Introduction to Applied Nonlinear Dynamical Systems and Chaos

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1 Stephen Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos Second Edition With 250 Figures 4jj Springer

2 I Series Preface v L I Preface to the Second Edition vii Introduction 1 1 Equilibrium Solutions, Stability, and Linearized Stability Equilibria of Vector Fields Stability of Trajectories 7 1.2a Linearization Maps a Definitions of Stability for Maps b Stability of Fixed Points of Linear Maps c Stability of Fixed Points of Maps via the Linear Approximation Some Terminology Associated with Fixed Points Application to the Unforced Duffing Oscillator Exercises 16 2 Liapunov Functions Exercises 25 3 Invariant Manifolds: Linear and Nonlinear Systems Stable, Unstable, and Center Subspaces of Linear, Autonomous Vector Fields a Invariance of the Stable, Unstable, and Center Subspaces b Some Examples Stable, Unstable, and Center Manifolds for Fixed Points of Nonlinear, Autonomous Vector Fields a Invariance of the Graph of a Function: Tangency of the Vector Field to the Graph Maps Some Examples Existence of Invariant Manifolds: The Main Methods of Proof, and How They Work 43

3 xii 3.5a Application of These Two Methods to a Concrete Example: Existence of the Unstable Manifold Time-Dependent Hyperbolic Trajectories and their Stable and Unstable Manifolds a Hyperbolic Trajectories b Stable and Unstable Manifolds of Hyperbolic Trajectories Invariant Manifolds in a Broader Context Exercises 62 4 Periodic Orbits Nonexistence of Periodic Orbits for Two-Dimensional, Autonomous Vector Fields Further Remarks on Periodic Orbits Exercises 76 5 Vector Fields Possessing an Integral Vector Fields on Two-Manifolds Having an Integral Two Degree-of-Freedom Hamiltonian Systems and Geometry a Dynamics on the Energy Surface b Dynamics on an Individual Torus Exercises 85 6 Index Theory Exercises 89 7 Some General Properties of Vector Fields: Existence, Uniqueness, Differentiability, and Flows Existence, Uniqueness, Differentiability with Respect to Initial Conditions Continuation of Solutions Differentiability with Respect to Parameters Autonomous Vector Fields Nonautonomous Vector Fields a The Skew-Product Flow Approach b The Cocycle Approach c Dynamics Generated by a Bi-Infinite Sequence of Maps Liouville's Theorem a Volume Preserving Vector Fields and the Poincare Recurrence Theorem Exercises Asymptotic Behavior The Asymptotic Behavior of Trajectories 104

4 xiii 8.2 Attracting Sets, Attractors, and Basins of Attraction The LaSalle Invariance Principle Attraction in Nonautonomous Systems Ill 8.5 Exercises The Poincare-Bendixson Theorem Exercises Poincare Maps Case 1: Poincare Map Near a Periodic Orbit Case 2: The Poincare Map of a Time-Periodic Ordinary Differential Equation a Periodically Forced Linear Oscillators Case 3: The Poincare Map Near a Homoclinic Orbit Case 4: Poincare Map Associated with a Two Degree-of-Freedom Hamiltonian System a The Study of Coupled Oscillators via Circle Maps Exercises Conjugacies of Maps, and Varying the Cross-Section Case 1: Poincare Map Near a Periodic Orbit: Variation of the Cross-Section Case 2: The Poincare Map of a Time-Periodic Ordinary Differential Equation: Variation of the Cross-Section Structural Stability, Genericity, and Transversality Definitions of Structural Stability and Genericity Transversality Exercises Lagrange's Equations Generalized Coordinates Derivation of Lagrange's Equations a The Kinetic Energy The Energy Integral Momentum Integrals Hamilton's Equations Cyclic Coordinates, Routh's Equations, and Reduction of the Number of Equations Variational Methods a The Principle of Least Action b The Action Principle in Phase Space c Transformations that Preserve the Form of Hamilton's Equations d Applications of Variational Methods The Hamilton-Jacobi Equation 187

