Chaos and Integrability - University of Cambridge · 2008-05-01 · Concepts of Theoretical Physics Soliton The ’Wave of Translation’ itself was regarded as a curiosity until

Chaos and Integrability

Natalia BerloffDepartment of Applied Mathematics and Theoretical Physics

University of Cambridge

Concepts of Theoretical Physics

Chaos in deterministic equations

The development of high-speed computers made apparent the extraordinaryrichness of chaotic systems. What is chaos?

Consider a dynamical system which is started twice, but from very slightlydifferent initial conditions:

• Non-chaotic system: error in prediction which grows linearly with time.

• Chaotic system: error grows exponentially with time, so the state of thesystem is essentially unknown after a few characteristic times.

Chaos 6= Randomness!

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Edward Lorenz – father of chaos

Lorenz, a meteorologist, figured out in the 1960s that small differences ina dynamic system such as the atmosphere could set off enormous changes.He noted that minute variables in initial conditions would result in widelydiffering patterns, known as the butterfly effect. In 1972 he presented astudy entitled ”Predictability: Does the Flap of a Butterfly’s Wings in

Brazil Set Off a Tornado in Texas?”

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Lorenz used the simplified equations that describe the interaction betweentemperature variations and convective motion

xt = −σx + σy,

yt = xz + rx − y,

zt = xy − bz,

where σ, r and b are constants.

Minimum conditions for chaotic dynamics(1) at least three dynamical variables;(2) a non-linear term coupling several variables together.

Features of chaotic behaviour:(1) trajectories of the system in three dimensions diverge;(2) motion is confined to a finite region of the phase space of the dynamicalvariables;(3) each trajectory is unique and does not intersect any other trajectory.

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Example: Damped driven pendulum

Consider a damped sinusoidally driven pendulum of length l with a bob ofmass m and damping constant γ.The driving angular frequency is ωD.From Newton’s second law

mlθtt + γlθt + mg sin θ = A cos ωDt.

Changing variables gives θtt + qθt + sin θ = α cos ωDt that can be rewrittenas

ωt = −qω − sin θ + α cos φ,

θt = ω,

φt = ωD,

where ω is the instantaneous angular velocity, θ is the angle of pendulumwith respect to its vertical equilibrium position, φ is the phase of the drivingforce.

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Depending on the parameters α and q the trajectory follows a limit cycle,a double loop, 2n loop or becomes chaotic (see class demonstrations).

How do we know from the phase diagram of the chaotic trajectory thatthe pendulum isn’t just going through a lot of swings instead of an infinitenumber?

A solution to this problem is to use a Poincare section.This is done by taking ”snapshots” of the phase space at time intervalsequal to tn = 2nπ/ωD, where n is an integer.

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Regular motion with a single period means that only one point is plotted.A period doubling adds another point, and each extra period doubling addstwo more points. It is when a chaotic state is reached that the Poincaresection looks really interesting. Instead of points, long ”wavy” lines are nowseen. If we were to magnify a section of these lines we would see that theywere actually composed of bunches of lines. If we magnified one of theselines we would see that this was also composed of another bunch of lines....

For chaotic motion, the structure of the Poincare sections is self-similar,that is, the same on different scales.

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Scaling laws and self-organised criticality

Some remarkable scaling laws are found in complex systems.

(i) Gutenberg-Richter law: the annual probability of occurence ofearthquakes with magnitude greater than m on the Richter scale is apower law in the energy S.

Analysis of 330,000 Californian eathquakes from 1984 to 2000 confirms thislaw.

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(ii) Mandelbrot “Fractal Geometry of Nature”:“How Long Is the Coast of Britain?” In this paperMandelbrot examines the surprising property that themeasured length of a stretch of coastline depends on thescale of measurement. Empirical evidence suggests thatthe smaller the increment of measurement, the longer themeasured length becomes. If you were to measure a stretchof coastline with a yardstick, you would get a shorter resultthan if you were to measure the same stretch with a smallruler. This is because you would be laying the ruler alonga more curvilinear route than that followed by the yardstick.

The measured length L(δ) of various country borders is afunction of the measurement scale δ

L(δ) ∼ δ1−D D − fractal dimension

D = 1.25 for the west coast of Britain, D = 1.15 for theland frontier of Germany, D = 1.52 for the coastline ofNorway.

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Other examples

(iii) The frequency with which words are used in the English languagefollows a power law;

(iv) scaling laws in the large-scale distribution of galaxies in the Universe;

(v) city population obeys a power-law distribution;

(vi) noise distribution of many processes is 1/f , f is the frequency;

(vii) formation of large-scale structures – self-organization, eg.Belousov-Zhabotinsky Reaction:

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Bacterial growth

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Cellular automaton

A clue in understanding these phenomena may be found in behaviour ofcellular automata.

