Series: Mathematical Methods and Modeling for Complex Phenomena.
Eds.: A.C.J. Luo and V. Afraimovich.

Hyperbolic Chaos: A Physicist's View

Sergey P. Kuznetsov


This book is devoted to studies aimed at identifying or design of physical systems, in which chaotic dynamics occurs associated with uniformly hyperbolic attractors, such as the Plykin attractor or the Smale – Williams solenoid. Basic notions of the relevant mathematical theory are discussed, as well as approaches proposed for constructing systems with hyperbolic attractors. In particular, we consider models driven with periodic pulses; dynamics consisted of periodically repeated stages, each of which corresponds to specific form of differential equations; design of systems of alternately excited oscillators transmitting excitation each other; the use of parametric excitation of oscillations; introduction of the delayed feedback. Examples of maps, differential equations, as well as simple mechanical and electronic systems are presented manifesting chaotic dynamics due to the occurrence of the uniformly hyperbolic attractors.



Part I. Basic Notions and Review

1 Dynamical Systems and Hyperbolicity
2 Possible Occurrence of Hyperbolic Attractors

Part II. Low-Dimensional Models

3 Kicked Mechanical Models and Differential Equations with Periodic Switch
4 Non-Autonomous Systems of Coupled Self-Oscillators
5 Autonomous low-dimensional systems with uniformly hyperbolic attractors in the Poincare maps
6 Parametric Generators of Hyperbolic Chaos
7 Recognizing the Hyperbolicity: Cone Criterion and Other Approaches

Part III. Higher-Dimensional Systems and Phenomena

8 Systems of four alternately excited non-autonomous oscillators
9 Autonomous systems based on dynamics close to heteroclinic circle
10 Systems with time-delay feedback
11 Chaos in cooperative dynamics of alternately synchronized ensembles of globally coupled self-oscillators

Part IV. Experimental Studies

12 Electronic device with attractor of Smale – Williams type
13 Delay-time electronic devices generating trains of oscillations with phases governed by chaotic maps

Conclusion: Prospects and Research Directions


List of references