5 xiv 13.8a Applications of the Hamilton-Jacobi Equation Exercises Hamiltonian Vector Fields Symplectic Forms a The Relationship Between Hamilton's Equations and the Symplectic Form Poisson Brackets a Hamilton's Equations in Poisson Bracket Form Symplectic or Canonical Transformations a Eigenvalues of Symplectic Matrices b Infinitesimally Symplectic Transformations c The Eigenvalues of Infinitesimally Symplectic Matrices d The Flow Generated by Hamiltonian Vector Fields is a One-Parameter Family of Symplectic Transformations Transformation of Hamilton's Equations Under Symplectic Transformations a Hamilton's Equations in Complex Coordinates Completely Integrable Hamiltonian Systems Dynamics of Completely Integrable Hamiltonian Systems in Action-Angle Coordinates a Resonance and Nonresonance b Diophantine Frequencies c Geometry of the Resonances Perturbations of Completely Integrable Hamiltonian Systems in Action-Angle Coordinates Stability of Elliptic Equilibria Discrete-Time Hamiltonian Dynamical Systems: Iteration of Symplectic Maps a The KAM Theorem and Nekhoroshev's Theorem for Symplectic Maps Generic Properties of Hamiltonian Dynamical Systems Exercises Gradient Vector Fields Exercises Reversible Dynamical Systems The Definition of Reversible Dynamical Systems Examples of Reversible Dynamical Systems Linearization of Reversible Dynamical Systems a Continuous Time b Discrete Time 238

6 xv 16.4 Additional Properties of Reversible Dynamical Systems Exercises Asymptotically Autonomous Vector Fields Exercises Center Manifolds Center Manifolds for Vector Fields Center Manifolds Depending on Parameters The Inclusion of Linearly Unstable Directions Center Manifolds for Maps Properties of Center Manifolds Final Remarks on Center Manifolds Exercises Normal Forms Normal Forms for Vector Fields a Preliminary Preparation of the Equations b Simplification of the Second Order Terms c Simplification of the Third Order Terms Id The Normal Form Theorem Normal Forms for Vector Fields with Parameters a Normal Form for The Poincare-Andronov-Hopf Bifurcation Normal Forms for Maps a Normal Form for the Naimark-Sacker Torus Bifurcation Exercises The Elphick-Tirapegui-Brachet-Coullet-Iooss Normal Form a An Inner Product on Hk b The Main Theorems c Symmetries of the Normal Form d Examples e The Normal Form of a Vector Field Depending on Parameters Exercises Lie Groups, Lie Group Actions, and Symmetries a Examples of Lie Groups b Examples of Lie Group Actions on Vector Spaces c Symmetric Dynamical Systems Exercises Normal Form Coefficients Hamiltonian Normal Forms 316

7 xvi 19.10a General Theory b Normal Forms Near Elliptic Fixed Points: The Semisimple Case c The Birkhoff and Gustavson Normal Forms d The Lyapunov Subcenter Theorem and Moser's Theorem e The KAM and Nekhoroshev Theorem's Near an Elliptic Equilibrium Point lOf ^Hamiltonian Normal Forms and Symmetries &*Final Remarks Exercises Conjugacies and Equivalences of Vector Fields a An Application: The Hartman-Grobman Theorem b An Application: Dynamics Near a Fixed Point-Sositaisvili's Theorem Final Remarks on Normal Forms Bifurcation of Fixed Points of Vector Fields A Zero Eigenvalue a Examples b What Is A "Bifurcation of a Fixed Point"? c The Saddle-Node Bifurcation Id The Transcritical Bifurcation le The Pitchfork Bifurcation f Exercises A Pure Imaginary Pair of Eigenvalues: The Poincare-Andronov-Hopf Bifurcation a Exercises Stability of Bifurcations Under Perturbations The Idea of the Codimension of a Bifurcation a The "Big Picture" for Bifurcation Theory b The Approach to Local Bifurcation Theory: Ideas and Results from Singularity Theory c The Codimension of a Local Bifurcation d Construction of Versal Deformations e Exercises Versal Deformations of Families of Matrices a Versal Deformations of Real Matrices b Exercises The Double-Zero Eigenvalue: the Takens-Bogdanov Bifurcation a Additional References and Applications for the Takens-Bogdanov Bifurcation b Exercises 446

