Nonlinear Physical Science

Series Editors

Albert C. J. Luo , Department of Mechanical and Industrial Engineering,Southern Illinois University Edwardsville, Edwardsville, IL, USA

Dimitri Volchenkov , Department of Mathematics and Statistics, Texas TechUniversity, Lubbock, TX, USA

Advisory Editors

Eugenio Aulisa , Department of Mathematics and Statistics, Texas TechUniversity, Lubbock, TX, USA

Jan Awrejcewicz , Department of Automation, Biomechanics and Mechatronics,Lodz University of Technology, Lodz, Poland

Eugene Benilov , Department of Mathematics, University of Limerick, Limerick,Limerick, Ireland

Maurice Courbage, CNRS UMR 7057, Universite Paris Diderot, Paris 7, Paris,France

Dmitry V. Kovalevsky , Climate Service Center Germany (GERICS),Helmholtz-Zentrum Geesthacht, Hamburg, Germany

Nikolay V. Kuznetsov , Faculty of Mathematics and Mechanics, Saint PetersburgState University, Saint Petersburg, Russia

Stefano Lenci , Department of Civil and Building Engineering and Architecture(DICEA), Polytechnic University of Marche, Ancona, Italy

Xavier Leoncini, Case 321, Centre de Physique Théorique, MARSEILLE CEDEX09, France

Edson Denis Leonel , Departamento de Física, São Paulo State University, RioClaro, São Paulo, Brazil

Marc Leonetti, Laboratoire Rhéologie et Procédés, Grenoble Cedex 9, Isère, France

Shijun Liao, School of Naval Architecture, Ocean and Civil Engineering, ShanghaiJiao Tong University, Shanghai, China

Josep J. Masdemont , Department of Mathematics, Universitat Politècnica deCatalunya, Barcelona, Spain

Dmitry E. Pelinovsky , Department of Mathematics and Statistics, McMasterUniversity, Hamilton, ON, Canada

Sergey V. Prants , Pacific Oceanological Inst. of the RAS, Laboratory ofNonlinear Dynamical System, Vladivostok, Russia

Laurent Raymond , Centre de Physique Théorique, Aix-Marseille University,Marseille, France

Victor I. Shrira, School of Computing and Maths, Keele University, Keele,Staffordshire, UK

C. Steve Suh , Department of Mechanical Engineering, Texas A&M University,College Station, TX, USA

Jian-Qiao Sun, School of Engineering, University of California, Merced, Merced,CA, USA

J. A. Tenreiro Machado , ISEP-Institute of Engineering, Polytechnic of Porto,Porto, Portugal

Simon Villain-Guillot , Laboratoire Ondes et Matière d’Aquitaine, Université deBordeaux, Talence, France

Michael Zaks , Institute of Physics, Humboldt University of Berlin, Berlin,Germany

Nonlinear Physical Science focuses on recent advances of fundamental theories andprinciples, analytical and symbolic approaches, as well as computational techniquesin nonlinear physical science and nonlinear mathematics with engineeringapplications.

Topics of interest in Nonlinear Physical Science include but are not limited to:

• New findings and discoveries in nonlinear physics and mathematics• Nonlinearity, complexity and mathematical structures in nonlinear physics• Nonlinear phenomena and observations in nature and engineering• Computational methods and theories in complex systems• Lie group analysis, new theories and principles in mathematical modeling• Stability, bifurcation, chaos and fractals in physical science and engineering• Discontinuity, synchronization and natural complexity in physical sciences• Nonlinear chemical and biological physics

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Edson Denis Leonel

Scaling Laws in DynamicalSystems

Edson Denis LeonelDepartamento de FísicaSão Paulo State UniversityRio Claro, São Paulo, Brazil

ISSN 1867-8440 ISSN 1867-8459 (electronic)Nonlinear Physical ScienceISBN 978-981-16-3543-4 ISBN 978-981-16-3544-1 (eBook)https://doi.org/10.1007/978-981-16-3544-1

Jointly published with Higher Education PressThe print edition is not for sale in China Mainland. Customers from China Mainland please order theprint book from: Higher Education Press.

© Higher Education Press 2021This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whetherthe whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publishers, the authors, and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publishers nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made. The publishers remain neutral with regard to jurisdictionalclaims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd.The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721,Singapore

To my son Gustavo

Preface

The main goal of this book is to present and discuss many of the common scalingproperties observed in some nonlinear dynamical systems described by mappings.The unpredictability of the time evolution of two nearby initial conditions in thephase space together with the exponential divergence from each other as time goesby lead to the concept of chaos. Some of the observables in nonlinear systems exhibitcharacteristics of scaling invariance being then described via scaling laws.

From the variation of control parameters, physical observables in the phase spacemay be characterized by using power laws that many times yield into universalbehavior. The application of such a formalism has been well accepted in the scientificcommunity of nonlinear dynamics. Therefore I had in mind when writing this bookwas to bring together few of the research results in nonlinear systems using scalingformalism that could be treated either in under-graduation as well as in the post-graduation in the several exact programs but no earlier requirements were neededfrom the students unless the basic physics and mathematics. At the same time, thebook must be original enough to contribute to the existing literature but with noexcessive superposition of the topics already dealt with in other textbooks. Themajority of the chapters present a list of exercises. Some of them are analytic andothers are numeric with few presenting some degree of computational complexity.

In Chap. 1 we discuss the fundamental concepts and the main definitions usedalong the book and that are also known in nonlinear dynamics theory.

Chapter 2 is dedicated to a discussion of discrete mapping, emerging from theidea of Poincaré surface of section. After introducing the concept of mapping, thefixed points and their stability are discussed and an application involving the logisticmap is made.

