目录
 前辅文
 1 Initial Discussion
  1.1 Objectives
  1.2 Initial Concepts
  1.3 Summary
 2 The Concept of Attractor
  2.1 Objectives
  2.2 Initial Concepts
  2.3 The Damped Oscillator
   2.3.1 Overdamping
   2.3.2 Critical Damping
   2.3.3 Underdamped Case
  2.4 Van der Pol Oscillator
  2.5 Chaotic Attractor
   2.5.1 The Lorenz System
   2.5.2 Duffing Equation
  2.6 Strange Nonchaotic Attractor
  2.7 Concept of Attractor
  2.8 Summary
  2.9 Proposed Exercises
 3 Stability of Fixed Points
  3.1 Objectives
  3.2 First-Order Linear Differential Equation
  3.3 Linear Systems
  3.4 Nonlinear Systems
   3.4.1 Example 1
   3.4.2 Example 2
   3.4.3 Example 3
  3.5 Summary
  3.6 Proposed Exercises
 4 Some Local Bifurcations
  4.1 Objectives
  4.2 Local Bifurcations
  4.3 Saddle-Node Bifurcation
   4.3.1 Example of Saddle-Node Bifurcation
  4.4 Transcritical Bifurcation
   4.4.1 Example of Transcritical Bifurcation
  4.5 Supercritical Pitchfork Bifurcation
   4.5.1 Example of Supercritical Pitchfork Bifurcation
  4.6 Subcritical Pitchfork Bifurcation
  4.7 Normal Forms
  4.8 Summary
  4.9 Proposed Exercises
 5 Scaling Analysis in Local Bifurcations
  5.1 Objectives
  5.2 Convergence to Fixed Points
  5.3 Convergence to the Fixed Point: A Phenomenological Description
   5.3.1 Scaling Hypotheses
  5.4 Scaling Analysis in the Saddle-Node Bifurcation
  5.5 Scaling Analysis in the Transcritical Bifurcation
  5.6 Scaling Analysis in the Supercritical Pitchfork Bifurcation
  5.7 Summary
  5.8 Proposed Exercises
 6 One-Dimensional Discrete Maps
  6.1 Objectives
  6.2 Introduction
  6.3 The Concept of Stability
   6.3.1 Asymptotically Stable Fixed Point
   6.3.2 Neutral Stability
   6.3.3 Unstable Fixed Point
  6.4 Applications of Fixed-Point Calculation in the Logistic Map
  6.5 Bifurcations
   6.5.1 Transcritical Bifurcation
   6.5.2 Period-Doubling Bifurcation
   6.5.3 Tangent Bifurcation
  6.6 Summary
  6.7 Proposed Exercises
 7 Some Dynamical and Statistical Properties of the Logistic Map
  7.1 Objectives
  7.2 Convergence to the Steady State
   7.2.1 Transcritical Bifurcation
   7.2.2 Period-Doubling Bifurcation
   7.2.3 Route to Chaos via Period Doubling
   7.2.4 Tangent Bifurcation
  7.3 Lyapunov Exponents
  7.4 Summary
  7.5 Proposed Exercises
 8 The Logistic-Like Map
  8.1 Objectives
  8.2 The Mapping Equation
  8.3 Transcritical Bifurcation
   8.3.1 Analytical Determination of the Exponents α, β, z, and δ
   8.3.2 Critical Exponents in the Period-Doubling Bifurcation
  8.4 Extension of Results to Other Maps
   8.4.1 Hassell Map
   8.4.2 Maynard Map
  8.5 Summary
  8.6 Proposed Exercises
 9 Introduction to Two-Dimensional Discrete Maps
  9.1 Objectives
  9.2 Linear Maps
  9.3 Nonlinear Maps
  9.4 Applications of Two-Dimensional Maps
   9.4.1 Hénon Map
   9.4.2 Lyapunov Exponents
   9.4.3 Ikeda Map
  9.5 Summary
  9.6 Proposed Exercises
 10 The Fermi Accelerator Model: Non-dissipative Version
  10.1 Objectives
  10.2 The Fermi-Ulam Model
   10.2.1 Jacobian Matrix for Indirect Collisions
   10.2.2 Jacobian Matrix for Multiple Collisions
   10.2.3 Fixed Points
   10.2.4 Phase Space
   10.2.5 Measure Preservation in Phase Space
  10.3 Simplified Version of the Fermi-Ulam Model