A cellular automaton is a discrete model that consists of a regular grid ofcells, each in one of a finite number of states.The grid can be in any finite number of dimensions.Time is also discrete, and the state of a cell at time t is a function of thestates of a finite number of cells (called its neighborhood) at time t − 1.These neighbors are a selection of cells relative to the specified cell.Every cell has the same rule for updating, based on the values in thisneighbourhood.Each time the rules are applied to the whole grid a new generation iscreated.

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Example: The Game of Life

This game became widely known when it was mentioned in an articlepublished by Scientific American in 1970. It consists of a collectionof cells which, based on a few mathematical rules, can live, die ormultiply. Depending on the initial conditions, the cells form various patternsthroughout the course of the game.

The Rules:

• For a space that is ’populated’:

– Each cell with one or no neighbors dies, as if by loneliness.– Each cell with four or more neighbors dies, as if by overpopulation.– Each cell with two or three neighbors survives.

• For a space that is ’empty’ or ’unpopulated’

– Each cell with three neighbors becomes populated.

Life is one of the simplest examples of what is sometimes called ”emergentcomplexity” or ”self-organizing systems.”

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Example: Forest-fire model

Set of rules: A cell can be empty, occupied by a

tree, or burning.

1. A burning cell turns into an empty cell

2. A tree will burn if at least one neighbor is

burning

3. A tree ignites with probability f even if no

neighbor is burning

4. An empty space fills with a tree with

probability p

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Integrability

Certain nonlinear problems have a surprisingly simple underlying

structure, and can be solved by essentially linear methods.

These problems can have simple travelling wave solutions that interact witheach other elastically. The only effect of interactions is a phase shift.Because of the analogy with particles these are referred to as solitons.

In 1834, a young Scottish engineer John Scott Russell was observing a boat being drawn

along ’rapidly’ by a pair of horses. When the boat suddenly stopped Scott Russell noticed

that the bow wave continued forward ”at great velocity, assuming the form of a large

solitary elevation, a well-defined heap of water which continued its course along the

channel apparently without change of form or diminution of speed”.

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Soliton

The ’Wave of Translation’ itself was regarded as a curiosity until the 1960s when scientists

began to use modern digital computers to study nonlinear wave propagation. Then an

explosion of activity occurred when it was discovered that many phenomena in physics,

electronics and biology can be described by the mathematical and physical theory of the

’soliton’.

Korteweg-de Vries (KdV) equation

KdV equation governs moderately small in amplitude, shallow water waves:

K(u) = ut + 6uux + uxxx = 0.

This equation has a solitary wave solution u = 2k2sech2k(x− 4k2t−x0), where k and

x0 are constants.

The KdV equation has an infinite number of conservation laws.

One may obtain the general solution that evolves from arbitrary initial data.

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KdV and mKdV equations

Consider the modified KdV (mKdV) equation

M(v) = vt − 6v2vx + vxxx = 0.

If v is a solution of the mKdV equation, then u = −(v2 + vx) is a solution of the KdV

equation.

The Miura Transformation

x′= x +

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ǫ2t, t

′= t, u(x, t) = u

′(x

′, t

′) −

1

ǫ2

leaves the KdV equation invariant, whereas setting

v(x, t) = −ǫw(x′, t

′) +

1

ǫ

transforms the mKdV equation into

wt′ +∂

∂x′(3w

2+ 2ǫ

2w

3+ wx′x′) = 0.

ClearlyR ∞

−∞w dx′ is a conserved quantity.

The Miura transformation yields u′ = 2w + ǫwx′ − ǫ2w2.

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Thinking of ǫ ≪ 1, we may solve this equation recursevely for w as a function of u′

w = w0 + ǫw1 + ǫ2w2 + · · · =

u′

2−

ǫ

4u′

x′ +ǫ2

4

„

u′

x′x′

2+ u

′2

«

+ · · ·.

This allows us to obtain an infinite number of conserved quantities.

Inverse Scattering Transform

The transform u = −(v2 + vx) may be viewed as a Riccati equation for v in terms of u.

The transform v = Ψx/Ψ leads to a linear equation

Ψxx + uΨ = 0 or Ψxx + (λ + u)Ψ = 0.

This gives an implicit linearization of the KdV equation in the form of the time-independent

Schrodinger equation of quantum mechanics.

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Chaos and Integrability - University of Cambridge · 2008-05-01 · Concepts of Theoretical Physics Soliton The ’Wave of Translation’ itself was regarded as a curiosity until