In Chap. 3 some dynamical and statistical properties for the logistic map arediscussed. The investigation is started from the convergence to the stationary state atand near the bifurcations. Using a set of scaling hypothesis and a homogeneous andgeneralized function an analytic expression involving the three critical exponents isobtained leading to a scaling law. A route to chaos is discussed via period doublingbifurcation where a ratio between the control parameters identifying the perioddoubling bifurcation lead to the Feigenbaum exponent. An algorithm to discuss theLyapunov exponent calculation is also presented.

vii

viii Preface

Chapter 4 is dedicated to a discussion of a generalized version of the logisticmap which is referred to as the logistic-like. Some dynamical properties for themapping are discussed including the fixed point determination, their stability, thetypes of bifurcation observed and also a careful discussion on the behavior of theconvergence to the stationary state and near the bifurcations. It is shown that thecritical exponents characterize a scaling law for the convergence to the fixed pointfor the transcritical or supercritical pitchfork are not universal and do indeed dependon the nonlinearity of the mapping. On the other hand the exponents measured inthe period doubling bifurcation are universal and independent on the nonlinearity ofthe mapping. Two different approaches are considered where one of them considersa phenomenological description with scaling hypotheses while the other takes intoconsideration a procedure that transforms the equation of differences into an ordinarydifferential equation whose solution gives analytically all the critical exponents.

A generalization to discuss two-dimensional mappings is made in Chap. 5starting with the linear mappings obtaining and classifying the fixed points. Thenthe nonlinear mappings are introduced as well the procedure used to classify thestability of the fixed points. Two examples of nonlinear mappings are given: (i) theHénon map and; (ii) the Ikeda map. A procedure to obtain the Lyapunov exponentsfor two-dimensional mappings is also presented.

Chapter 6 is dedicated to discuss the Fermi accelerator model. A historical back-ground is presented followed by a careful construction of the equations describing thedynamics, the properties of the phase space including fixed point determination andchaotic sea investigations leading to a scaling invariance for the chaotic diffusion.

In Chap. 7 some dynamical properties of the dissipative Fermi accelerator modelare discussed. Different types of dissipation are taken into consideration includinginelastic collisions leading to a fraction loss of energy upon collision with thewalls. Depending on the control parameters the stable and unstable manifolds of asaddle fixed point can cross each other leading to a destruction of a chaotic attractorproducing hence a boundary crisis. Other type of dissipation considered is a dragforce that consists of a particle crossing a media with a fluid reducing the energy ofthe particle along its trajectory. Three types of drag forces are considered, namely, (i)proportional to the velocity of the particle; (ii) proportional to the squared velocityof the particle; (iii) proportional to a power of the velocity which is not linear norquadratic. Then a stochastic perturbation to the boundary is considered leading to aninteresting scaling observation.

An alternative version of the Fermi accelerator model, often known as a bouncer isdiscussed in Chap. 8. The reinjection mechanism of a particle for a further collisionwith the wall is made by a constant gravitational field. An interesting property of thebouncer model is that depending on the combination of control parameters and initialconditions, unlimited energy growth can be observed leading to Fermi acceleration.

Chapter 9 discusses a procedure that uses a connection with the standardmappingto localize the position of the first invariant spanning curve above of the chaotic sea fora family of area preserving mapping which angles diverge in the limit of vanishinglyaction. The idea is to use a transition from local to global chaos present in the standard

Preface ix

map to obtain the position of the first invariant spanning curve and hence, describethe limit of the chaotic diffusion.

In Chap. 10 three different procedures to described the chaotic diffusion for afamily of area preserving mappings are described. The first of them considers aphenomenological description which is obtained from scaling hypotheses leadingto a homogeneous and generalized function and hence to a scaling law involvingthree critical exponents. The second one considers a transformation of the equationof differences of the mapping into an ordinary differential equation which is solvedanalytically allowing a determination of one of the critical exponents and also toan excellent agreement of the theory with the numerical results. The localization ofthe first invariant spanning curve plays a major rule in defining one of the criticalexponents of the scaling invariance. Finally a third one considers the solution of thediffusion equation giving the probability to observe a particle at a certain positionin the phase space at a specific time. From the knowledge of the probability, all theaverage observables are determined leading to the three critical exponents.

The discussions of the scaling properties for a dissipative standard mapping aremade in Chap. 11. We concentrate in the scaling invariance for chaotic orbits neara transition from unlimited to limited diffusion, which is explained via the analyt-ical solution of the diffusion equation. Indeed it gives the probability of observinga particle with a specific action at a given time. The momenta of the probability aredetermined and the behavior of the average squared action is obtained. The limits ofsmall and large time recover the results known in the literature from the phenomeno-logical approach while a scaling for intermediate time is obtained as dependent onthe initial action.

The elementary concepts of billiards are introduced in Chap. 12. In a billiard, aclassical particle or, in an equivalent way an ensemble of non-interacting particles,move inside a closed domain to where they collide with the boundary. The dynamicaldescription is made by the use of nonlinear mappings that define the position of theparticle at the boundary and the orientation of the trajectory after the collision. Threetypes of billiards are considered and the structure of the phase space depends on theshape of the boundary. One of them is the circle billiard. Another one is the ellipticaland finally a third one which has an oval shape. Both the circle and elliptical haveintegrable dynamics while the oval has mixed phase space leading to the observationof the chaos, invariant spanning curves and periodic islands.

