本书讨论了用映射描述的非线性系统中的标度的普适特性，并刻画了混沌动力学的两运动状态的转变。混沌动力学是由相空间中两个非常接近的初始条件的时间演化的不可预测性确定的。随着时间的推移，它会产生指数级的差异。对于混沌扩散的研究得到了连续运动状态转变的标度不变性。本书讨论了两种不同类型的运动状态变化：一种考虑在二维、非线性和保面积映射中观察到的从可积性到不可积性的转变，因此在变量作用和角度中是保守的动力学；另一种也考虑了使用非线性映射的动力学，并描述了对于耗散标准映射的无限混沌扩散的抑制以及时变台球中Fermi加速度抑制的等效转变。
本书可以使读者理解标度理论对非线性系统中常见的运动状态变化和其他临界动力学的普适性。这包括从可积性到不可积性的转变以及从有限扩散到无限扩散的转变，也可以应用于能量的扩散，从而应用于Fermi加速度。后者是台球动力学研究的热点，在过去几年中有许多重要的成果发表。本书可供动力系统和控制工程、数学、物理、机械和电气工程的高年级本科生和研究生阅读，也可供相关专业的研究人员参考。

1 Posing the Problems
1.1 Initial Concepts
1.2 Summary
2 A Hamiltonian and a Mapping
2.1 The Mapping
2.2 Specific Applications
2.2.1 Standard Mapping
2.2.2 The Fermi-Ulam Model
2.2.3 The Bouncer Model
2.2.4 A Hybrid Fermi-Ulam Bouncer Model
2.2.5 Family of Mappings
2.3 Summary
3 A Phenomenological Description for Chaotic Diffusion
3.1 Dynamical Properties for the Chaotic Sea: A Phenomenological Description
3.2 Summary
4 A Semi-phenomenological Description for Chaotic Diffusion
4.1 A Semi-phenomenological Approach
4.2 Summary
5 A Solution of the Diffusion Equation
5.1 A Solution of the Diffusion Equation
5.2 Summary
6 Characterization of a Continuous Phase Transition in an Area-Preserving Map
6.1 Evidence of a Phase Transition
6.2 Break of Symmetry
6.3 Order Parameter
6.4 Topological Defects
6.5 Elementary Excitations
6.6 Summary
7 Scaling Invariance for Chaotic Diffusion in a Dissipative Standard Mapping
7.1 A Dissipative Standard Mapping
7.2 Analytical Description of the Diffusion
7.3 Summary
8 Characterization of a Transition from Limited to Unlimited Diffusion
8.1 Pieces of Evidence of a Phase Transition
8.2 Break of Symmetry
8.3 Order Parameter
8.4 Topological Defects
8.5 Elementary Excitations
8.6 Summary
9 Billiards with Moving Boundary
9.1 A Billiard Dynamical System
9.2 Unlimited Diffusion of the Velocity for the Oval Billiard
9.3 Summary
10 Suppression of Fermi Acceleration in Oval Billiard
10.1 Dissipation in a Billiard
10.1.1 Long-Stand Dynamics
10.1.2 Time-Dependent Regime
10.1.3 Discussion of the Phenomenological Results
10.1.4 Time Average
10.1.5 Determination of the Critical Exponents
10.1.6 Distribution of Velocities
10.2 Summary
11 Suppressing the Unlimited Energy Gain: Shreds of Evidence of a Phase Transition
11.1 Elements of a Phase Transition
11.2 Break of Symmetry
11.3 Order Parameter
11.4 Topological Defects
11.5 Elementary Excitations
11.6 Summary
References