Ergodic theory of chaos and strange attractors.

*(English)*Zbl 0989.37516
Rev. Mod. Phys. 57, No. 3, Part 1, 617-656 (1985); addendum ibid. 57, No. 4, 1115 (1985).

Summary: Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.

In the addendum, further references are given, the choice of an embedding dimension for the computation of the information dimension is discussed, and the algorithms proposed in the paper will be available in the form of diskettes.

In the addendum, further references are given, the choice of an embedding dimension for the computation of the information dimension is discussed, and the algorithms proposed in the paper will be available in the form of diskettes.

##### MSC:

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |