本书讨论了在可以用常微分方程或映射来描述的非线性动力系统中观察到的许多常见的标度特性。相空间中两个相邻初始条件的时间演化的不可预测性以及随着时间的推移相互之间的指数发散性引出了混沌的概念。非线性系统中的一些可观测物表现出标度不变性的特征，因而可以通过标度律来描述。 从控制参数的变化来看，相空间中的物理观测可以用多次服从普遍行为的幂律来表示。这种形式化的应用在非线性动力学领域已被广泛接受。因此，作者试图把非线性系统中的一些研究成果与标度形式化的方法结合起来。书中的方法既可以在本科阶段学习，也在可以研究生阶段学习。本书只要求基础物理和数学知识，大多数章节提供了充足的解析、数值练习题。

前辅文
1 Introductio
  1.1 Initial Concept
  1.2 Summar
2 One-Dimensional Mapping
  2.1 Introduction
  2.2 The Concept of Stability
   2.2.1 Asymptotically Stable Fixed Poin
   2.2.2 Neutral Stability
   2.2.3 Unstable Fixed Poin
  2.3 Fixed Points to the LogisticMap
  2.4 Bifurcations
   2.4.1 Transcritical Bifurcation
   2.4.2 Period Doubling Bifurcatio
   2.4.3 Tangent Bifurcatio
  2.5 Summar
  2.6 Exercise
3 Some Dynamical Properties for the Logistic Ma
  3.1 Convergence to the Stationary Stat
   3.1.1 Transcritical Bifurcation
   3.1.2 Period Doubling Bifurcatio
   3.1.3 Route to Chaos via Period Doublin
   3.1.4 Tangent Bifurcatio
  3.2 Lyapunov Exponen
  3.3 Summar
  3.4 Exercise
4 The Logistic-Like Map
  4.1 The Mappin
  4.2 Transcritical Bifurcatio
   4.2.1 Analytical Approach to Obtain α, β, z and δ
   4.2.2 Critical Exponents for the Period Doubling Bifurcatio
  4.3 Extensions to Other Mapping
   4.3.1 Hassell Mapping
   4.3.2 Maynard Mappin
  4.4 Summar
  4.5 Exercise
5 Introduction to Two Dimensional Mappings
  5.1 Linear Mapping
  5.2 Nonlinear Mapping
  5.3 Applications of Two Dimensional Mapping
   5.3.1 Hénon Ma
   5.3.2 Lyapunov Exponent
   5.3.3 IkedaMap
  5.4 Summar
  5.5 Exercise
6 A Fermi Accelerator Mode
  6.1 Fermi-Ulam Model
   6.1.1 Jacobian Matrix for the Indirect Collision
   6.1.2 Jacobian Matrix for the Direct Collision
   6.1.3 Fixed Point
   6.1.4 Phase Spac
   6.1.5 Phase Space Measure Preservatio
  6.2 A Simplified Version of the Fermi-Ulam Model
  6.3 Scaling Properties for the Chaotic Se
  6.4 Localization of the First Invariant Spanning Curv
  6.5 The Regime of Growt
  6.6 Summar
  6.7 Exercise
7 Dissipation in the Fermi-Ulam Model
  7.1 Dissipation via Inelastic Collision
   7.1.1 Jacobian Matrix for the Direct Collision
   7.1.2 Jacobian Matrix for the Indirect Collision
   7.1.3 The Phase Space
   7.1.4 Fixed Point
   7.1.5 Construction of theManifolds
   7.1.6 Transient and Manifold Crossings Determinatio
   7.1.7 Determining the Exponent δ from the Eigenvalues of the Saddle Poin
  7.2 Dissipation by Drag Force
   7.2.1 Drag Force of the Type F = −˜η
   7.2.2 Drag Force of the Type F = ±˜η
   7.2.3 Drag Force of the Type F = −˜ηv
  7.3 Summar
  7.4 Exercise
8 Dynamical Properties for a Bouncer Model
  8.1 The Model
  8.2 Complete Version of the Bouncer Model
   8.2.1 Successive Collision
   8.2.2 Indirect Collision
   8.2.3 Jacobian Matrix
   8.2.4 The Phase Space
  8.3 A Simplified Version of the Bouncer Mode
  8.4 Numerical Investigation on the Simplified Versio
  8.5 Approximation of Continuum Tim
  8.6 Summar
  8.7 Exercise
9 Localization of Invariant Spanning Curves
  9.1 The Standard Mappin
  9.2 Localization of the Curves
  9.3 Rescale in the Phase Spac
  9.4 Summar
  9.5 Exercise
10 Chaotic Diffusion in Non-Dissipative Mapping
  10.1 A Family of Discrete Mappings
  10.2 Dynamical Properties for the Chaotic Sea:A Phenomenological Description
  10.3 A Semi Phenomenological Approac
  10.4 Determination of the Probability via the Solution of the Diffusion Equation
  10.5 Summar
  10.6 Exercise
11 Scaling on a Dissipative Standard Mapping
  11.1 The Model
  11.2 A Solution for the Diffusion Equatio
  11.3 Specific Limit
  11.4 Summar
  11.5 Exercise
12 Introduction to Billiard Dynamic
  12.1 The Billiard
   12.1.1 The Circle Billiar
   12.1.2 The Elliptical Billiar
   12.1.3 The Oval Billiard
  12.2 Summar
  12.3 Exercise
13 Time Dependent Billiard
  13.1 The Billiard
   13.1.1 The LRA Conjectur
  13.2 The Time Dependent Elliptical Billiard
  13.3 The Oval Billiar
  13.4 Summar
  13.5 Exercise
14 Suppression of Fermi Acceleration in the Oval Billiar
  14.1 The Model and the Mappin
  14.2 Results for the Case of F ∝ −V
  14.3 Results for the Case of F ∝ ±V2
  14.4 Results for the Case of F ∝ −Vδ
  14.5 Summar
  14.6 Exercise
15 A Thermodynamic Model for Time Dependent Billiards
  15.1 Motivation
  15.2 Heat Transference
  15.3 The Billiard Formalis
   15.3.1 Stationary Estate
   15.3.2 Dynamical Regim
   15.3.3 Numerical Simulations
   15.3.4 Average Velocity over n
   15.3.5 Critical Exponent
   15.3.6 Distribution of Velocitie
  15.4 Connection Between the Two Formalis
  15.5 Summar
  15.6 Exercise
Appendix A: Expressions for the Coefficients
Appendix B: Change of Referential Frame
Appendix C: Solution of the Diffusion Equation
Appendix D: Heat Flow Equatio
Appendix E: Connection Between t and n in a Time Dependent Oval Billiar
Appendix F: Solution of the Integral to Obtain the Relation Between n and t in the Time Dependent Oval Billiard
Bibliograph