ABSTRACT
This chapter presents some defi nitions and relevant results about the statistical properties and classifi cation of chaotic attractors. Chaos can be defi ned as follows: a chaotic strange attractor is a non-trivial attracting set that contains a dense orbit with at least one positive Lyapunov exponent. Generally, chaos refers to the existence of a Smale horseshoe with a hyperbolic structure. In Section 1.1 several criterions for measuring chaos in dynamical systems are presented and discussed in some details. This includes: the Lebesgue measure, the physical (or Sinai-Ruelle-Bowen) measure, the Hausdorff dimension, the topological entropy, Lyapunov exponents, ergodic theory and its importance in quantifying and understanding behavior of chaotic attractors. Different correlations are defi ned such as autocorrelation function (ACF) discussed in Section 1.1.7 and the decay of correlations presented in Section 1.1.9.1 and the Central Limit Theorem in Section 1.1.9.2. A hyperbolicity test is given in Section 1.1.8 in order to clarify the nature of chaos in a Hénon map. A classifi cation of strange attractors of dynamical systems is given in Section 1.2. This classifi cation contains the so-called hyperbolic attractors, Lorenz-type attractors and quasi-attractors.
