we mention the two-resonance approximation, the local-quadraticapproximation, the centered-resonance approximation. It seems to be rather difficult tounderstand how a particular approximation can influence the threshold to widespreadchaos and thus, how the accuracy of the estimate of the threshold can be controlled.Advances in this direction would explain essential deviations of the outcome of the RGapproach from the numerical estimates in some cases and could considerably improve theresults given by the renormalization theory.

Appendix A

The Normal Form for a HamiltonianSystem with Two Degrees ofFreedom

Let us consider the Hamiltonian system

H(I1, I2, θ1, θ2) = H0(I1, I2) + εH1(I1, I2, θ1, θ2), (A.1)

H1(I1, I2, θ1, θ2) =∑

n1,n2∈Z

Vn1,n2(I1, I2) cos(n1θ1 + n2θ2 + γn1,n2). (A.2)

Given two resonances (m1,m2) and (p1, p2), we take into account only the correspondingtwo terms in H1. This approximation is called the two-resonance approximation. Further,we fix a KAM torus with the frequencies given by a Diophantine frequency vector ω =(ω1, ω2) on the energy surface H = h. The KAM torus is supposed to lie between theresonances (m1,m2) and (p1, p2) on the energy surface H = h. The position of theunperturbed torus on the energy surface H0 = h of the integrable part H0 is givenby a vector IT = (IT1 , I

T2 ). Denote the corresponding positions of the resonances by

IM = (IM1 , IM2 ) and IP = (IP1 , IP2 ). In what follows we replace the amplitudes εVm1,m2 and

εVp1,p2 of the resonances by their values M = εVm1,m2(IM1 , IM2 ) and P = εVp1,p2(IP1 , IP2 ) at

IM and IP . In other words we use the so-called centered-resonance approximation.Note that

ω1 =∂H0

∂I1

(IT1 , IT2 ), ω2 =

∂H0

∂I2

(IT1 , IT2 ). (A.3)

Expand the function H0 in a Taylor series at the point (IT1 , IT2 ) to the second order:

H ′0(I1, I2) = H0(IT1 , IT2 ) + ω1(I1 − IT1 ) + ω2(I2 − IT2 ) +

+a(I1 − IT1 )2 + 2b(I1 − IT1 )(I2 − IT2 ) + c(I2 − IT2 )2, (A.4)

where

a =1

2

∂2H0

∂I21

(IT1 , IT2 ), b =

1

2

∂2H0

∂I1∂I2

(IT1 , IT2 ), c =

1

2

∂2H0

∂I22

(IT1 , IT2 ). (A.5)

In the following we consider the function H ′0 instead of H0. We refer to this approximationas the local-quadratic approximation. We have

H ′(I1, I2, θ1, θ2) = H ′0(I1, I2)+M cos(m1θ1 +m2θ2 +γM)+P cos(p1θ1 +p2θ2 +γP ). (A.6)

91

92 APPENDIX A. THE NORMAL FORM

Note that by shifting θi + const → θi, we can get rid of the terms γM and γP . Thus, wecan assume that γM = 0, γP = 0. Adding a constant to a Hamilton function does notchange the equations of motion. Thus, we can suppose that h = H0(IT1 , I

T2 ) = 0. Shifting

Ii − ITi → Ii, we obtain

H(I1, I2, θ1, θ2) = ω1I1+ω2I2+aI21 +2bI1I2+cI2

2 +M cos(m1θ1+m2θ2)+P cos(p1θ1+p2θ2).(A.7)

Introduce the matrix

R =

(m1 m2

p1 p2

). (A.8)

We perform the canonical transformation defined by the formulae(θ′1θ′2

)= R

(θ1

θ2

), (A.9)(

I ′1I ′2

)= (Rt)−1

(I1

I2

). (A.10)

In the new coordinates, the coefficients of the linear and quadratic terms become(ω′1ω′2

)= R

(ω1

ω2

), (A.11)

(a′ b′

b′ c′

)= R

(a bb c

)Rt. (A.12)

The Hamiltonian reads

H ′(I ′1, I′2, θ′1, θ′2) = ω′1I

′1 + ω′2I

′2 + a′I ′21 + 2b′I ′1I

′2 + c′I ′22 +M cos θ′1 + P cos θ′2. (A.13)

We refer to the Hamiltonian (A.13) as the non-scaled normal form. Replace the Hamil-tonian H ′ by λ1H

′. Trajectories are invariant under such changes of the time. We takeλ1 = 1/ω′2. Then the frequency vector becomes (ω, 1), where ω = ω′1/ω

′2. Using, if

necessary, the linear canonical transformation which interchanges the action coordinates,we can assume that the absolute value of ω is less than one. With the help of the lin-ear canonical transformation which changes the sign of the first action variable, we canfurther suppose that ω is positive.

Note that, without changing the equations of motion, we can substitute the functionλ2H

′(I ′1/λ2, I′2/λ2, θ

′1, θ′2) for the HamiltonianH ′, where λ2 is an arbitrary nonzero number.

Thus, we can assume without loss of generality that the numbers a′, b′, c′ satisfy therelation

a′2 + b′2 + c′2 = 1. (A.14)

We can further assume that the coefficient a′ is positive. Indeed, if a′ is negative, weapply to the Hamiltonian H ′ the linear canonical transformation which changes the signsof both action coordinates, and then multiply the resulting Hamiltonian by (−1).

Omitting primes, we have now the following formula for the Hamiltonian

H(I1, I2, θ1, θ2) = ωI1 + I2 + aI21 + 2bI1I2 + cI2

2 +M cos θ1 + P cos θ2,

0 < ω < 1, 0 < a, a2 + b2 + c2 = 1. (A.15)

93

We will call Hamiltonian (A.15) the normal form for a Hamiltonian system with twodegrees of freedom. Recall that the parameters ω, a, b, c, M , P of the normal formdepend on the value of perturbation parameter ε, the energy level h, the resonances(m1,m2), (p1, p2), and the winding ratio of the KAM torus.

