Dynamical Systems and Chaos Henk Broer and Floris Takens January 28, 2008 i Preface The discipline of Dynamical Systems provides the mathematical language describ- ing the time dependence of deterministic systems. Since the past four decades there is a ongoing theoretical development. This book starts from the phenomological point of view, reviewing examples, some of which have been guiding in the de- velopment of the theory. So we discuss oscillatiors, like the pendulum in all its variations concerning damping and periodic forcing and the Van der Pol system, the H´enon and Logistic families, the Newton algorithm seen as a dynamical system and the Lorenz and the R¨ossler systems. The Doubling map on the circle and the Thom map (a.k.a. the Arnold cat map) on the 2-dimensional torus are useful toy models to illustrate theoretical ideas like symbolic dynamics. In an appendix the 1963 Lorenz model is derived from appropriate partial differential equations. The phenomena concern equilibrium, periodic, quasi-periodic and chaotic dynam- ics as these occur in all kinds of modelling and are met in computer simulations and in experiments. The application area varies from celestial mechanics and eco- nomical evolutions to population dynamics and climate variability. One general, motivating question is how one should give intelligent interpretations to data that come from computer simulations or experiments. For this thorough theoretical in- vestigations are needed. One key idea is that the dynamical systems used for modelling should be ‘robust’, which means that relevant qualitative properties should be persistent under small perturbations. Here we can distinguish between variations of the initial state or of system parameters. In the latter case one may think of the coefficients in the equations of motion. One of the strongest forms of persistence, called structural stability will be discussed, but many weaker forms often are more relevant. Instead of individual evolutions one also has to consider invariant manifolds, like stable and unstable manifolds of equilibria and periodic orbits and invariant tori as these often mark the geometrical organisation of the state space. Here a notion like (normal) hyperbolicity come into play. In fact, looking more globally at the structure of dynamical systems, we consider attractors and basin boundaries, mainly based on the examples of the Solenoid and the Horseshoe map. The central concept of the theory is chaos, to be defined in terms of unpredictabil- ity. The prediction principle we use is coined l’histoire se rep´ ete` , i.e., by matching with past observations. The unpredictability then will be expressed in a dispersion exponent. Structural stability turns out to be useful in proving that the chaotic- ity of the Doubling and Thom maps is persistent under small perturbations. The ideas on predictability will also be used in the development of Reconstruction The- ory, where important dynamical invariants like the box-counting and correlation dimensions, which often are fractal, of an attractor and related forms of entropy ii are re-constructed from time series based on a segment of the past evolution. Also numerical estimation methods of the invariants will be discussed; these are impor- tant for applications, e.g., in controlling chemical processes and in early warning systems regarding epilepsia. Another central subject is formed by multi- and quasi-periodic dynamics. Here the dispersion exponent vanishes and quasi-periodicity to some extent forms the ‘order in between the chaos’. We deal with circle dynamics near rigid rotations and similar settings for area preserving and holomorphic maps. Persistence of quasi-periodicity is part of Kolmogorov-Arnold-Moser (or KAM) Theory. In the main text we focus on the so-called 1 bite small divisor problem that can directly be solved by Fourier series. In an appendix we show how this linear problem is used to prove a KAM Theorem by Newton iterations. In another appendix we discuss transitions from ordery to more complicated dynam- ics upon variation of system parameters, in particular Hopf, Hopf-Ne˘ımark-Sacker and quasi-periodic bifurcations, indicating how this yields a unified theory for the onset of turbulence in fluid dynamics according to Hopf-Landau-Lifschitz-Ruelle- Takens. Also we indicate a few open problems regarding both conservative and dissipative chaos. This book is written as a textbook for either undergraduate or graduate students, depending on their level. In any case a good knowledge of ordinary differential equations is required. Also some knowledge of metric spaces, topology and mea- sure theory will help and an appendix on these subjects has been included. The book also is directed towards the applied researcher who likes to get a better un- derstanding of the data that computer simulation or experiment may provide him with. We maintain a wide scope, but given the wealth of material on this subject, we cannot possibly aim for completeness. However, the book