Chapter 13 is dedicated to the discussion of some properties of time dependentbilliard that is a billiard which boundary moves in time. The nonlinear mappingdescribing the dynamics of the particle is constructed furnishing the dynamical vari-ables at each impact using that the velocity is obtained by themomentumconservationlaw. After the collision, the energy of the particle changes, consequently a new pairof variables must be included to the traditional ones describing the dynamics for thestatic boundary, namely, the velocity of the particle and the instant of the collision.The Loskutov-Ryabov-Akinshin (LRA) conjecture, which claims that the chaoticdynamics for a static billiard is a sufficient condition for Fermi acceleration whena time perturbation to the boundary is introduced, is discussed. The conjecture wastested for the oval billiard leading then to unlimited energy growth. In the elliptic

x Preface

billiard, which is integrable for the static case, an introduction of a time dependenceto the boundary leads the separtrix curve presented in the phase space to transforminto a stochastic layer and hence producing the needed condition to observe Fermiacceleration.

In Chap. 14 we introduce a drag force in the dynamics of the oval billiard.From the discussion made in Chap. 13 we saw from the LRA conjecture, the ovalbilliard exhibits unlimited energy growth when a time perturbation to the boundaryis introduced. The essence of Chap. 14 is to investigate the dynamics of the ovalbilliard under three different types of drag force, namely, (i) F ∝ −V ; (ii) F ∝ ±V 2

and; (iii) F ∝ −V δ with δ �= 1 and δ �= 2 and we show the presence of dissipationsuppresses the unlimited energy growth for the bouncing particles. This is a clearevidence the Fermi acceleration seems not to be a robust phenomena.

In Chap. 15 we discuss some thermodynamic properties for a set of particlesmoving inside a time dependent oval billiard. Two different approaches will beconsidered. One of them considers the heat flow transfer obtained from the solu-tion of the Fourier equation leading to an expression of the temperature. The otherone considers the time evolution for an ensemble of particles by using the billiardevolution. A connection with the equipartition theorem and the knowledge of theaverage squared velocity allows the determination of the temperature of the gas.

All of these notes were typed by myself since from the title until the last word ofthe references using LaTeX. As graphical editors I used xmgrace and gimp, in almostall figures.

Rio Claro, São Paulo, BrazilApril 2021

Edson Denis Leonel

Acknowledgements

Themainmotivation to write this book comes from a request of a group of students inboth under-graduation and graduation in Physics at Unesp—São Paulo State Univer-sity, at the city of Rio Claro, to course a discipline in nonlinear dynamics. The coursewas composed of part in nonlinear dynamics and part presenting some of the resultsinvolving scaling formalism long investigated in my research group. I offered thenthe course more than once and noticed there was space in the literature to construct astandard textbook joining the topics. At the same time, thewrittenmaterial should notoverlap the existing literature well settled in the community for a long while. Afterrunning the course few times and a good compilation of the material this monographemerged.

I acknowledgemy students for taking part on the course particularlyCéliaMayumiKuwana, Joelson Dayvison Veloso Hermes, Felipe Augusto Oliveira Silveira, AnneKétri Pasquinelli da Fonseca, Lucas Kenji Arima Miranda, Yoná Hirakawa Huggler,Raphael Moratta Vieira Rocha, Laura Helena Pozzo and Danilo Rando for activelyparticipation, careful reading and valuable suggestions on the text.

I am also very grateful to Professors Paulo Cesar Rech, Juliano Antonio deOliveira, Ricardo Luiz Viana and Antonio Marcos Batista for a critical reading onthe material.

I kindly acknowledge the Department of Physics of Unesp in Rio Claro forproviding the needed conditions for the construct and edition of the present material.

xi

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Initial Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 One-Dimensional Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 The Concept of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Asymptotically Stable Fixed Point . . . . . . . . . . . . . . . . . . 192.2.2 Neutral Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.3 Unstable Fixed Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Fixed Points to the Logistic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.1 Transcritical Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4.2 Period Doubling Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . 242.4.3 Tangent Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Some Dynamical Properties for the Logistic Map . . . . . . . . . . . . . . . . . 293.1 Convergence to the Stationary State . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Transcritical Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1.2 Period Doubling Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . 353.1.3 Route to Chaos via Period Doubling . . . . . . . . . . . . . . . . . 353.1.4 Tangent Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Lyapunov Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 The Logistic-Like Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 The Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Transcritical Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.1 Analytical Approach to Obtain α, β, z and δ . . . . . . . . . . 49

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xiv Contents

4.2.2 Critical Exponents for the Period DoublingBifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Extensions to Other Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.1 Hassell Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3.2 Maynard Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Introduction to Two Dimensional Mappings . . . . . . . . . . . . . . . . . . . . . 575.1 Linear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2 Nonlinear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3 Applications of Two Dimensional Mappings . . . . . . . . . . . . . . . . . 60

5.3.1 Hénon Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3.2 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3.3 Ikeda Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6 A Fermi Accelerator Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.1 Fermi-Ulam Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.1.1 Jacobian Matrix for the Indirect Collisions . . . . . . . . . . . 736.1.2 Jacobian Matrix for the Direct Collisions . . . . . . . . . . . . . 746.1.3 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.1.4 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.1.5 Phase Space Measure Preservation . . . . . . . . . . . . . . . . . . 75