Note that Eq. (A.11) suggests the following choice of the winding ratio for the lastKAM torus between given resonances (m1,m2) and (p1, p2). Define the new angle coor-dinates by the relation(

1 −11 0

)(θ′1θ′2

)=

(m1 m2

p1 p2

)(θ1

θ2

). (A.16)

The perturbation part in the new coordinates is given by

H ′1(θ′1, θ′2) = M cos(θ′1 − θ′2) + P cos θ′1. (A.17)

The winding ratios of the resonances are W ′M = 1 and W ′

P = 0. Perturbation (A.17)corresponds to the perturbation in the well-known paradigm Hamiltonian of Escande andDoveil [16]. The last KAM torus is of the winding ratio W ′ = g with g = (

√5−1)/2 being

the golden mean. The winding ratio W of this torus in the old coordinates (I1, I2, θ1, θ2)can be found with the help of Eq. (A.11). Denote the matrices(

1 −11 0

),

(m1 m2

p1 p2

)(A.18)

by R0 and R respectively. Then(ω1

ω2

)= R−1R0

(ω′1ω′2

). (A.19)

We have

R−1R0 =1

m1p2 −m2p1

(p2 −m2 −p2

m1 − p1 p1

). (A.20)

Thus,

W =ω1

ω2

=(p2 −m2)ω′1 − p2ω

′2

(m1 − p1)ω′1 + p1ω′2=

(p2 −m2)W ′ − p2

(m1 − p1)W ′ + p1

. (A.21)

In the case W ′ = g we obtain

W =(p2 −m2)g − p2

(m1 − p1)g + p1

. (A.22)

94 APPENDIX A. THE NORMAL FORM

Appendix B

Classical Perturbation Theory

In this section we apply classical perturbation theory to the normal form for Hamiltoniansystems with two degrees of freedom, see Appendix A. In our discussion we follow [34].

Consider a perturbed Hamiltonian system

H(I,θ) = H(I1, I2, θ1, θ2) = H0(I1, I2) + ε∑

(n1,n2)

Vn1,n2(I1, I2) cos(n1θ1 + n2θ2)

= H0(I) + ε∑n

Vn(I) cosn · θ. (B.1)

We see that the Hamiltonian of the system contains resonances. In general, it is notpossible to get rid of all the resonances with the help of canonical transformations. Ouraim now is to introduce new canonical coordinates such that the Hamiltonian in thesecoordinates has no resonances in first order with respect to the perturbation parameterε. To do that, we use the canonical transformation (I1, I2, θ1, θ2) → (J1, J2, φ1, φ2) withthe generating function

F (J ,θ) = θ · J + εΦ(J ,θ), (B.2)

where the function Φ(J ,θ) is still to be determined. The new angles and the old actionsare given by

φ =∂F

∂J= θ + ε

∂Φ

∂J, (B.3)

I =∂F

∂θ= J + ε

∂Φ

∂θ. (B.4)

The expression for the new Hamilton function to the first order in ε reads

H(J ,φ) = H(I(J ,φ),θ(J ,φ))

= H0(I(J ,φ)) + ε∑n

Vn(I(J ,φ)) cosn · θ(J ,φ)

≈ H0(J) +∂H0

∂I(J) · ε∂Φ

∂θ+ ε

∑n

Vn(J) cosn · θ(J ,φ). (B.5)

To get rid of resonances, we demand that

∂H0

∂I(J) · ∂Φ

∂θ= −

∑n

Vn(J) cosn · θ(J ,φ). (B.6)

95

96 APPENDIX B. CLASSICAL PERTURBATION THEORY

Let us expand the function Φ in a Fourier series.

Φ(J ,θ) =∑

Wn(J) sinn · θ. (B.7)

Then∂Φ

∂θ=(∑

Wn(J) cosn · θ)n. (B.8)

To satisfy Eq. (B.6), we choose the function Φ in the following way:

Φ(J ,θ) = −∑n

Vn(J)

n · ω(J)sinn · θ, (B.9)

where ω = ∂H0/∂I. The new angles are

φ = θ + ε∂Φ

∂J=

= θ − ε∑ ∂Vn(J)

∂J

1

n · ω(J)sinn · θ(J ,φ) +

+ε∑ Vn(J)

(n · ω(J))2(n · ∂ω

∂J) sinn · θ(J ,φ)

= θ − ε∑(

∂Vn(J)

∂J− Vn(J)

n · ω(J)n · σ

)sinn · θ(J ,φ)

n · ω(J). (B.10)

The old momenta read

I = J + ε∂Φ

∂θ

= J − ε∑

nVn(J)

n · ω(J)cosn · θ(J ,φ), (B.11)

where σ = ∂ω/∂I is the Hessian of the function H0.We are interested in the terms of the second order in the new Hamilton function.

H(J ,θ) = H0(J) + ε(ω · ∂Φ

∂θ) +

ε2

2

(∂Φ

∂θ, σ∂Φ

∂θ

)+

+ ε∑

Vn(J) cosn · θ +

+ ε2∑ ∂Vn

∂J· ∂Φ

∂θcosn · θ. (B.12)

The function Φ(J ,θ) is chosen to satisfy Eq. (B.6). Thus,

H(J ,θ) = H0(J) + ε · 0 +

+ ε2

(1

2

(∂Φ

∂θ, σ∂Φ

∂θ

)+∑n

(∂Vn∂J· ∂Φ

∂θ

)cosn · θ

)

= H0(J) + ε2∑mn

(1

2

Vm(m · ω)

(m · σ)− ∂Vm∂J

)×

× Vn(n · ω)

n cosm · θ cosn · θ. (B.13)

97

Using the notation

Smn(J) =Vm

(m · ω)

(1

2(m, σn)

Vn(n · ω)

−(m · ∂Vn

∂J

)), (B.14)

Tn(J) = (T 1n, T

2n) =

1

(n · ω)

(∂Vn∂J− Vn

(n · ω)(n · σ)

), (B.15)

rewrite the formulas for the new angle variables and the Hamiltonian:

φ = θ − ε∑

Tn(J) sinn · θ, (B.16)

H(J ,θ) = H0(J) +ε2

2

∑mn

Smn(J) (cos ((m− n) · θ) + cos ((m+ n) · θ))

= H0(J) +ε2

2

∑m

Smm(J) +ε2

2

∑m

Smm(J) cos (2m · θ) +

+ε2

2

∑m 6=n

Smn(J) (cos ((m− n) · θ) + cos ((m+ n) · θ)) . (B.17)

From now on, we restrict ourselves to the case of only two resonances m = (m1,m2) andp = (p1, p2).