does aim to be a guide to the dynamical systems literature. Below we present a few general references at textbook, handbook or encyclopædia level. A brief historical note. Occasionally in the text some historical note is presented and here we give a birdseye perspective. One historical line starts around 1900 with Poincar´e, originating from celestial mechanics, in particular from his studies of the unwieldy 3-body probem [39], which later turned out to have chaotic evolutions. Poincar´e, among other things, introduced geometry in the theory of ordinary dif- ferential equations; instead of studying only one evolution, trying to compute it in one way or the other, he was the one that proposed to consider the structure and organisation of all possible evolutions of a dynamical system. This qualitative line was picked up later by the Fields medallists Thom and Smale and by a lot of others in the 1960’s and 70’s and this development leaves many traces in the present book. iii Around this same time a really significant input to ‘chaos’ theory came from outside mathematics. We mention Lorenz’s meteorological contribution with the celebrated Lorenz attractor [129, 130], the work of May on population dynamics [133] and the input of H´enon-Heiles from astronomy, to which in the 1976 the famous H´enon at- tractor was added [108]. It should be said that during this time the computer became an important tool for lengthy computations, simulations and for graphical represen- tations, which had a tremendous effect on the field. Many of these developments also will be dealt with below. From the early 1970’s on these two lines merged, leading to the discipline of non- linear dynamical systems as known now, which is an exciting area of research and which has many application both in the classical sciences, in the life sciences, in meteorology, etc. Guide to the literature. As said before, we could not aim for completeness, but we do aim, among other things, to be an entrance to the literature. We have ordered the bibliography at the end, subdividing it as follows. - The references [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] form general contributions to the theory of nonlinear dynamical systems at a textbook level; - More in the direction of bifurcation theory we like to point at the textbooks [12, 13, 14] and to [15, 16]; - For textbooks on the ergodic theory of nonlinear dynamics see [17, 18]; - A few handbooks on dynamical systems are [19, 20, 21, 22], where in the latter reference we like to point at the chapters [23, 24, 25, 26, 27]; - We also mention the Russian Encyclopædia of Mathematics as published by Springer, of which we refer explicitly to [28]. Acknowledgements. Henk Broer and Floris Takens Groningen, Spring 2008 iv Contents 1 Examples and definitions of dynamical phenomena 1 1.1 Thependulumasadynamicalsystem . 2 1.1.1 Thefreependulum ...................... 2 1.1.1.1 Thefree,undampedpendulum . 2 1.1.1.2 Thefree,dampedpendulum. 7 1.1.2 Theforcedpendulum. 9 1.2 Generaldefinitionofdynamicalsystems . 14 1.2.1 Differentialequations. 16 1.2.2 Constructionsofdynamicalsystems . 19 1.2.2.1 Restriction . 19 1.2.2.2 Discretisation . 20 1.2.2.3 Suspension. .. .. .. .. .. .. 21 1.2.2.4 Poincar´emap . 23 1.3 Examples ............................... 25 1.3.1 AHopfbifurcationintheVanderPolequation . 25 1.3.1.1 TheVanderPolequation . 25 1.3.1.2 Hopfbifurcation . 26 1.3.2 TheH´enonmap;saddlepointsandseparatrices . 28 1.3.2.1 Saddlepointsandseparatrices. 31 1.3.2.2 Numericalcomputationofseparatrices . 31 1.3.3 TheLogisticsystem;bifurcationdiagrams . 32 1.3.4 TheNewtonalgorithm . 38 1.3.4.1 R as a circle: stereographic projection . 40 1.3.4.2 ApplicabilityoftheNewtonalgorithm∪ {∞} . 41 1.3.4.3 Non-convergingNewtonalgorithm . 41 1.3.4.4 Newtonalgorithmeinhigherdimensions . 43 1.3.5 Dynamical systems defined by partial differential equations 45 1.3.5.1 The1-dimensionalwaveequation . 45 1.3.5.2 Solutionof the 1-dimensionalwave equation . 46 1.3.5.3 The1-dimensionalheatequation . 46 1.3.6 TheLorenzattractor . 49 vi CONTENTS 1.3.6.1 Discussion. .. .. .. .. .. .. 49 1.3.6.2 Numericalsimulations. 50 1.3.7 TheR¨osslerattractor;Poincar´emap . 51 1.3.7.1 TheR¨osslersystem . 52 1.3.7.2 TheattractorofthePoincar´emap . 53 1.3.8 TheDoublingmapandsymbolicdynamics . 54 1.3.8.1 TheDoublingmapontheinterval . 54 1.3.8.2 TheDoublingmaponthecircle . 55 1.3.8.3 TheDoublingmapinsymbolicdynamics . 55 1.3.8.4 Analysisof the Doublingmap insymbolicform . 57 1.4 Exercises ............................... 60 2 Qualitative properties and predictability of evolutions 67 2.1 Stationaryandperiodicevolutions . 68 2.1.1 Predictability of periodic and stationary motions . .... 68 2.1.2 Asymptotically and eventually periodic evolutions . .... 70 2.2 Multi-andquasi-periodicevolutions . .. 71 2.2.1 The n-dimensionaltorus . 73 2.2.2 Translationsonatorus . 74 2.2.2.1 Translation systems on the 1-dimensional torus . 75 2.2.2.2 Translation systems on the 2-dimensional torus with time set R ..................