6.2 A Simplified Version of the Fermi-Ulam Model . . . . . . . . . . . . . . . 776.3 Scaling Properties for the Chaotic Sea . . . . . . . . . . . . . . . . . . . . . . . 806.4 Localization of the First Invariant Spanning Curve . . . . . . . . . . . . 846.5 The Regime of Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7 Dissipation in the Fermi-Ulam Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 937.1 Dissipation via Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.1.1 Jacobian Matrix for the Direct Collisions . . . . . . . . . . . . . 947.1.2 Jacobian Matrix for the Indirect Collisions . . . . . . . . . . . 957.1.3 The Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967.1.4 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.1.5 Construction of the Manifolds . . . . . . . . . . . . . . . . . . . . . . 987.1.6 Transient and Manifold Crossings Determination . . . . . . 997.1.7 Determining the Exponent δ from the Eigenvalues

of the Saddle Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.2 Dissipation by Drag Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2.1 Drag Force of the Type F = −η̃v . . . . . . . . . . . . . . . . . . . 1047.2.2 Drag Force of the Type F = ±η̃v2 . . . . . . . . . . . . . . . . . . 1067.2.3 Drag Force of the Type F = −η̃vγ . . . . . . . . . . . . . . . . . . 109

Contents xv

7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8 Dynamical Properties for a Bouncer Model . . . . . . . . . . . . . . . . . . . . . . 1158.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1158.2 Complete Version of the Bouncer Model . . . . . . . . . . . . . . . . . . . . . 116

8.2.1 Successive Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.2.2 Indirect Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.2.3 Jacobian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198.2.4 The Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8.3 A Simplified Version of the Bouncer Model . . . . . . . . . . . . . . . . . . 1208.4 Numerical Investigation on the Simplified Version . . . . . . . . . . . . 1238.5 Approximation of Continuum Time . . . . . . . . . . . . . . . . . . . . . . . . . 1308.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

9 Localization of Invariant Spanning Curves . . . . . . . . . . . . . . . . . . . . . . 1359.1 The Standard Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359.2 Localization of the Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369.3 Rescale in the Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

10 Chaotic Diffusion in Non-Dissipative Mappings . . . . . . . . . . . . . . . . . . 14310.1 A Family of Discrete Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14310.2 Dynamical Properties for the Chaotic Sea:

A Phenomenological Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 14710.3 A Semi Phenomenological Approach . . . . . . . . . . . . . . . . . . . . . . . . 15210.4 Determination of the Probability via the Solution

of the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15510.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15810.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

11 Scaling on a Dissipative Standard Mapping . . . . . . . . . . . . . . . . . . . . . . 16311.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16311.2 A Solution for the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . 16511.3 Specific Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16611.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16911.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

12 Introduction to Billiard Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17112.1 The Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

12.1.1 The Circle Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17312.1.2 The Elliptical Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17412.1.3 The Oval Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

12.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17712.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

xvi Contents

13 Time Dependent Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18113.1 The Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

13.1.1 The LRA Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18413.2 The Time Dependent Elliptical Billiard . . . . . . . . . . . . . . . . . . . . . . 18513.3 The Oval Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18713.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18913.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

14 Suppression of Fermi Acceleration in the Oval Billiard . . . . . . . . . . . 19114.1 The Model and the Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19114.2 Results for the Case of F ∝ −V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19314.3 Results for the Case of F ∝ ±V 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 19514.4 Results for the Case of F ∝ −V δ . . . . . . . . . . . . . . . . . . . . . . . . . . . 19914.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20214.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

15 A Thermodynamic Model for Time Dependent Billiards . . . . . . . . . . 20515.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20515.2 Heat Transference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20715.3 The Billiard Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

15.3.1 Stationary Estate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21315.3.2 Dynamical Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21315.3.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21415.3.4 Average Velocity over n . . . . . . . . . . . . . . . . . . . . . . . . . . . 21615.3.5 Critical Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21715.3.6 Distribution of Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . 218

15.4 Connection Between the Two Formalism . . . . . . . . . . . . . . . . . . . . 21915.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22015.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Appendix A: Expressions for the Coefficients j . . . . . . . . . . . . . . . . . . . . . . . 223

Appendix B: Change of Referential Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Appendix C: Solution of the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . 231

Appendix D: Heat Flow Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Appendix E: Connection Between t and n in a Time DependentOval Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Appendix F: Solution of the Integral to Obtain the RelationBetween n and t in the Time Dependent Oval Billiard . . . . 239

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

List of Figures

Fig. 1.1 Illustration of a damped oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 3Fig. 1.2 Illustration of a pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Fig. 1.3 Figure illustrating a Poincaré section . . . . . . . . . . . . . . . . . . . . . . . 4Fig. 1.4 Plot of the orbit diagram for the logistic map . . . . . . . . . . . . . . . . 7Fig. 1.5 Plot of the orbit diagram obtained for the logistic map

considering a finite transient. The number of iterationsconsidered were: a n = 10; b n = 100, c n = 1000and d n = 10000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Fig. 1.6 Illustration of the Fermi–Ulam model. Here l correspondsto the distance of the fixed wall up to the origin of the system . . 10

Fig. 1.7 Plot of the phase space of the Fermi–Ulam model. Axes arerepresented by the velocity of the particle V and the phaseof the moving wall φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Fig. 1.8 Sketch of a billiard and its dynamical variables . . . . . . . . . . . . . . 12Fig. 1.9 a Plot of the phase space and b and c show typical orbits

of the circle billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Fig. 1.10 a Plot of the phase space; and illustration of the typical

orbits for the elliptical billiard considering: b rotationalorbits and c librational orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Fig. 1.11 Plot of the phase space for the oval billiard. The parametersused were p = 2 and: a ε = 0.05 and b ε = 0.1 . . . . . . . . . . . . . 14

Fig. 2.1 Pictorial illustration of a Poincaré surface of sectionand the sequence of points x0 → x1 → x2 → x3 · · ·that can be described by a discrete mapping . . . . . . . . . . . . . . . . . 18

Fig. 2.2 Illustration of the two types of monotonic convergenceto the fixed point x∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Fig. 2.3 Illustration of an alternating convergence to the fixedpoint x∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

xvii

xviii List of Figures

Fig. 2.4 Graphical analysis showing the convergence to the fixedpoint. In (a) a monotonic convergence usingxn+1 = f (xn) = 2xn(1 − xn) while in (b) an alternatingconvergence for xn+1 = f (xn) = 2.8xn(1 − xn) . . . . . . . . . . . . . 20