H(J ,θ) = H0(J) +ε2

2(Smm(J) + Spp(J)) +

+ε2

2(Smm(J) cos (2m · θ) + Spp(J) cos (2p · θ)) +

+ε2

2(Smp(J) + Spm(J)) (cos ((m− p) · θ) + cos ((m+ p) · θ)) . (B.18)

The Hamiltonian given by (B.18) is a function of the new momenta J1, J2 and old anglesθ1, θ2. Our aim is to represent the Hamiltonian as a function of the new momenta J1, J2

and new angles φ1, φ2. In this representation the Hamiltonian can be expressed by

H(J ,φ) = H0(J) +ε2


+ ε2

∞∑a=0

∞∑b=−∞

Kab cos ((am+ bp) · φ) , (B.19)

where a, b are integers and the coefficients Kab are still to be determined. The coefficientK00 is assumed to be zero. Consider the expansion of the term cos ((αm+ βp) · θ) in aFourier series with respect to the new angles φ1, φ2:

cos ((αm+ βp) · θ) =∞∑a=0

∞∑b=−∞

Aαβab cos ((am+ bp) · φ) , (B.20)


where the coefficients Aαβab are given by

Aαβab =1

2π2

∫dφ cos ((αm+ βp) · θ) cos ((am+ bp) · φ) . (B.21)

The coefficients Kab read

Kab =1

2

(SmmA

20ab + SppA

02ab + (Smp + Spm)(A1,1

ab + A1,−1ab )

). (B.22)

To find the coefficients Aαβab we make use of the change (φ1, φ2) → (θ1, θ2) under theintegral in Eq. (B.21). We obtain

Aαβab =1

2π2

∫dθ

∣∣∣∣∂φ∂θ∣∣∣∣ cos ((αm+ βp) · θ) cos (((am+ bp) · θ)

− ε ((am+ bp) · (Tm(J) sinm · θ + Tp(J) sinp · θ)))

with the Jacobian matrix ∂φ/∂θ = ∂(φ1, φ2)/∂(θ1, θ2) being

∂φ

∂θ=

(∂φ1

∂θ1

∂φ1

∂θ2∂φ2

∂θ1

∂φ2

∂θ2

)(B.23)

∂φ1

∂θ1

= 1− ε(T 1mm1 cos(m1θ1 +m1θ2) + T 1

pp1 cos(p1θ1 + p1θ2)),

∂φ1

∂θ2

= −ε(T 1mm2 cos(m1θ1 +m1θ2) + T 1


∂φ2

∂θ1

= −ε(T 2mm1 cos(m1θ1 +m1θ2) + T 2


∂φ2

∂θ2

= 1− ε(T 2mm2 cos(m1θ1 +m1θ2) + T 2

pp2 cos(p1θ1 + p1θ2)).

The determinant of the matrix ∂φ/∂θ is∣∣∣∣∂φ∂θ∣∣∣∣ =

(1− ε(T 1

mm1 cosm · θ + T 1pp1 cosp · θ)

)×

×(1− ε(T 2

mm2 cosm · θ + T 2pp2 cosp · θ)

)−

− ε2(T 1mm2 cosm · θ + T 1

pp2 cosp · θ)×

×(T 2mm1 cosm · θ + T 2

pp1 cosp · θ)

= 1− ε ((Tm ·m) cosm · θ + (Tp · p) cosp · θ) + O(ε2). (B.24)

We rewrite the determinant given by (B.24) as follows:∣∣∣∣∂φ∂θ∣∣∣∣ = 1− ε

(1

(m · ω)

((∂Vm∂J·m)− Vm

(m · ω)(m, σm)

)cosm · θ+

+1

(p · ω)

((∂Vp∂J· p)− Vp

(p · ω)(p, σp)

)cosp · θ

)+ O(ε2). (B.25)

99

Consider again Eq. (B.24). We are interested in the term of the second order in ε. It isgiven by

T 1mm1T

2mm2 cos2 (m · θ) + T 1

pp1T2pp2 cos2 (p · θ) +

+T 1mm1T

2pp2 cosm · θ cosp · θ + T 1

pp1T2mm2 cosm · θ cosp · θ −

−T 1mm2T

2mm1 cos2 (m · θ)− T 1

pp2T2pp1 cos2 (p · θ)−

−T 1mm2T

2pp1 cosm · θ cosp · θ − T 1

pp2T2mm1 cosm · θ cosp · θ

= −((m2p1 −m1p2)T 1

mT2p cosm · θ cosp · θ−

−(m2p1 −m1p2)T 2mT

1p cosm · θ cosp · θ

)= −(m2p1 −m1p2) cosm · θ cosp · θ(T 1

mT2p − T 2

mT1p))

= −(m2p1 −m1p2) cosm · θ cosp · θ ×

×

1

(m · ω)

(∂Vm∂J1

− Vm(m · ω)

(m · σ)1

)× 1

(p · ω)

(∂Vp∂J2− Vp

(p · ω)(p · σ)2

)− 1

(m · ω)

(∂Vm∂J2

− Vm(m · ω)

(m · σ)2

)1

(p · ω)

(∂Vp∂J1− Vp

(p · ω)(p · σ)1

)= +

ε2(m2p1 −m1p2) cosm · θ cosp · θ(m · ω) (p · ω)

×

×(−∂Vm∂J1

∂Vp∂J2

+∂Vm∂J2

∂Vp∂J1

)+

+VmVp

(m · ω) (p · ω)

((m · σ)2 (p · σ)1 − (m · σ)1 (p · σ)2)+

+Vm

(m · ω)

((m · σ)1 ∂Vp

∂J2− (m · σ)2 ∂Vp

∂J1

)−

− Vp(p · ω)

((p · σ)1 ∂Vm

∂J2− (p · σ)2 ∂Vm

∂J1

). (B.26)


Having found the determinant of the matrix ∂φ/∂θ, we obtain

Aαβab = Cαβab +

ε

2 (m · ω)

((m · σ)m

(m · ω)Vm −

(∂Vm∂J·m))

×(Cα+1,βab Cα−1,β

ab

)+

ε

2 (p · ω)

((p · σ)p

(p · ω)Vp −

(∂Vp∂J· p))(

Cα,β+1ab Cα,β−1

ab

)+

ε2(m2p1 −m1p2)

4 (m · ω) (p · ω)×

×(∂Vm

∂J2

∂Vp∂J1− ∂Vm∂J1

∂Vp∂J2

)+

+VmVp

(m · ω) (p · ω)(m2p1 −m1p2) detσ +

+Vm

(m · ω)

((m · σ)1 ∂Vp

∂J2− (m · σ)2 ∂Vp

∂J1

)−

− Vp(p · ω)

((p · σ)1 ∂Vm

∂J2− (p · σ)2 ∂Vm

∂J1

)×(

Cα−1,β+1ab Cα+1,β−1

ab + Cα−1,β−1ab Cα+1,β+1

ab

). (B.27)

The coefficients Cα,βab are given by

Cαβab =

1

2π2

∫dθ cos ((αm+ βp) · θ)×

× cos(((am+ bp) · θ)− yab sinm · θ − zab sinp · θ), (B.28)

where

yab = ε ((am+ bp) · Tm(J)) (B.29)

zab = ε ((am+ bp) · Tp(J)) . (B.30)

Note that

cos((r · θ)− yab sinm · θ − zab sinp · θ)

= cos((r · θ)) cos(yab sinm · θ − zab sinp · θ) +

+ sin((r · θ)) sin(yab sinm · θ − zab sinp · θ)

= cos((r · θ)) cos(yab sinm · θ) cos(zab sinp · θ)−− cos((r · θ)) sin(yab sinm · θ) sin(zab sinp · θ) +