  10.4 Scaling Properties of the Chaotic Sea
  10.5 Location of the First Invariant Spanning Curve
  10.6 The Growth Regime
  10.7 Summary
  10.8 Proposed Exercises
 11 Dissipation in the Fermi Accelerator Model
  11.1 Objectives
  11.2 Dissipation via Inelastic Collisions
   11.2.1 Jacobian Matrix for Multiple Collisions
   11.2.2 Jacobian Matrix for Indirect Collisions
   11.2.3 Phase Space
   11.2.4 Fixed Points
   11.2.5 Construction of the Manifolds
   11.2.6 Determination of the Manifold Crossing and the Transient
   11.2.7 Determining the Exponent δ from the Eigenvalues of the Saddle Point
  11.3 Dissipation via Viscous Drag
   11.3.1 Drag Force F =−˜ηv
   11.3.2 Drag Force F =−˜ηv2
   11.3.3 Drag Force F =−˜ηvγ
  11.4 Summary
  11.5 Proposed Exercises
 12 Dynamical Properties of the Bouncer Model
  12.1 Objectives
  12.2 The Model
  12.3 Full Version of the Bouncer Model
   12.3.1 Successive Collisions
   12.3.2 Indirect Collisions
   12.3.3 Jacobian Matrix
   12.3.4 Phase Space
  12.4 Simplified Version of the Bouncer Model
  12.5 Numerical Investigation in the Simplified Version
  12.6 Continuous-Time Approximation
  12.7 Summary
  12.8 Proposed Exercises
 13 Localization of Invariant Curves
  13.1 Objectives
  13.2 The Standard Map
  13.3 Localization of the Curves
  13.4 Rescaling in Phase Space
  13.5 Summary
  13.6 Proposed Exercises
 14 Chaotic Diffusion in Non-dissipative Maps
  14.1 Objectives
  14.2 A Family of Discrete Maps
  14.3 Properties of the Chaotic Sea: A Phenomenological Description
  14.4 A Semi-phenomenological Approach
  14.5 Obtaining the Probability via the Diffusion Equation
  14.6 Summary
  14.7 Proposed Exercises
 15 Introduction to Billiard Dynamics
  15.1 Objectives
  15.2 The Billiard
   15.2.1 Circular Billiard
   15.2.2 Elliptical Billiard
   15.2.3 Ovoid Billiard
  15.3 Summary
  15.4 Proposed Exercises
 16 Time-Dependent Billiards
  16.1 Objectives
  16.2 The Billiard
   16.2.1 LRA Conjecture
  16.3 Time-Dependent Elliptical Billiard
  16.4 Oval Billiard
  16.5 Summary
  16.6 Proposed Exercises
 17 Suppression of Fermi Acceleration in the Oval Billiard
  17.1 Objectives
  17.2 The Model and the Mapping
  17.3 Results for the Case Fα−V
  17.4 Results for the Case Fα−V2
  17.5 Results for the Case Fα−Vδ
  17.6 Summary
  17.7 Proposed Exercises
 18 A Thermodynamic Model for Time-Dependent Billiards
  18.1 Objectives
  18.2 Motivation
  18.3 Heat Transfer
  18.4 Billiard Formalism
   18.4.1 Steady State
   18.4.2 Dynamical Regime
   18.4.3 Numerical Simulations
   18.4.4 Velocity Average Over n
   18.4.5 Critical Exponents
   18.4.6 Velocity Distribution
  18.5 Connection Between the Two Formalisms
  18.6 Summary
  18.7 Proposed Exercises
 Appendix A: Euler Relations
 Appendix B: Numerical Integration Methods
 Appendix C: Expressions for the Coefficients j in the Dynamical Approach
 Appendix D: Change of Reference Frame
 Appendix E: Solution of the Diffusion Equation
 Appendix F: Heat Flux Equation
 Appendix G: Connection Between t and n in the Time-Dependent Ovoid Billiard
 Appendix H: Solution of the Integral for the Relation Between n and t in the Time-Dependent Ovoid Billiard
 References
 Index
