Reviews of topical problems

Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics

Kotel'nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Saratov Branch, ul. Zelenaya 38, Saratov, 410019, Russian Federation

Research is reviewed on the identification and construction of physical systems with chaotic dynamics due to uniformly hyperbolic attractors (such as the Plykin attraction or the Smale-Williams solenoid). Basic concepts of the mathematics involved and approaches proposed in the literature for constructing systems with hyperbolic attractors are discussed. Topics covered include periodic pulse-driven models; dynamics models consisting of periodically repeated stages, each described by its own differential equations; the construction of systems of alternately excited coupled oscillators; the use of parametrically excited oscillations; and the introduction of delayed feedback. Some maps, differential equations, and simple mechanical and electronic systems exhibiting chaotic dynamics due to the presence of uniformly hyperbolic attractors are presented as examples.

Text can be downloaded in Russian. English translation is available on IOP Science.
PACS: 05.45.−a, 45.50.−j, 84.30.−r (all)
DOI: 10.3367/UFNe.0181.201102a.0121
Citation: Kuznetsov S P "Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics" Phys. Usp. 54 119–144 (2011)
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Оригинал: Кузнецов С П «Динамический хаос и однородно гиперболические аттракторы: от математики к физике» УФН 181 121–149 (2011); DOI: 10.3367/UFNr.0181.201102a.0121

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