Front Matter
Part I Basic Notions and Review
1 Dynamical Systems and Hyperbolicity
1.1 Dynamical systems: basic notions
1.2 Model examples of chaotic attractors
1.3 Notion of hyperbolicity
1.4 Content and conclusions of the hyperbolic theory
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
3.4 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
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
6.3 Parametric generator of hyperbolic chaos based on four coupled oscillators with pump modulation
References
7 Recognizing the Hyperbolicity: Cone Criterion and Other Approaches
7.1 Verification of transversality for manifolds
7.2 Visualization of invariant measures
7.3 Cone criterion and examples of its application
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
8.3 Hyperchaos and synchronous chaos in a system of coupled non-autonomous oscillators
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
10.3 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
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