Fig. 2.5 Schematic illustration of the monotonic divergenceof the fixed point x∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Fig. 2.6 Schematic illustration of the alternating divergenceof the fixed point x∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Fig. 2.7 Plot of the orbit diagram for the logistic map obtainedfrom Eq. (2.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Fig. 2.8 Plot of the orbit diagram obtained for the logistic mapgiven by Eq. (2.7) emphasizing the period 3 windowcoming from a tangent bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . 25

Fig. 2.9 (a) Plot of xn+3 versus xn for three different controlparameters. (b) Amplification of the central region of (a)emphasizing the approximation of xn+3 to the equationxn+3 = xn with the control parameter given by R < Rc

before, at R = Rc and R > Rc after. (c) Channel formedby the function xn+3 and the equation xn+3 = xn and timeevolution of an orbit near the channel. . . . . . . . . . . . . . . . . . . . . . . 26

Fig. 3.1 (a) Plot of x versus n for R = 1 and different valuesof the initial condition x0, as shown in the figure.(b) Overlap of the curves shown in (a) onto a singleand universal plot after the following scalingtransformations x → x/xα

0 and n → n/xz0 with α = 1and z = −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Fig. 3.2 Plot of nx versus x0. A power law fitting gives z = −1 . . . . . . . . 32Fig. 3.3 Plot of x versus n for x0 = 0.1 and two different values

of the control parameter namely R = 0.99 and R = 0.999 . . . . . 32Fig. 3.4 Plot of τ versus μ considering tol = 10−10. A power law

fitting gives δ = −0.994(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Fig. 3.5 (a) Plot of x(n) − x∗ versus n

for different initial conditions, as shown inthe figure. A power law fit gives β = −0.49939(7).(b) Overlap of the curves shown in (a) onto a singleand universal curve after the following scalingtransformations (x(n) − x∗) → (x(n) − x∗)/(x0 − x∗)αand n → n/(x0 − x∗)z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Fig. 3.6 Plot of the cascade of bifurcations in the logistic mapshowing the period doubling sequence . . . . . . . . . . . . . . . . . . . . . 37

Fig. 3.7 (a) Plot of λ versus n considering R = 4 and x0 = 0.499for the logistic map. (b) Amplification of the box shownin (a) illustrating the fluctuations of the Lyapunov exponentfor small values of n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

List of Figures xix

Fig. 3.8 Example of a computational code written in Fortranto calculate the Lyapunov exponent applying a convergencecriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Fig. 3.9 Plot of λ versus R in the logistic map using the initialcondition as x0 = 0.499 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Fig. 4.1 (a) and (c) Plot of x versus n for R = 1, γ = 1and γ = 3/2 respectively and different values of x0,as shown in the figures. (b) and (d) show the overlapof the curves plotted in (a) and (c) into a single and henceuniversal curve. The scaling transformations used arex → x/xα

0 and n → n/xz0 with α = 1 and z = −1 for (b)and α = 1 and z = −3/2 for (d) . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Fig. 4.2 Plot of nx versus x0 for γ = 1 and γ = 2. Power lawfittings give z = −1.0002(3) and z = −2.001(2) . . . . . . . . . . . . 48

Fig. 4.3 Plot of τ versus μ for γ = 1 and γ = 3/2. A power lawfitting gives δ = −1 and is independent of γ . . . . . . . . . . . . . . . . 49

Fig. 4.4 Plot of the coefficient j6 versus γ evaluated

at x∗ = (1 − 1/R)1/γ and Rc = 2+γ

γ. . . . . . . . . . . . . . . . . . . . . . . 52

Fig. 5.1 Illustration of the chaotic attractor generatedfrom the evolution of the initial condition(x0, y0) = (0.1, 0.1) for the control parametersa = 1.4 and b = 0.3. The region in white correspondsto the basin of attraction of the chaotic attractorwhile the region in gray marks the initial conditionsthat diverge to x → −∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Fig. 5.2 Plot of the convergence of the positive Lyapunov exponentfor the Hénon map given by Eq. (5.14). We considered5 different initial conditions in the basin of attractionof the chaotic attractor. The average value for large enoughtime was λ = 0.4192(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Fig. 5.3 Plot of the chaotic attractor produced from the iterationof the Ikeda map using the initial condition(x0, y0) = (0.1, 0.1) for the control parameters p = 1,B = 0.9, k = 0.4 and α̃ = 6. The white regionidentifies the basin of attraction of the chaotic attractorshown in the figure while the gray region showsthe basin of attraction of the attracting fixed point whichthe coordinates are not shown in the scale of the figure . . . . . . . . 65