+ sin((r · θ)) sin(yab sinm · θ) cos(zab sinp · θ)−− sin((r · θ)) cos(yab sinm · θ) sin(zab sinp · θ). (B.31)

B.1. APPLICATION TO THE NORMAL FORM 101

In order to calculate the coefficients Aα,βab , we make use of the Fourier expansions

cos(z sin θ) = J0(z) + 2∞∑k=1

J2k(z) cos(2kθ), (B.32)

sin(z sin θ) = 2∞∑k=0

J2k+1(z) sin((2k + 1)θ) (B.33)

with J±ν(z) being the Bessel function. The Bessel function J±ν(z) is the solution of thefollowing ordinary differential equation

z2d2w

dz2+ z

dw

dz+ (z2 − ν2)w = 0. (B.34)

Using relations (B.31), it can be shown that

Cαβab = Ja+α(yab)Jb+β(zab) + Ja−α(yab)Jb−β(zab). (B.35)

B.1 Application to the normal form

We apply classical perturbation theory to the normal form

H(I1, I2, θ1, θ2) = ω1I1 + ω2I2 + αI21 + 2βI1I2 + γI2

2 +M cos θ1 + P cos θ2. (B.36)

Recall that ω2 = 1. We write the parameters of the normal form as (ω, α, β, γ,M, P ) withω being equal to ω1. The resonances are given bym = (1, 0), p = (0, 1). The Hamiltonianin the new variables reads

H(J ,φ) = H0(J) +ε2


+ ε2

∞∑a=0

∞∑b=−∞

Kab cos(aφ1 + bφ2). (B.37)

We choose the Hamiltonian

H0(J) = H0(J) +ε2

2(Smm(J) + Spp(J)), (B.38)

as a new integrable part. The function

ε2

∞∑a=0

∞∑b=−∞

Kab cos(aφ1 + bφ2) (B.39)

is considered to be perturbation to the integrable Hamiltonian system governed by H0(J).The Hessian σ is expressed by

1

2σ =

1

2

∂2H0

∂I2 =

(α ββ γ

). (B.40)


The terms Smm(J), Smp(J), Spm(J), Spp(J) read

Smm(J) =M

ω1(J)

((1, 0)

(α ββ γ

)(10

)M

ω1(J)

)=

αM 2

ω21(J)

, (B.41)

Spp(J) =P

−ω2(J)

((0, 1)

(α ββ γ

)(01

)P

ω2(J)

)=

γP 2

ω22(J)

, (B.42)

Smp(J) =M

ω1(J)

((1, 0)

(α ββ γ

)(01

)P

ω2(J)

)=

MPβ

ω1(J)ω2(J), (B.43)

Spm(J) =P

ω2(J)

((0, 1)

(α ββ γ

)(10

)M

ω1(J)

)=

MPβ

ω1(J)ω2(J), (B.44)

with

ω1(J) = ω1 + 2αI1 + 2βI2, (B.45)

ω2(J) = ω2 + 2βI1 + 2γI2. (B.46)

The Fourier coefficients Aαβab are given by

Aαβab = Cαβab +

εαM

ω21(J)

(Cα+1,βab + Cα−1,β

ab ) +εγP

ω22(J)

(Cα,β+1ab + Cα,β−1

ab )

+ε2MP (αγ − β2)

ω21(J)ω2

2(J)(Cα−1,β+1

ab + Cα+1,β−1ab + Cα+1,β+1

ab + Cα−1,β−1ab ), (B.47)

with

Cα,βab = Ja+α(yab)Jb+β(zab) + Ja−α(yab)Jb−β(zab). (B.48)

Using the relations

Tm(J) =1

ω1(J)

(0− M

ω1(J)(1, 0)

(2α 2β2β 2γ

))= − M

ω21(J)

(2α, 2β), (B.49)

Tp(J) =1

ω2(J)

(0− P

ω2(J)(0, 1)

(2α 2β2β 2γ

))= − P

ω22(J)

(2β, 2γ), (B.50)

we find the points yab and zab

yab = ε(a, b)−Mω2

1(J)

(αβ

)= − εM

ω21(J)

(aα− bβ), (B.51)

zab = ε(a,−b) −Pω2

2(J)

(−β−γ

)=

εP

ω22(J)

(aβ − bγ). (B.52)

Let W = ω1/ω2 be the winding ratio of the KAM torus under study. We assume that0 < W < 1. Consider the continued fraction expansion of W :

W = a0 +1

a1 + 1a2+ 1

a3+...

=1

a1 + 1a2+ 1

a3+...

. (B.53)

B.1. APPLICATION TO THE NORMAL FORM 103

The resonances (m1,m2) and (p1, p2) correspond to the first two rational approximants ofthe irrational number W . As a new pair of resonances we take (m′1,m

′2) = (a1a2 + 1,−a2)

and (p′1, p′2) = (a1a2a3 + a1 + a3,−a2a3 − 1) which correspond to the third and forth

rational approximants of W :

P3

Q3

=1

a1 + 1a2

,P4

Q4

=1

a1 + 1a2+ 1

a3

. (B.54)

The two-resonance perturbation is of the form

H1 = ε2(Km′1m

′2

cos(m′1φ1 +m′2φ2) +Kp′1p′2

cos(p′1φ1 + p′2φ2)). (B.55)

The coefficients Km′1m′2

and Kp′1p′2

can be found using Eq. (B.22).The Hamiltonian reads

H = ω1J1 + ω2J2 + αJ21 + 2βJ1J2 + γJ2

2 +ε2

2(Smm(J) + Spp(J))

+ε2(Km′1m

′2(J) cos(m′1φ1 +m′2φ2) +Kp′1p

′2(J) cos(p′1φ1 + p′2φ2)

). (B.56)

Using the procedure described in Appendix A, the normal form for the Hamiltonian givenby (B.56) can now be found. Note that the winding ratio of the KAM torus is ω1/ω2, theresonances are (m′1,m

′2) and (p′1, p

′2), the energy level is h = 0. Let the parameters of the

new normal form be (ω′, α′, β′, γ′,M ′, P ′). Recall that the parameters of the normal formfor the initial Hamiltonian are (ω, α, β, γ,M, P ). The map

(ω, α, β, γ,M, P )→ (ω′, α′, β′, γ′,M ′, P ′) (B.57)

defines the renormalization map.The realization of the renormalization map in the Maple computer algebra system is

presented in Appendix D.


Appendix C

Derivatives of Implicit Functions

The frequencies of an integrable Hamiltonian system are given by the derivatives of theenergy with respect to actions. Often it is difficult to find an explicite expression for theenergy as a function of actions.