 前辅文
 1 Initial Discussion
  1.1 Objectives
  1.2 Initial Concepts
  1.3 Summary
 2 The Concept of Attractor
  2.1 Objectives
  2.2 Initial Concepts
  2.3 The Damped Oscillator
   2.3.1 Overdamping
   2.3.2 Critical Damping
   2.3.3 Underdamped Case
  2.4 Van der Pol Oscillator
  2.5 Chaotic Attractor
   2.5.1 The Lorenz System
   2.5.2 Duffing Equation
  2.6 Strange Nonchaotic Attractor
  2.7 Concept of Attractor
  2.8 Summary
  2.9 Proposed Exercises
 3 Stability of Fixed Points
  3.1 Objectives
  3.2 First-Order Linear Differential Equation
  3.3 Linear Systems
  3.4 Nonlinear Systems
   3.4.1 Example 1
   3.4.2 Example 2
   3.4.3 Example 3
  3.5 Summary
  3.6 Proposed Exercises
 4 Some Local Bifurcations
  4.1 Objectives
  4.2 Local Bifurcations
  4.3 Saddle-Node Bifurcation
   4.3.1 Example of Saddle-Node Bifurcation
  4.4 Transcritical Bifurcation
   4.4.1 Example of Transcritical Bifurcation
  4.5 Supercritical Pitchfork Bifurcation
   4.5.1 Example of Supercritical Pitchfork Bifurcation
  4.6 Subcritical Pitchfork Bifurcation
  4.7 Normal Forms
  4.8 Summary
  4.9 Proposed Exercises
 5 Scaling Analysis in Local Bifurcations
  5.1 Objectives
  5.2 Convergence to Fixed Points
  5.3 Convergence to the Fixed Point: A Phenomenological Description
   5.3.1 Scaling Hypotheses
  5.4 Scaling Analysis in the Saddle-Node Bifurcation
  5.5 Scaling Analysis in the Transcritical Bifurcation
  5.6 Scaling Analysis in the Supercritical Pitchfork Bifurcation
  5.7 Summary
  5.8 Proposed Exercises
 6 One-Dimensional Discrete Maps
  6.1 Objectives
  6.2 Introduction
  6.3 The Concept of Stability
   6.3.1 Asymptotically Stable Fixed Point
   6.3.2 Neutral Stability
   6.3.3 Unstable Fixed Point
  6.4 Applications of Fixed-Point Calculation in the Logistic Map
  6.5 Bifurcations
   6.5.1 Transcritical Bifurcation
   6.5.2 Period-Doubling Bifurcation
   6.5.3 Tangent Bifurcation
  6.6 Summary
  6.7 Proposed Exercises
 7 Some Dynamical and Statistical Properties of the Logistic Map
  7.1 Objectives
  7.2 Convergence to the Steady State
   7.2.1 Transcritical Bifurcation
   7.2.2 Period-Doubling Bifurcation
   7.2.3 Route to Chaos via Period Doubling
   7.2.4 Tangent Bifurcation
  7.3 Lyapunov Exponents
  7.4 Summary
  7.5 Proposed Exercises
 8 The Logistic-Like Map
  8.1 Objectives
  8.2 The Mapping Equation
  8.3 Transcritical Bifurcation
   8.3.1 Analytical Determination of the Exponents α, β, z, and δ
   8.3.2 Critical Exponents in the Period-Doubling Bifurcation
  8.4 Extension of Results to Other Maps
   8.4.1 Hassell Map
   8.4.2 Maynard Map
  8.5 Summary
  8.6 Proposed Exercises
 9 Introduction to Two-Dimensional Discrete Maps
  9.1 Objectives
  9.2 Linear Maps
  9.3 Nonlinear Maps
  9.4 Applications of Two-Dimensional Maps
   9.4.1 Hénon Map
   9.4.2 Lyapunov Exponents
   9.4.3 Ikeda Map
  9.5 Summary
  9.6 Proposed Exercises
 10 The Fermi Accelerator Model: Non-dissipative Version
  10.1 Objectives
  10.2 The Fermi-Ulam Model
   10.2.1 Jacobian Matrix for Indirect Collisions
   10.2.2 Jacobian Matrix for Multiple Collisions
   10.2.3 Fixed Points
   10.2.4 Phase Space
   10.2.5 Measure Preservation in Phase Space
  10.3 Simplified Version of the Fermi-Ulam Model