Fig. 6.1 Illustration of the Fermi-Ulam model. The motionof the moving wall is given by xw(t) = ε cos(ωt). Thefixed wall is placed at x = l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

xx List of Figures

Fig. 6.2 Plot of the phase space for the Fermi-Ulam modelobtained from the Mapping (6.9) for the control parameterε = 10−3. The position of the first invariant spanningcurve is shown. The stability islands and other invariantcurves are also shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Fig. 6.3 Illustration of the area evolution in the phase spacefrom the instant n to the instant (n + 1). One can noticesthat the area of the phase space in the instant (n + 1)is given by the area of the phase space in the instantn through the determinant of the Jacobian matrix, i.e.d An+1 = det Jnd An . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Fig. 6.4 Plot of the convergence of the Lyapunov exponent λ

versus n for the control parameter ε = 10−3, the same usedin Fig. 6.2 for the Fermi-Ulammodel given by theMapping6.9. The average value of the positive Lyapunov exponentfor sufficiently large time is λ = 0.728(1) considering5 different initial conditions along the chaotic sea,as mentioned in the figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Fig. 6.5 Plot of the phase space for the simplified Fermi-Ulammodel given by Mapping (6.37) for the control parameterε = 10−3. The position of the lowest velocity invariantspanning curve is illustrated by red dots and is identifiedas fisc. Periodic islands and other invariant curves arealso shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Fig. 6.6 (a) Plot of Vrms versus n considering the parametersε = 10−4, ε = 10−3 and ε = 10−2 for an initial velocityV0 = 10−3ε at each curve. (b) The same curves shown in(a) after a transformation n → nε2. The numerical fittinggives β = 0.4921(5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Fig. 6.7 Plot of Vsat versus ε. A power law fitting givesα = 0.516(5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Fig. 6.8 Overlap of the curves presented in Fig. 6.6aonto a single and universal plot after doing the followingtransformations: (i) Vrms → Vrms/ε

α and; (ii) n → n/εz . . . . . . 84Fig. 6.9 Plot of Vrms versus n for the control parameter

ε = 10−4 considering numerical simulation (symbols)and the analytical result given by Eq. (6.65) . . . . . . . . . . . . . . . . . 88

Fig. 6.10 Sketch of the Fermi-Ulam model with the wallmoving according to the equations(t) = R cos(wt) +

√L2 − R2 sin2(wt) . . . . . . . . . . . . . . . . . . . . 89

Fig. 6.11 Illustration of a periodically corrugated waveguideand the dynamical variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

List of Figures xxi

Fig. 7.1 Illustration of a chaotic attractor and an asymptoticallystable fixed point for the following combination of controlparameters: α = 0.93624, β = 1 and ε = 0.04. The curveshows the lower limit for the chaotic attractor. A saddlefixed point is also shown in the figure . . . . . . . . . . . . . . . . . . . . . . 97

Fig. 7.2 Plot of the stable (gray) and unstable (black) manifoldsoriginated from the same saddle point S. The controlparameters used were α = 0.93624, β = 1 and ε = 0.04 . . . . . . 99

Fig. 7.3 Plot of the basin of attraction for the chaotic attractor(black) and for the attracting fixed point (gray). Theboundary between the two is limited by the stablemanifoldsemanating from the saddle point, marked by a star. Theasymptotically fixed point is marked by a bullet. Oneof the two branches of the unstable manifold convergesto the attracting fixed point spiraling counterclockwisewhile the other evolves towards the chaotic attractor.The control parameters used are β = 1, α = 0.93624and ε = 0.04 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Fig. 7.4 Plot of V versus φ considering the control parametersβ = 1, ε = 0.04 and α = 0.9375. The black dots identifythe region of the phase space where the chaotic attractorexisted (transient) prior the crisis while the circles showthe convergence to the asymptotically stable fixed point.The doted line was added only as a guide to the eye . . . . . . . . . . 101

Fig. 7.5 Plot of the stable and unstable manifolds from the samesaddle point for the control parameters ε = 0.04,β = 1 and α = 0.9375. Black shows the unstablebranch departing from the saddle point convergingtowards the attracting fixed point. The dots identifythe other branch passing in the region of the phase spacewhere the chaotic attractor existed prior the crisis. Thestable manifolds are also visible. The box shows the severalcrossings between the manifolds confirming the boundarycrisis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Fig. 7.6 Plot of τ versusμ. A power law fitting gives δ = −2.01(2).We considered an ensemble of 5 × 103 differentinitial conditions in the region of the phase spacewhere the chaotic attractor existed prior the crisis. Thecontrol parameters used were β = 1 and ε = 0.04 while α

was varied around αc = 0.93624 . . . . . . . . . . . . . . . . . . . . . . . . . . 103

xxii List of Figures

Fig. 7.7 Plot of V versus n for the control parameter ε = 10−2

and η = 10−3. A linear fitting furnishes a slopeof −0.0002 = −2η, in well agreement with the analyticalapproximation. The inset corresponds to an amplificationof the regime of the decay, showing the behaviorof the decay in a smaller scale of time, illustratingthe oscillations at small window of time . . . . . . . . . . . . . . . . . . . . 106

Fig. 7.8 Plot of: (a) V versus n for the parameter ε = 10−2

and drag coefficient η = 10−3. An exponential fittinggives a slope −0.002 = −2η, in well agreementwith the analytical description. (b) Plot of the phase spacefor the non-dissipative model overlapped for the timeevolution of the dissipative case showing the approximationto the asymptotically stable fixed point identified as starat V f

∼= 0.321 . . .. The inset plot of (a) shows the timeevolution of V versus n near the region of the fixed point . . . . . . 108

Fig. 7.9 Plot of V versus n for the control parametersε = 10−2 and η = 10−2. A polynomial fitting givesV (n) = V0 + αn + βn2 where α = −0.001257(1)and β = 9.998 × 10−8 with V0 = 9.902 ∼= 10 . . . . . . . . . . . . . . . 112

Fig. 8.1 Sketch of a bouncer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Fig. 8.2 Plot of the phase space for Mapping (8.7) considering

γ = 1 and the following control parameters:(a) ε = 0.1; (b) ε = 0.2; (c) ε = 0.3 and; (d) ε = 0.4 . . . . . . . . . 120

Fig. 8.3 Plot of the phase space for the Mapping (8.11)considering γ = 1 and the following control parameters:(a) ε = 0.1; (b) ε = 0.2; (c) ε = 0.3 and; (d) ε = 0.4 . . . . . . . . . 121