The following situation can be encountered. The energy surface H0 = h is implicitlygiven by the relation

I2 = I2(I1, h) = I2(I1, h(I1, I2)). (C.1)

Our aim is to find the first and second derivatives of the energy h with respect toactions I1 and I2 using Eq. (C.1). Taking the derivatives of Eq. (C.1) with respect to I1

and I2, we obtain

∂I2

∂I1

= 0 =∂I2(I1, h)

∂I1

+∂I2(I1, h)

∂h

∂h(I1, I2)

∂I1

, (C.2)

∂I2

∂I2

= 1 =∂I2(I1, h)

∂h

∂h(I1, I2)

∂I2

. (C.3)

Note that on the left-hand side the quantity I2 is assumed to be an independent variable.On the right-hand side I2 is considered to be a function of I1 and h. It follows now that

∂h

∂I1

=∂h(I1, I2)

∂I1

= −∂I2(I1, h)/∂I1

∂I2(I1, h)/∂h, (C.4)

∂h

∂I2

=∂h(I1, I2)

∂I1

=1

∂I2(I1, h)/∂h. (C.5)

105

106 APPENDIX C. DERIVATIVES OF IMPLICIT FUNCTIONS

Taking the derivatives of Eq. (C.1) with respect to I1 and I2 one time more, we find

∂2I2

∂I21

= 0 =∂2I2(I1, h)

∂I21

+∂2I2(I1, h)

∂I1∂h

∂h

∂I1

+∂2I2(I1, h)

∂h2

∂h

∂I1

∂h

∂I1

+∂I2(I1, h)

∂h

∂2h

∂I21

, (C.6)

∂2I2

∂I1∂I2

= 0 =∂2I2(I1, h)

∂I1∂h

∂h

∂I2

+∂2I2(I1, h)

∂h2

∂h

∂I2

∂h

∂I1

+∂I2(I1, h)

∂h

∂2h

∂I1∂I2

, (C.7)

∂2I2

∂I22

= 0 =∂2I2(I1, h)

∂h2

∂h

∂I2

∂h

∂I2

+∂I2(I1, h)

∂h

∂2h

∂I22

. (C.8)

From these formulae it follows now that

∂2h

∂I21

=−1

∂I2/∂h

(∂2I2

∂I21

+ 2∂2I2

∂I1∂h

(−∂I2/∂I1

∂I2/∂h

)∂2I2

∂h2

(−∂I2/∂I1

∂I2/∂h

)2), (C.9)

∂2h

∂I1∂I2

=−1

∂I2/∂h

(∂2I2

∂I1∂h

1

∂I2/∂h

∂2I2

∂h2

(1

∂I2/∂h

)(−∂I2/∂I1

∂I2/∂h

)), (C.10)

∂2h

∂I22

=−1

∂I2/∂h

(∂2I2

∂h2

(1

∂I2/∂h

)2). (C.11)

We have omitted the arguments (I1, h) of the function I2(I1, h) for brevity.

Appendix D

Maple Program

The following program is a realization of the renormalization group (RG) approach to thestudy of the breakup of invariant tori in Hamiltonian systems with two degrees of free-dom. The program is written using the Maple1 computer algebra system and consists oftwo parts. The first part contains general procedures concerning the RG operator. Theprocedures RenormalizationOperator, scale, and numericalApproximation implement therenormalization map and can be used independently from the other procedures. The otherprocedures in this part simplify the analysis of the iterations of a given normal form underthe renormalization map. The second part is devoted to some applications.

> restart:with(linalg):with(plots):

D.1 Renormalization operator

D.1.1 Operator

The procedure computes the image under the renormalization map of the normal form givenby a list Parameters, see Appendix B for details. The input list consists of eight componentscorresponding to two coefficients of the linear part of the normal form, three coefficientsof the quadratic part, a value of the perturbation parameter, and amplitudes of the tworesonances. The parameters of the normal form are not assumed to be scaled.

1Maple and Maple V are registered trademarks of Waterloo Maple Inc.

107

108 APPENDIX D. MAPLE PROGRAM

> RenormalizationOperator:=proc(Parameters)> local omega,omega1,omega2,alpha,beta,gamma,epsilon,M,P,> x,lambda,H0,H0_I1,H0_I2,myParameters,w1,w2,m1,m2,p1,p2,> a1,a2,a3,H1,C,A,Km,Kp,K,Y,Z,Smm,Smp,Spm,Spp;> myParameters:=scale(Parameters);> omega1:=evalf(myParameters[1]);> omega2:=evalf(myParameters[2]);> alpha:=evalf(myParameters[3]);> beta:=evalf(myParameters[4]);> gamma:=evalf(myParameters[5]);> epsilon:=evalf(myParameters[6]);> M:=evalf(myParameters[7]);> P:=evalf(myParameters[8]);> omega:=omega1/omega2;> a1:=floor(1./omega);> a2:=floor(1./(1./omega-a1));> a3:=floor(1/(1./(1./omega-a1)-a2));> w1:=omega1+2*alpha*I1+2*beta*I2;> w2:=omega2+2*beta*I1+2*gamma*I2;> Smm:=(alpha)*M^2/w1^2;> Spp:=(gamma)*P^2/(-1*w2)^2;> Smp:=M*P*beta/(w1*w2);> Spm:=beta*M*P/(w1*w2);> H0:=unapply(eval(omega1*I1+omega2*I2+alpha*I1^2+2*beta*I1*I2+gamma*I2^> 2+> epsilon^2/2.*(Smm+Spp)),I1,I2);> Y:=unapply(eval(-2.0*epsilon*M*(a*alpha-b*beta)/(w1^2)),a,b);> Z:=unapply(eval(2*epsilon*P*(a*beta-b*gamma)/(w2^2)),a,b);> C:=unapply(> eval(BesselJ(a+i1,Y(a,b))*BesselJ(b+i2,Z(a,b))+> BesselJ(a-i1,Y(a,b))*BesselJ(b-i2,Z(a,b))),a,b,i1,i2);> A:=unapply(eval(C(a,b,i1,i2)+> (epsilon*alpha*M)*(C(a,b,i1+1,i2)+C(a,b,i1-1,i2))/(w1^2)+> (epsilon*gamma*P)*(C(a,b,i1,i2+1)+C(a,b,i1,i2-1))/(w2^2)+> (epsilon^2*M*P*(alpha*gamma-beta^2))*> (C(a,b,i1-1,i2+1)+C(a,b,i1+1,i2-1)+> C(a,b,i1-1,i2-1)+C(a,b,i1+1,i2+1))/(w1^2*w2^2)),a,b,i1,i2);> K:=unapply(eval((1/2)*(Smm*A(a,b,2,0)+Spp*A(a,b,0,2)+> (Smp+Spm)*(A(a,b,1,1)+A(a,b,1,-1)))),a,b);> p1:=a1*a2+1;p2:=-a2;m1:=a1*(a2*a3+1)+a3;m2:=-(a2*a3+1);> H1:=unapply(eval(> epsilon^2*(evalf(K(m1,m2))*cos(m1*theta1+m2*theta2)+> evalf(K(p1,p2))*cos(p1*theta1+p2*theta2))),I1,I2,theta1,theta2);> numericalApproximation(H0,1,H1,0,m1,m2,p1,p2,omega);> end:

D.1.2 Scaling

Scales the coefficients of the normal form given by a list Parameters=[ omega1, omega2,alpha, beta, gamma, epsilon, M, P ] so that alphaˆ2+betaˆ2+gammaˆ2 =1, 0<omega1<1,omega2 =1, epsilon=1, see Appendix A for details.