  10.4 Scaling Properties of the Chaotic Sea
  10.5 Location of the First Invariant Spanning Curve
  10.6 The Growth Regime
  10.7 Summary
  10.8 Proposed Exercises
 11 Dissipation in the Fermi Accelerator Model
  11.1 Objectives
  11.2 Dissipation via Inelastic Collisions
   11.2.1 Jacobian Matrix for Multiple Collisions
   11.2.2 Jacobian Matrix for Indirect Collisions
   11.2.3 Phase Space
   11.2.4 Fixed Points
   11.2.5 Construction of the Manifolds
   11.2.6 Determination of the Manifold Crossing and the Transient
   11.2.7 Determining the Exponent δ from the Eigenvalues of the Saddle Point
  11.3 Dissipation via Viscous Drag
   11.3.1 Drag Force F =−˜ηv
   11.3.2 Drag Force F =−˜ηv2
   11.3.3 Drag Force F =−˜ηvγ
  11.4 Summary
  11.5 Proposed Exercises
 12 Dynamical Properties of the Bouncer Model
  12.1 Objectives
  12.2 The Model
  12.3 Full Version of the Bouncer Model
   12.3.1 Successive Collisions
   12.3.2 Indirect Collisions
   12.3.3 Jacobian Matrix
   12.3.4 Phase Space
  12.4 Simplified Version of the Bouncer Model
  12.5 Numerical Investigation in the Simplified Version
  12.6 Continuous-Time Approximation
  12.7 Summary
  12.8 Proposed Exercises
 13 Localization of Invariant Curves
  13.1 Objectives
  13.2 The Standard Map
  13.3 Localization of the Curves
  13.4 Rescaling in Phase Space
  13.5 Summary
  13.6 Proposed Exercises
 14 Chaotic Diffusion in Non-dissipative Maps
  14.1 Objectives
  14.2 A Family of Discrete Maps
  14.3 Properties of the Chaotic Sea: A Phenomenological Description
  14.4 A Semi-phenomenological Approach
  14.5 Obtaining the Probability via the Diffusion Equation
  14.6 Summary
  14.7 Proposed Exercises
 15 Introduction to Billiard Dynamics
  15.1 Objectives
  15.2 The Billiard
   15.2.1 Circular Billiard
   15.2.2 Elliptical Billiard
   15.2.3 Ovoid Billiard
  15.3 Summary
  15.4 Proposed Exercises
 16 Time-Dependent Billiards
  16.1 Objectives
  16.2 The Billiard
   16.2.1 LRA Conjecture
  16.3 Time-Dependent Elliptical Billiard
  16.4 Oval Billiard
  16.5 Summary
  16.6 Proposed Exercises
 17 Suppression of Fermi Acceleration in the Oval Billiard
  17.1 Objectives
  17.2 The Model and the Mapping
  17.3 Results for the Case Fα−V
  17.4 Results for the Case Fα−V2
  17.5 Results for the Case Fα−Vδ
  17.6 Summary
  17.7 Proposed Exercises
 18 A Thermodynamic Model for Time-Dependent Billiards
  18.1 Objectives
  18.2 Motivation
  18.3 Heat Transfer
  18.4 Billiard Formalism
   18.4.1 Steady State
   18.4.2 Dynamical Regime
   18.4.3 Numerical Simulations
   18.4.4 Velocity Average Over n
   18.4.5 Critical Exponents
   18.4.6 Velocity Distribution
  18.5 Connection Between the Two Formalisms
  18.6 Summary
  18.7 Proposed Exercises
 Appendix A: Euler Relations
 Appendix B: Numerical Integration Methods
 Appendix C: Expressions for the Coefficients j in the Dynamical Approach
 Appendix D: Change of Reference Frame
 Appendix E: Solution of the Diffusion Equation
 Appendix F: Heat Flux Equation
 Appendix G: Connection Between t and n in the Time-Dependent Ovoid Billiard
 Appendix H: Solution of the Integral for the Relation Between n and t in the Time-Dependent Ovoid Billiard
 References
 Index