Fig. 8.4 Plot of V versus n for the control parameters γ = 1and ε = 10 considering: (a) a simplified versionand; (b) a complete version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Fig. 8.5 Plot of V versus n for ε = 10 and γ = 0.999 . . . . . . . . . . . . . . . . 124Fig. 8.6 (a) Plot of Vrms versus n. (b) Same of (a)

after the transformation n → nε2, hence a plot of Vrms

versus nε2. The control parameters are shown in the figure . . . . . 125Fig. 8.7 (a) Plot of V sat versus (1 − γ ) and (b) V sat versus ε. The

numerical values for the exponents are α1 = 0.998(8)and α2 = −0.4987(8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Fig. 8.8 Plot of nx versus (1 − γ ) for a fixed value of ε. A powerlaw fit gives z2 = −0.998(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Fig. 8.9 (a) Plot of different curves of average velocity as a functionof n. (b) Overlap of all curves shown in (a) onto a singleand universal plot after the scaling transformations givenby Eqs. (8.33) and (8.34) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Fig. 8.10 Plot of Vrms versus n considering γ = 0.999. Thetheoretical result is given by Eq. (8.52) . . . . . . . . . . . . . . . . . . . . . 132

List of Figures xxiii

Fig. 9.1 Plot of the phase space for Mapping (9.1) consideringthe control parameters: (a) K = 0.5; (b) K = 0.75; (c)K = 0.97 and; (d) K = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Fig. 9.2 Plot of the phase space for Mapping (9.2) consideringF(I ) = 1

I γ for the control parameters ε = 0.01and: (a) γ = 1 and (b) γ = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Fig. 9.3 Plot of I ∗ versus ε. Continuous lines correspondto the theoretical result given by Eq. (9.6) while symbolstogether with their uncertainty represented by the errorbars denote the numerical simulation. Circles correspondto the parameter γ = 1 while squares are obtained for γ = 2 . . . 139

Fig. 9.4 Plot of the phase space shown in Fig. 9.2after the transformation I → I

I ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 140Fig. 10.1 Plot of the phase space for theMapping (10.11) considering

the control parameters ε = 0.01 and γ = 1. The symbolsidentify the elliptic fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Fig. 10.2 Plot of Irms as a function of: (a) n, and (b) nε2. The controlparameters used were γ = 1 considering ε = 10−4,ε = 5 × 10−4 and ε = 10−3, as shown in the figure . . . . . . . . . . . 148

Fig. 10.3 Plot of Irms,sat versus ε for: (a) γ = 1 and (b) γ = 2. Thecritical exponents obtained are: (a) α = 0.508(4) and (b)α = 0.343(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Fig. 10.4 Plot of nx versus ε for: (a) γ = 1 and (b) γ = 2. Thecritical exponent obtained was: (a) z = −0.98(2) and (b)z = −1.30(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Fig. 10.5 (a) Plot of Irms versus n for γ = 1 and different values of ε

as shown in the Figure. (b) Overlap of the curves shownin (a) onto a single and hence universal plot after the scalingtransformations Irms → Irms/ε

α and n → n/εz . . . . . . . . . . . . . . 151Fig. 10.6 Plot of Irms(n) versus n for different control parameters.

The symbols denote the numerical simulationswhile the continuous curves correspond to the Equation(10.55) with the same control parameters as usedin the numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Fig. 10.7 Sketch of the potential V (x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Fig. 11.1 (a) Plot of the phase space for a dissipative standard

mapping considering the parameters ε = 100and γ = 10−3. (b) Normalized probability distributionfor the chaotic attractor shown in (a) . . . . . . . . . . . . . . . . . . . . . . . 164

xxiv List of Figures

Fig. 11.2 (a) Plot of Irms versus n for different control parametersand initial conditions, as labeled in the figure. Symbolsare for numerical simulation, while continuous curvesare analytical. (b) Overlap of the curves shown in (a)onto a single and universal plot after the appropriatescaling transformations. Inset of (b) shows an exponentialdecay to the attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Fig. 12.1 Illustration of the angles describing the billiard. Thetrajectory of the particle is drawn by the line segmentsand change after the impacts with the boundary . . . . . . . . . . . . . . 172

Fig. 12.2 (a) Plot of the phase space for the circle billiard. (b) and (c)Illustrate a trajectory in the billiard with different lengthof time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Fig. 12.3 (a) Illustration of the phase space for the elliptical billiard.(b) Example of a rotating orbit and (c) a librating orbit . . . . . . . . 175

Fig. 12.4 Plot of the phase space for the oval billiard consideringthe control parameters: (a) ε = 0.05 and (b) ε = 0.1; (c)ε = 0.2 and (d) ε = 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Fig. 12.5 Plot of the periodic orbits in the oval billiard: (a) period 2and; (b) period 4. The control parameters used were p = 2and: (a) ε = 0.05; (b) ε = 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Fig. 12.6 Plot of the positive Lyapunov exponent for the chaoticregions shown in Fig. 12.4(c), (d). The control parametersused were p = 2 and: (a) ε = 0.2 and (b) ε = 0.3 . . . . . . . . . . . . 177

Fig. 12.7 Illustration of the stadium billiard with parabolicboundaries and the unfolding mechanism . . . . . . . . . . . . . . . . . . . 179

Fig. 13.1 Plot of four collisions of a particle with a time dependentboundary. The position of the boundary is drawnat the instant of the impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Fig. 13.2 Plot of the phase space for the elliptical billiardtogether with a sketch of the stochastic layer producedby the destruction of the separatrix curve. The controlparameters used were: (a) static case e = 0.4, q = 1; (b)time dependent boundary e = 0.4, a = 0.01 with V0 = 1with 104 collisions of the particle with the boundary . . . . . . . . . . 185