D.1. RENORMALIZATION OPERATOR 109

> scale:=proc(Parameters)> local omega1,omega2,alpha,beta,gamma,epsilon,M,P,> lambda,x;> omega1:=Parameters[1];> omega2:=Parameters[2];> alpha:=Parameters[3];> beta:=Parameters[4];> gamma:=Parameters[5];> epsilon:=Parameters[6];> M:=Parameters[7];> P:=Parameters[8];> if (abs(omega1)>abs(omega2)) then> x:=omega1;> omega1:=omega2;> omega2:=x;> x:=M;M:=P;P:=x;> x:=alpha;alpha:=gamma;gamma:=x;> fi:> lambda:=1/omega2;> omega1:=omega1*lambda;> omega2:=omega2*lambda;> alpha:=alpha*lambda;> beta:=beta*lambda;> gamma:=gamma*lambda;> epsilon:=epsilon; #No BUG> M:=M*lambda;> P:=P*lambda;> if (omega1<0.0) then> omega1:=-omega1;> beta:=-beta;> fi:> lambda:=sqrt(alpha^2+beta^2+gamma^2);> alpha:=alpha/lambda;> beta:=beta/lambda;> gamma:=gamma/lambda;> M:=M*lambda;> P:=P*lambda;> if (alpha<0.0) then alpha:=-alpha:beta:=-beta:gamma:=-gamma:fi:> if (M>0.0)then M:=-M: fi:> if (P>0.0)then P:=-P: fi:> [omega1,omega2,alpha,beta,gamma,epsilon,M,P];> end:

D.1.3 Numerical approximation

Given a near-integrable system H0+epsilon*H1 in action-angle variables, an energy levelenergy, a winding ratio omega, and a pair of resonances (m1,m2 ), (p1,p2 ), find the cor-responding normal form. Only for the use in the procedure Operator. In particular, theperturbation part is assumed to contain no term of the form sin(k1*theta1+k2*theta2). SeeAppendix A.


> numericalApproximation:=proc(H0,epsilon,H1,energy,m1,m2,p1,p2,omega)> local omega1,omega2,alpha,beta,gamma,M,P,> eqs,vars,grenze,res,sols,H0_I1,H0_I2,H0_I11,H0_I12,H0_I22,> omegaV,A,R;> H0_I1:=unapply(diff(H0(I1,I2),I1),I1,I2);> H0_I2:=unapply(diff(H0(I1,I2),I2),I1,I2);> H0_I11:=unapply(diff(H0_I1(I1,I2),I1),I1,I2);> H0_I12:=unapply(diff(H0_I1(I1,I2),I2),I1,I2);> H0_I22:=unapply(diff(H0_I2(I1,I2),I2),I1,I2);> vars:=I1,I2;> grenze:=I1=-1000..1000,I2=-1000..1000;> res:=[1,-omega];> eqs:=H0(I1,I2)=energy,res[1]*H0_I1(I1,I2)+res[2]*H0_I2(I1,I2)=0;> sols:=fsolve(eqs,vars,grenze);> omega1:=H0_I1(subs(sols,I1),subs(sols,I2));> omega2:=H0_I2(subs(sols,I1),subs(sols,I2));> alpha:=H0_I11(subs(sols,I1),subs(sols,I2))/2;> beta:=H0_I12(subs(sols,I1),subs(sols,I2))/2;> gamma:=H0_I22(subs(sols,I1),subs(sols,I2))/2;> res:=[m1,m2];> eqs:=H0(I1,I2)=energy,res[1]*H0_I1(I1,I2)+res[2]*H0_I2(I1,I2)=0;> sols:=fsolve(eqs,vars,grenze);> M:=evalf((1/(2.*Pi^2))*int(> int(cos(res[1]*theta1+res[2]*theta2)*> H1(subs(sols,I1),subs(sols,I2),theta1,theta2),> theta1=0..2.*Pi),theta2=0..2.*Pi));> res:=[p1,p2];> eqs:=H0(I1,I2)=energy,res[1]*H0_I1(I1,I2)+res[2]*H0_I2(I1,I2)=0;> sols:=fsolve(eqs,vars,grenze);> P:=evalf((1/(2.*Pi^2))*int(int(> cos(res[1]*theta1+res[2]*theta2)*> H1(subs(sols,I1),subs(sols,I2),theta1,theta2),> theta1=0..2.*Pi),theta2=0..2.*Pi));> omegaV:=vector([omega1,omega2]);> A:=matrix([[alpha,beta],[beta,gamma]]);> R:=matrix([[m1,m2],[p1,p2]]);> omegaV:=multiply(R,omegaV);> A:=multiply(R,multiply(A,transpose(R)));> omega1:=omegaV[1];> omega2:=omegaV[2];> alpha:=A[1,1];> beta:=A[1,2];> gamma:=A[2,2];> [omega1,omega2,alpha,beta,gamma,epsilon,M,P];> end:

D.1.4 Threshold

Procedures ratio and newRatio return 2 if the normal form defined by a list Parameters re-mains at least at the same distance from the origin after some number of iterations of therenormalization map. Otherwise the procedures return 0. The distance is defined in proceduredistance and is chosen to be Mˆ2+Pˆ2. The number of iterations is 3 for procedure ratio andis governed by global variable Num for procedure newRatio. Procedure threshold finds thethreshold to widespread chaos between resonances (m1,m2) and (p1,p2) on the energy surfaceH=energy of a near integrable system H0+epsilon*H1. The breakup of the KAM torus of wind-

D.1. RENORMALIZATION OPERATOR 111

ing ratio omega is studied. The procedure makes use of ratio. The initial value epsilon forthe perturbation parameter is assumed to be larger than the threshold. For some test value ofthe perturbation parameter, we find the corresponding normal form with the help of procedurenumericalApproximation and apply to it procedure ratio. Using the bisection method, we canfind the critical value. The accuracy of the bisection method is step.