8 xvii 20.7 A Zero and a Pure Imaginary Pair of Eigenvalues: the Hopf-Steady State Bifurcation a Additional References and Applications for the Hopf-Steady State Bifurcation b Exercises Versal Deformations of Linear Hamiltonian Systems a Williamson's Theorem b Versal Deformations of Jordan Blocks Corresponding to Repeated Eigenvalues c Versal Deformations of Quadratic Hamiltonians of Codimension < d Versal Deformations of Linear, Reversible Dynamical Systems e Exercises Elementary Hamiltonian Bifurcations a One Degree-of-Freedom Systems b Exercises c Bifurcations Near Resonant Elliptic Equilibrium Points d Exercises Bifurcations of Fixed Points of Maps An Eigenvalue of a The Saddle-Node Bifurcation b The Transcritical Bifurcation c The Pitchfork Bifurcation An Eigenvalue of 1: Period Doubling a Example b The Period-Doubling Bifurcation A Pair of Eigenvalues of Modulus 1: The Naimark-Sacker Bifurcation The Codimension of Local Bifurcations of Maps a One-Dimensional Maps b Two-Dimensional Maps Exercises Maps of the Circle a The Dynamics of a Special Class of Circle Maps-Arnold Tongues b Exercises On the Interpretation and Application of Bifurcation Diagrams: A Word of Caution 552

9 xviii 23 The Smale Horseshoe Definition of the Smale Horseshoe Map Construction of the Invariant Set Symbolic Dynamics The Dynamics on the Invariant Set Chaos Final Remarks and Observations Symbolic Dynamics The Structure of the Space of Symbol Sequences The Shift Map Exercises The Conley Moser Conditions, or "How to Prove That a Dynamical System is Chaotic" The Main Theorem Sector Bundles Exercises Dynamics Near Homoclinic Points of Two-Dimensional Maps Heteroclinic Cycles Exercises Orbits Homoclinic to Hyperbolic Fixed Points in Three-Dimensional Autonomous Vector Fields The Technique of Analysis Orbits Homoclinic to a Saddle-Point with Purely Real Eigenvalues a Two Orbits Homoclinic to a Fixed Point Having Real Eigenvalues b Observations and Additional References Orbits Homoclinic to a Saddle-Focus a The Bifurcation Analysis of Glendinning and Sparrow b Double-Pulse Homoclinic Orbits c Observations and General Remarks Exercises Melnikov's Method for Homoclinic Orbits in Two-Dimensional, Time-Periodic Vector Fields The General Theory Poincare Maps and the Geometry of the Melnikov Function Some Properties of the Melnikov Function 713

10 xix 28.4 Homoclinic Bifurcations Application to the Damped, Forced Duffing Oscillator Exercises Liapunov Exponents Liapunov Exponents of a Trajectory Examples Numerical Computation of Liapunov Exponents Exercises Chaos and Strange Attractors Exercises Hyperbolic Invariant Sets: A Chaotic Saddle Hyperbolicity of the Invariant Cantor Set A Constructed in Chapter a Stable and Unstable Manifolds of the Hyperbolic Invariant Set Hyperbolic Invariant Sets in W a Sector Bundles for Maps onm A Consequence of Hyperbolicity: The Shadowing Lemma a Applications of the Shadowing Lemma Exercises Long Period Sinks in Dissipative Systems and Elliptic Islands in Conservative Systems Homoclinic Bifurcations Newhouse Sinks in Dissipative Systems Islands of Stability in Conservative Systems Exercises Global Bifurcations Arising from Local Codimension Two Bifurcations The Double-Zero Eigenvalue A Zero and a Pure Imaginary Pair of Eigenvalues Exercises Glossary of Frequently Used Terms 793 Bibliography 809 Index 836

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