Fig. 13.3 Plot of the average velocity: (a) average over the orbitand considering an ensemble of different initial conditions,and; (b) average over the orbit. The control parametersused were a = 0.1 and: (a) e = 0.1, e = 0.2, e = 0.3,e = 0.4 and e = 0.5; (b) e = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Fig. 13.4 Plot of the average velocity V versus n for the controlparameters: ε = 0.08, p = 3 and η = 0.5. The initialvelocities are shown in the figure . . . . . . . . . . . . . . . . . . . . . . . . . . 187

List of Figures xxv

Fig. 13.5 Plot of the curves shown in Fig. 13.4 onto a singleand universal curve after the following scalingtransformations: V → V /V0

α and n → n/V z0 . The control

parameters used were: ε = 0.08, p = 3 and η = 0.5. Theinitial velocities are shown in the figure . . . . . . . . . . . . . . . . . . . . 188

Fig. 14.1 (a) Plot of V versus n. The control parameters consideredwere ε = 0.1, η = 0.1, p = 3 and η̃ = 10−3 starting thedynamics with the initial velocity V0 = 10. (b) Linearfitting for the decay of the average velocity as a function of η̃ . . 194

Fig. 14.2 (a) Plot of the average velocity V versus n consideringthe initial velocity V0 = 10. The control parameters usedwere ε = 0.1, η = 0.1, p = 3 and η̃ = 10−3. (b) A linearfit for the decay of the velocity as a function of η̃ . . . . . . . . . . . . . 196

Fig. 14.3 (a) Plot of the average velocity for large values of nas a function of the control parameter η̃. The controlparameters used were ε = 0.1, η = 0.1 and p = 3. (b)Plot of nc versus η̃ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Fig. 14.4 (a) Plot of the average velocity V versus n for threedifferent control parameters η̃, as shown in the figure. Theinitial velocity was V0 = 10−2 and the control parametersconsidered ε = 0.1, η = 0.1 and p = 3. (b) Plot of V sat

versus η̃. A power law fitting gives α = −0.5005(4). (c)Plot of nx versus η̃ with a fitting giving z = −1.027(1) . . . . . . . 198

Fig. 14.5 Same plot of Fig. 14.4(a) with the rescaled axis showingan universal curve. The control parameters used areε = 0.1, η = 0.1 and p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Fig. 14.6 (a) Plot of r versus t for different values of the exponentδ, as shown in the figure. The initial velocity usedwas V0 = 10−3. (b) Same plot of (a) but with initialvelocity V0 = 10−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Fig. 14.7 Plot of f versus δ. For the parameter δ > 1.48and considering 105 collisions with the boundary, noneof the particles have their energy completely dissipated.The control parameter used were p = 3, ε = 0.1, η = 0.1and the drag coefficient used was η̃ = 10−3 . . . . . . . . . . . . . . . . . 201

Fig. 14.8 Decay of the velocity for the particle considering δ = 1.5.The control parameters used were p = 3, ε = 0.1,η̃ = 0.1 and η = 10−3. The best fit gives a decaydescribed by a second degree polynomial function givenby V (n) = 10.02(1) − 0.00485(1)n + 5.871(1) × 10−7n2 . . . . . 202

Fig. 15.1 Sketch of a set of particles moving in a billiardwith time dependent boundary. The highlighted areacorresponds to the collision zone and defines the domainto where the particles can collide with the boundary . . . . . . . . . . 206

xxvi List of Figures

Fig. 15.2 Sketch of the region where heat transference may beobserved. The arrows identify the direction of the heat fluxwhen the temperature of the gas is T < Tb . . . . . . . . . . . . . . . . . . 208

Fig. 15.3 Illustration of 4 collisions of the particle with the boundaryof the billiard. Each color corresponds to a given collision.The boundary position is ploted at the instant of the collision . . . 211

Fig. 15.4 (a) Plot of < V > versus n for different valuesof γ and two different combinations of ηε. (b)Overlap of the curves shown in (a) onto a singleand universal plot after the application of the followingscaling transformations: n → n/[(1 − γ )z1(ηε)z2 ]and < V >→< V > /[(1 − γ )α1(ηε)α2 ]. The continuouslines give the theoretical results obtained from Eq. (15.42) . . . . . 215

Fig. 15.5 Plot of: (a) < V sat > and (b) nx as a function of (1 − γ ).The inner plots show the behavior of < V sat > and nx

for different values of εη . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216Fig. 15.6 Plot of the probability distribution for an ensemble

of 105 particles in a dissipative and stochastic versionof the oval billiard. Blue was obtained for 10 collisionswith the boundary while red was obtained for 100collisions. The inner figure was obtained for 50, 000collisions. The initial velocity considered was V0 = 0.2and the control parameters used were εη = 0.02and γ = 0.999 for p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

Fig. A.1 Plot of the coefficients j4 (left) and j6 (right),both as function of γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Fig. A.2 Plot of the coefficients j7 (left) and j8 (right),both as a function of γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Fig. B.1 Position of a particle measured by two referential frameas inertial (left) and non-inertial (right) . . . . . . . . . . . . . . . . . . . . . 228

List of Tables

Table 3.1 Table showing the order of the bifurcation, the periodof the orbit, the numerical values of the parametersand an estimation of the exponent δ f . . . . . . . . . . . . . . . . . . . . . . . 37

Table 3.2 Table showing the critical exponents α, β, z and δ

for the three bifurcations of fixed points observedin the logistic map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

xxvii