> distance:=proc(Parameters)> Parameters[6]^2*(Parameters[7]^2+Parameters[8]^2):> end:

> ratio:=proc(Parameters)> local flag,myParameters,r_ini,r_fin,i,num,rSmall:> ## fork method: returns 0 for left, 2 for right> rSmall:=1e-50:> r_ini:=distance(scale(Parameters)):> num:=3:flag:=0:> i:=0:> myParameters:=Parameters:> while i<num do> i:=i+1:> myParameters:=scale(RenormalizationOperator(myParameters)):> r_fin:=distance(myParameters):> if (r_fin>r_ini) then i:=num: flag:=2: fi:> if ((abs(myParameters[7])<rSmall) or (abs(myParameters[8])<rSmall))> then i:=num:fi:> od:> #print(myParameters);> flag:> end:

> newRatio:=proc(Parameters)> local flag,myParameters,r_ini,r_fin,i,num:> ## fork method: returns 0 for left, 2 for right> r_ini:=distance(scale(Parameters)):> num:=_Num:flag:=0:> i:=0:> myParameters:=Parameters:> while i<num do> i:=i+1:> myParameters:=scale(RenormalizationOperator(myParameters)):> r_fin:=distance(myParameters):> #print(r_ini,r_fin);> if (r_fin>100*r_ini) then i:=num: flag:=2: fi:> od:> if (r_fin>r_ini) then flag:=2: fi:> flag:> end:


> threshold:=proc(H0,epsilon,H1,energy,m1,m2,p1,p2,omega,step)> local f,i,Parameters,left,right,point,flag;> Parameters:=scale(numericalApproximation(H0,epsilon,H1,energy,m1,m2,p1> ,p2,omega)):> f:=ratio(Parameters):> if f=0 then RETURN(epsilon):fi:> Parameters:=scale(numericalApproximation(H0,0.0,H1,energy,m1,m2,p1,p2,> omega)):> f:=ratio(Parameters):> if f=2 then RETURN(0.0):fi:> left:=0.0;> right:=epsilon;> while abs(right-left)>step do> point:=(right+left)/2.;> #print(point):> Parameters:=scale(numericalApproximation(H0,point,H1,energy,m1,m2,p1,p> 2,omega)):> if ratio(Parameters)=2 then right:=point:else left:=point: fi:> od:> #print(energy,point):> point:> end:

D.1.5 Output

Using procedure threshold, procedure bild finds the threshold to widespread chaos betweenresonances (m1,m2) and (p1,p2) of a near-integrable system H0+epsilon*H1 on energylevels E0+i*(E1-E0)/numberE, i=0..numberE. The KAM torus of winding ratio omega isstudied.

> epsScale:=1.0:

> bild:=proc(H0,H1,m1,m2,p1,p2,omega,E0,E1,numberE,stepEps)> local E,epsilon,low,high,energie,n,i,iniE,finE:> n:=numberE:> E:=array(1..n+1,1..2):> iniE:=E0:> finE:=E1:> for i from 1 to n+1 do> energie:=iniE+(i-1)*(finE-iniE)/n:> E[i,1]:=energie:> E[i,2]:=threshold(H0,energie*epsScale,H1,energie,m1,m2,p1,p2,omega,ste> pEps);> od:> E:> end:

D.2 Application to the paradigm Hamiltonian of Es-

cande and Doveil

Define the paradigm Hamiltonian. The case M=P is discussed. The perturbation parameteris the stochasticity parameter S squared.

D.3. APPLICATION TO A MODEL OF THE ETHANE MOLECULE 113

> H:=(I1,I2,theta1,theta2,epsilon)->H0(I1,I2,theta1,theta2)+epsilon*H1(> I1,I2,theta1,theta2):> H0:=(I1,I2)-> 1/2*I1^2+I2;> H1:=(I1,I2,theta1,theta2)->> -1/16*cos(theta1)-1/16*cos(k*(theta1-theta2));

H0 := (I1 , I2 )→ 12

I1 2 + I2

H1 := (I1 , I2 , θ1, θ2)→ − 116

cos(θ1)− 116

cos(k (θ1− θ2))

Consider the case k=1. The resonances are (1,-1) and (1,0). The winding ratio of thelast KAM torus is omega=g. The critical value can now be found with the help of procedurethreshold.

> k:=1:> g:=(sqrt(5.)-1)/2.;> m1:=1:m2:=0:p1:=k:p2:=-k:> m1:=k:m2:=-k:p1:=1:p2:=0:> omega:=((p2-m2)*g-p2)/((m1-p1)*g+p1);> #_Num:=2;> energy:=1.:epsilon:=4.:> Parameters:=(numericalApproximation(H0,1.,H1,1.,k,-k,1,0,omega));> #Parameters:=scale(numericalApproximation(H0,epsilon,H1,energy,k,-k,1,> 0,omega));> #Parameters:=scale(numericalApproximation(H0,epsilon,H1,energy,4,-4,1,> 0,omega));> #print(threshold(H0,epsilon,H1,energy,1,0,k,-k,omega,0.01));> print(k,threshold(H0,epsilon,H1,energy,k,-k,1,0,omega,0.01));> #print(threshold(H0,epsilon,H1,energy,1,0,1,-1,omega,0.01));

g := .6180339890ω := .6180339890

Parameters := [−.3819660110, .6180339890,12,

12,

12, 1., −.06250000002, −.06250000002]

1, .4765625000The result is Sˆ2=.4765625000> sqrt(.4765625000);

.6903350636

D.3 Application to a model of the ethane molecule

Define the Hamiltonian.> H0:=(I1,I2)->I1*sqrt(k)+I2*I2/2:> H1:=(I1,I2,theta1,theta2)->> (1+cos(3*theta2))/(1+sqrt(2*I1/sqrt(k))*cos(theta1)):> H:=(I1,I2,theta1,theta2,epsilon)->H0(I1,I2,theta1,theta2)+epsilon*H1(I> 1,I2,theta1,theta2):> g:=(sqrt(5.)-1)/2.:k:=3.:omega:=3./g:> energy:=1.:epsilon:=energy/10.:

The resonances are (0,-3) and (1,-3), the winding ratio of the last KAM torus is 3/g. Usingprocedure bild, we find the critical value of the perturbation parameter as a function of energyfor 1<energy<1.2. The accuracy of the bisection method is 0.0001

> epsScale:=0.1:n:=20:> E:=bild(H0,H1,0,-3,1,-3,omega,1.0,1.2,n,0.0001);


E := E

> points1:= [ seq([E[i,1],E[i,2]],i=1..n+1) ];

points1 := [[1.0, .01240234375], [1.010000000, .01213183594],[1.020000000, .01205273438], [1.030000000, .01181884765],[1.040000000, .01162890625], [1.050000000, .01143310547],[1.060000000, .01123144531], [1.070000000, .01102392578],[1.080000000, .01091601562], [1.090000000, .01069775390],[1.100000000, .01047363282], [1.110000000, .01035205078],[1.120000000, .01022656250], [1.130000000, .009986816407],[1.140000000, .009852539062], [1.150000000, .009602050780],[1.160000000, .009458984375], [1.170000000, .009312011718],[1.180000000, .009161132812], [1.190000000, .009006347656],[1.200000000, .008847656250]]

> s1:=pointplot(points1,style=point,symbol=box,color=red,> title=‘Ethane‘,titlefont=[TIMES,ROMAN,24],> labels=["E","D"],labelfont=[SYMBOL,24],> scaling=UNCONSTRAINED):> display(s1);

Ethane

0.009

0.0095

0.01

0.0105

0.011

0.0115

0.012

∆

1 1.05 1.1 1.15 1.2

Ε

D.4 Application to the Baggott Hamiltonian

The parameters for the normal form are read from a file. We are interested in energies between35000 and 40000.

