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Dynamical Systems and Hyperbolicity

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Hyperbolic Chaos

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Abstract

In this chapter we review basic notions of the theory of dynamical systems essential for understanding subsequent chapters of the book. Particularly, a definition of the dynamical system is discussed and some important classes like continuous-time and discrete-time systems, conservative and dissipativc systems, autonomous and non-autonomous systems are introduced. Interpretation of dynamics as evolution of a cloud of representative points in the state space is considered and implemented for explanation of chaos and, particularly, for the uniformly hyperbolic attractors, like Smale-Williams solenoid, DA attractor of Smale, and Plykin type attractors. Lyapunov exponents as a tool for quantitative approach to analyzing chaotic or regular dynamics are examined, and methodic of their computation is discussed. Main notions of the hyperbolic theory are considered, and its substational content is briefly reviewed.

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  1. Kotel’nikov’s Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, 410019, Zelenaya 38, Saratov, Russian Federation

    Sergey P. Kuznetsov

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  1. Sergey P. Kuznetsov
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© 2012 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg

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Kuznetsov, S.P. (2012). Dynamical Systems and Hyperbolicity. In: Hyperbolic Chaos. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23666-2_1

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