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Hyperbolic Chaos: A Physicist`s View双曲混沌 一个物理学家的观点

样章
作者：
Sergey P. Kuznetsov
定价：
69.00元
ISBN：
978-7-04-031964-4
版面字数：
400.000千字
开本：
16开
全书页数：
320页
装帧形式：
精装
重点项目：
暂无
出版时间：
2011-09-14
读者对象：
学术著作
一级分类：
自然科学
二级分类：
交叉学科

《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。

  • Front Matter
  • Part I Basic Notions and Review
    • 1 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 II Low-Dimensional Models
    • 3 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
    • 4 Non-Autonomous Systems of Coupled Self-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´e Maps
      • 5.1 Autonomous system of two coupled oscillators with self-regulating alternating excitation
      • 5.2 System constructed on a base of the predator-prey model
      • 5.3 Example of blue sky catastrophe accompanied by a birth of Smale-Williams attractor
      • References
    • 6 Parametric Generators of Hyperbolic Chaos
      • 6.1 Parametric excitation of coupled oscillatorsThree-frequency parametric generator and its operation
      • 6.2 Hyperbolic chaos in parametric oscillator with Q-switch and pump modulation
        • 6.2.1 Dynamical equations
        • 6.2.2 Qualitative explanation of the operation
        • 6.2.3 Numerical results
        • 6.2.4 Numerical results in the frame of method of slow complex amplitudes
      • 6.3 Parametric generator of hyperbolic chaos based on four coupled oscillators with pump modulation
        • 6.3.1 Model, operation principle and basic equations
        • 6.3.2 Chaotic dynamics: results of computer simulation
      • References
    • 7 Recognizing the Hyperbolicity: Cone Criterion and Other Approaches
      • 7.1 Verification of transversality for manifolds
        • 7.1.1 Visualization of the manifolds
        • 7.1.2 Distributions of angles of the manifold intersections
      • 7.2 Visualization of invariant measures
      • 7.3 Cone criterion and examples of its application
        • 7.3.1 Procedure of verification of the cone criterion
        • 7.3.2 Examples of application of the cone criterion
      • References
  • Part III Higher-Dimensional Systems and Phenomena
    • 8 Systems of Four Alternately Excited Non-autonomous Oscillators
      • 8.1 Arnold’s cat map dynamics in a system of coupled non-autonomous van der Pol oscillators
      • 8.2 Dynamics corresponding to hyperchaotic maps
        • 8.2.1 System implementing toral hyperchaotic map
        • 8.2.2 Model with cascade transfer of excitation upward the frequency spectrum
      • 8.3 Hyperchaos and synchronous chaos in a system of coupled non-autonomous oscillators
        • 8.3.1 Equations and basic modes of operation
        • 8.3.2 Equations for slow complex amplitudes
      • References
    • 9 Autonomous Systems Based on Dynamics Close to Heteroclinic Cycle
      • 9.1 Heteroclinic connection: an example of Guckenheimer and Holmes
      • 9.2 Attractor of Smale-Williams type in a system of three coupled self-oscillators
      • 9.3 Attractor with dynamics governed by the Arnold cat map
      • 9.4 Model with hyperchaos
      • 9.5 An autonomous system with attractor of Smale-Williams type with resonance transfer of excitation in a ring array of van der Pol oscillators
      • References
    • 10 Systems with Time-delay Feedback
      • 10.1 Some notions concerning differential equations with deviating argument
      • 10.2 Van der Pol oscillator with delayed feedback, parameter modulation and auxiliary signal
        • 10.2.1 Attractor of Smale-Williams type in the time-delayed system
        • 10.2.2 Hyperchaotic attractors
      • 10.3 Van der Pol oscillator with two delayed feedback loops and parameter modulation
      • 10.4 Autonomous time-delay system
      • References
    • 11 Chaos in Co-operative Dynamics of Alternately Synchronized Ensembles of Globally Coupled Self-oscillators
      • 11.1 Kuramoto transition in ensemble of globally coupled oscillators
      • 11.2 Model of two alternately synchronized ensembles of oscillators
        • 11.2.1 Collective chaos in ensemble of van der Pol oscillators
        • 11.2.2 Slow-amplitude approach
        • 11.2.3 Description of the dynamics in terms of ensembles of phase oscillators
      • References
  • Part IV Experimental Studies
    • 12 Electronic Device with Attractor of Smale-Williams Type
      • 12.1 Scheme of the device and the principle of operation
      • 12.2 Experimental observation of the Smale-Williams attractor
      • References
    • 13 Delay-time Electronic Devices Generating Trains of Oscillations with Phases Governed by Chaotic Maps
      • 13.1 Van der Pol oscillator with delayed feedback, parameter modulation and auxiliary signal
      • 13.2 Van der Pol oscillator with two delayed feedback loops and parameter modulation
      • References
    • 14 Conclusion
      • References
  • Appendix A Computation of Lyapunov Exponents:The Benettin Algorithm
    • References
  • Appendix B H´enon and Ikeda Maps
    • References
  • Appendix C Smale’s Horseshoe and Homoclinic Tangle
    • References
  • Appendix D Fractal Dimensions and Kaplan-Yorke Formula
    • References
  • Appendix E Hunt’s Model: Formal Definition
    • References
  • Appendix F Geodesics on a Compact Surface of Negative Curvature
    • References
  • AppendixG Effect of Noise in a System with a Hyperbolic Attractor
    • References
  • Index

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