> L:=readdata("parametersE35000t40000B1t1Imax25Jmax0N50.dat",8):

> print(L);

D.4. APPLICATION TO THE BAGGOTT HAMILTONIAN 115

[[−1346.69, 832.516, −266.139, −166.505, −219.358, 1., 27.131, 32.3432],[−1339.66, 828.152, −266.131, −166.519, −219.358, 1., 26.8562, 31.8232],[−1332.89, 823.601, −266.13, −166.501, −219.324, 1., 26.5694, 31.301],[−1325.51, 819.334, −266.151, −166.523, −219.365, 1., 26.2984, 30.7756],[−1318.19, 814.987, −266.165, −166.539, −219.392, 1., 26.0055, 30.2456],[−1311.78, 810.121, −266.151, −166.485, −219.301, 1., 25.7164, 29.716],[−1304.36, 805.737, −266.164, −166.505, −219.329, 1., 25.426, 29.1949],[−1296.34, 801.622, −266.194, −166.568, −219.43, 1., 25.1425, 28.6605],[−1288.92, 797.145, −266.197, −166.587, −219.447, 1., 24.8602, 28.1249],[−1282.19, 792.258, −266.191, −166.542, −219.374, 1., 24.5775, 27.588],[−1274.12, 788.028, −266.227, −166.596, −219.47, 1., 24.2882, 27.0519],[−1266.74, 783.383, −266.226, −166.602, −219.466, 1., 24.0096, 26.5115],[−1260.09, 778.305, −266.21, −166.542, −219.363, 1., 23.7111, 25.9708],[−1252.09, 773.889, −266.238, −166.583, −219.434, 1., 23.441, 25.434],[−1244.34, 769.282, −266.251, −166.6, −219.462, 1., 23.1371, 24.8915],[−1236.48, 764.685, −266.266, −166.627, −219.5, 1., 22.8668, 24.3557],[−1229.69, 759.471, −266.249, −166.562, −219.39, 1., 22.5834, 23.8202],[−1221.63, 754.867, −266.27, −166.596, −219.443, 1., 22.2905, 23.2668],[−1213.7, 750.141, −266.284, −166.617, −219.472, 1., 22.0038, 22.7259],[−1205.86, 745.31, −266.301, −166.623, −219.486, 1., 21.7157, 22.1852],[−1198.2, 740.323, −266.301, −166.615, −219.463, 1., 21.4301, 21.6407],[−1190.04, 735.539, −266.32, −166.641, −219.503, 1., 21.1485, 21.1022],[−1182.28, 730.486, −266.321, −166.632, −219.481, 1., 20.8551, 20.553],[−1174.1, 725.595, −266.337, −166.654, −219.511, 1., 20.5583, 20.0071],[−1165.8, 720.696, −266.35, −166.683, −219.547, 1., 20.2738, 19.471],[−1158., 715.476, −266.356, −166.658, −219.51, 1., 19.9855, 18.9292]]

> iMax:=25:jMax:=0:> crit:=array(0..iMax,1..2):> for i from 0 to iMax do crit[i,1]:=i:crit[i,2]:=0:od:

> _Num:=3:energyMin:=35000:energyMax:=40000:epsIni:=1.0:epsMin:=10.:eps> Max:=50.:epsStep:=.010:epsStart:=20.0:

A procedure similar to procedure bild is used in order to find the threshold to widespreadchaos as a function of energy.

> for i from 0 to iMax do> energy:=energyMin+i*(energyMax-energyMin)/iMax:> l0:=L[1+i]:> flag:=0:> epsilon:=epsMin:> l:=l0:l[7]:=l[7]*epsilon/epsIni;l[8]:=l[8]*epsilon/epsIni;> f:=newRatio(l);> if (f=2) then print("Chaos for the epsMin");flag:=1: fi:> #print(i,f,epsilon);


> epsilon:=epsMax:> l:=l0:l[7]:=l[7]*epsilon/epsIni;l[8]:=l[8]*epsilon/epsIni;> f:=newRatio(l);> if (f=0) then print("No chaos for the epsMax");flag:=1: fi:> #print(i,f,epsilon);> epsLeft:=epsMin:epsRight:=epsMax:> while (flag=0) do> epsilon:=(epsLeft+epsRight)/2.0:> l:=l0:l[7]:=l[7]*epsilon/epsIni;l[8]:=l[8]*epsilon/epsIni;> #print(l);> f:=newRatio(l);> #print(i,f,epsilon);> if (f=2) then epsRight:=epsilon;fi:> if (f=0) then epsLeft:=epsilon;fi:> if ((-epsLeft+epsRight)/2.0<epsStep) then flag:=1:fi:> od:> crit[i,1]:=energy:crit[i,2]:=epsilon:#print(crit[i,1],crit[i,2]);> od:

> points1:= [ seq([crit[i,1],crit[i,2]],i=0..iMax) ];

points1 := [[35000, 24.82421875], [35200, 24.78515625], [35400, 24.86328125],[35600, 25.17578125], [35800, 25.72265625], [36000, 25.64453125],[36200, 24.27734375], [36400, 25.99609375], [36600, 25.91796875],[36800, 30.33203125], [37000, 30.29296875], [37200, 30.52734375],[37400, 22.79296875], [37600, 26.30859375], [37800, 26.93359375],[38000, 26.66015625], [38200, 27.12890625], [38400, 27.51953125],[38600, 27.12890625], [38800, 27.28515625], [39000, 27.08984375],[39200, 27.67578125], [39400, 27.48046875], [39600, 28.06640625],[39800, 28.53515625], [40000, 28.14453125]]

> s1:=pointplot(points1,style=point,symbol=box,color=red,> title=‘Baggott Hamiltonian‘,titlefont=[TIMES,ROMAN,24],> labels=["E","b1"],labelfont=[SYMBOL,24],> scaling=UNCONSTRAINED):> display(s1);

Baggott Hamiltonian

24

26

28

30

β1

35000 36000 37000 38000 39000 40000

Ε

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