双曲混沌 : 一个物理学家的观点

Part 1 Basic Notions and Review

Dynamical Systems and Hyperbolicity

1.1 Dynamical systems： basic notions

1.1.1 Systems with continuous and discrete time， and their mutual relation

1.1.2 Dynamics in terms of phase fluid： Conservative and dissipative systems and attractors

1.1.3 Rough systems and structural stability

1.1.4 Lyapunov exponents and their computation

1.2 Model examples of chaotic attractors

1.2.1 Chaos in terms of phase fluid and baker's map

1.2.2 Smale-Williams solenoid

1.2.3 DA-attractor

1.2.4 Plykin type attractors

1.3 Notion of hyperbolicity

1.4 Content and conclusions of the hyperbolic theory

1.4.1 Cone criterion.

1.4.2 Instability

1.4.3 Transversal Cantor structure and Kaplan-Yorke dimension

1.4.4 Markov partition and symbolic dynamics

1.4.5 Enumerating of orbits and topological entropy

1.4.6 Structural stability

1.4.7 Invariant measure of Sinai-Ruelle-Bowen

1.4.8 Shadowing and effect of noise

1.4.9 Ergodicity and mixing

1.4.10 Kolmogorov-Sinai entropy

References

2 Possible Occurrence of Hyperbolic Attractors

2.1 The Newhouse-Ruelle-Takens theorem and its relation to the uniformly hyperbolic attractors

2.2 Lorenz model and its modifications

2.3 Some maps with uniformly hyperbolic attractors

2.4 From DA to the Plykin type attractor

2.5 Hunt's example： Suspending the Plykin type attractor

2.6 The triple linkage： A mechanical system with hyperbolic dynamics

2.7 A possible occurrence of a Plykin type attractor in Hindmarsh-Rose neuron model

2.8 Blue sky catastrophe and birth of the Smale-Williams attractor

2.9 Taffy-pulling machine

References

Part 2 Low-Dimensional Models

Kicked Mechanical Models and Differential Equations with Periodic Switch

3.1 Smale-Williams solenoid in mechanical model： Motion of a particle on a plane under periodic kicks

3.2 A set of switching differential equations with attractor of Smale-Williams type

3.3 Explicit dynamical system with attractor of Plykin type

3.3.1 Plykin type attractor on a sphere

3.3.2 Plykin type attractor on the plane

3.4 Plykin-like attractor in smooth non-autonomous system

References

Non-Autonomous Systems of Coupled Serf-Oscillators

4.1 Van der Pol oscillator

4.2 Smale-Williams attractor in a non-autonomous system of alternately excited van der Pol oscillators

4.3 System of alternately excited van der Pol oscillators in terms of slow complex amplitudes

4.4 Non-resonance excitation transfer

4.5 Plykin-like attractor in non-autonomous coupled oscillators

4.5.1 Representation of states on a sphere and equations of the model

4.5.2 Numerical results for the coupled oscillators

References

5 Autonomous Low-dimensional Systems with Uniformly Hyperbolic

Attractors in the Poincar~ Maps

5.1 Autonomous system of two coupled oscillators with self-regulating alternating excitation

……

Part 3 Higher-Dimensional Systems and Phenomena

Part 4 Experimental Studies

《双曲混沌：一个物理学家的观点》从物理学而不是数学概念的角度介绍了目前动力系统中均匀双曲吸引子研究的进展小结构稳定的吸引子表现出强烈的随机性，但是对于动力系统中函数和参数的变化不敏感。基于双曲混沌的特征，《双曲混沌：一个物理学家的观点》将展示如何找到物理系统中的双曲混沌吸引子，以及怎样设计具有双曲混沌的物理系统。
《双曲混沌：一个物理学家的观点》可以作为研究生和高年级本科生教材，也可以供大学教授以及物理学、机械学和工程学相关研究人员参考。