Strange Attractors

  • R. V. Plykin
  • E. A. Sataev
  • S. V. Shlyachkov
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 66)


The term “strange attractor” was introduced in (Ruelle and Tokens 1971) to denote a limit set of a dynamical system that is not a manifold and is consequently not a fixed point, limit cycle, invariant torus and so on. It immediately gained wide prevalence among physicists. Nowadays the notion of “strange attractor” has more the nature of a paradigm than a rigorously defined mathematical object. A dynamical system is considered to have a strange attractor if the phase space of the system has a limit set consisting of trajectories with chaotic behaviour. With regard to chaos, there are a number of “physical” criteria for chaotic behaviour. Among these we can single out the presence of homoclinics, the continuity of the spectrum, the presence of a positive Lyapunov exponent, the fractionality of some dimension, the presence of an infinite series of bifurcations, and period doubling. There are an enormous nimber of works in which for a given physical system one of the above criteria or some other criterion is used to deduce the presence of a strange attractor in the system. An extensive bibliography of works of this kind can be found in the book (Neimark and Landa 1987).


Unstable Manifold Stable Manifold Strange Attractor Lorenz System Topological Entropy 
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Authors and Affiliations

  • R. V. Plykin
  • E. A. Sataev
  • S. V. Shlyachkov

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