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。

1 Introduction
1.1 Initial Concepts
1.2 Summary
2 One-Dimensional Mappings
2.1 Introduction
2.2 The Concept of Stability
2.2.1 Asymptotically Stable Fixed Point
2.2.2 Neutral Stability
2.2.3 Unstable Fixed Point
2.3 Fixed Points to the Logistic Map
2.4 Bifurcations
2.4.1 Transcritical Bifurcation
2.4.2 Period Doubling Bifurcation
2.4.3 Tangent Bifurcation
2.5 Summary
2.6 Exercises
3 Some Dynamical Properties for the Logistic Map
3.1 Convergence to the Stationary State
3.1.1 Transcritical Bifurcation
3.1.2 Period Doubling Bifurcation
3.1.3 Route to Chaos via Period Doubling
3.1.4 Tangent Bifurcation
3.2 Lyapunov Exponent
3.3 Summary
3.4 Exercises
4 The Logistic-Like Map
4.1 The Mapping
4.2 Transcritical Bifurcation
4.2.1 Analytical Approach to Obtain α, β, z and δ
4.2.2 Critical Exponents for the Period Doubling Bifurcation
4.3 Extensions to Other Mappings
4.3.1 Hassell Mapping
4.3.2 Maynard Mapping
4.4 Summary
4.5 Exercises
5 Introduction to Two Dimensional Mappings
5.1 Linear Mappings
5.2 Nonlinear Mappings
5.3 Applications of Two Dimensional Mappings
5.3.1 Hénon Map
5.3.2 Lyapunov Exponents
5.3.3 Ikeda Map
5.4 Summary
5.5 Exercises
6 A Fermi Accelerator Model
6.1 Fermi-Ulam Model
6.1.1 Jacobian Matrix for the Indirect Collisions
6.1.2 Jacobian Matrix for the Direct Collisions
6.1.3 Fixed Points
6.1.4 Phase Space
6.1.5 Phase Space Measure Preservation
6.2 A Simplified Version of the Fermi-Ulam Model
6.3 Scaling Properties for the Chaotic Sea
6.4 Localization of the First Invariant Spanning Curve
6.5 The Regime of Growth
6.6 Summary
6.7 Exercises
7 Dissipation in the Fermi-Ulam Model
7.1 Dissipation via Inelastic Collisions
7.1.1 Jacobian Matrix for the Direct Collisions
7.1.2 Jacobian Matrix for the Indirect Collisions
7.1.3 The Phase Space
7.1.4 Fixed Points
7.1.5 Construction of the Manifolds
7.1.6 Transient and Manifold Crossings Determination
7.1.7 Determining the Exponent 8 from the Eigenvalues of the Saddle Point
7.2 Dissipation by Drag Force
7.2.1 Drag Force of the Type F = –ην
7.2.2 Drag Force of the Type F = ±ην2
7.2.3 Drag Force of the Type F = -ηνγ
7.3 Summary
7.4 Exercises
8 Dynamical Properties for a Bouncer Model
8.1 The Model
8.2 Complete Version of the Bouncer Model
8.2.1 Successive Collisions
8.2.2 Indirect Collisions
8.2.3 Jacobian Matrix
8.2.4 The Phase Space
8.3 A Simplified Version of the Bouncer Model
8.4 Numerical Investigation on the Simplified Version
8.5 Approximation of Continuum Time
8.6 Summary
8.7 Exercises
9 Localization of Invariant Spanning Curves
9.1 The Standard Mapping
9.2 Localization of the Curves
9.3 Rescale in the Phase Space
9.4 Summary
9.5 Exercises
10 Chaotic Diffusion in Non-Dissipative Mappings
10.1 A Family of Discrete Mappings
10.2 Dynamical Properties for the Chaotic Sea: A Phenomenological Description
10.3 A Semi Phenomenological Approach
10.4 Determination of the Probability via the Solution of the Diffusion Equation
10.5 Summary
10.6 Exercises
11 Scaling on a Dissipative Standard Mapping
11.1 The Model
11.2 A Solution for the Diffusion Equation
11.3 Specific Limits
11.4 Summary
11.5 Exercises
12 Introduction to Billiard Dynamics
12.1 The Billiard
12.1.1 The Circle Billiard
12.1.2 The Elliptical Billiard
12.1.3 The Oval Billiard
12.2 Summary
12.3 Exercises
13 Time Dependent Billiards
13.1 The Billiard
13.1.1 The LRA Conjecture
13.2 The Time Dependent Elliptical Billiard
13.3 The Oval Billiard
13.4 Summary
13.5 Exercises
14 Suppression of Fermi Accele