MATHEMATICAL BIOSCIENCES http://math.asu.edu/˜mbe/ AND ENGINEERING Volume 1, Number 1, June 2004 pp. 161–184 STATISTICAL PROPERTIES OF DYNAMICAL CHAOS Vadim S. Anishchenko Institute of Nonlinear Dynamics, Department of Physics, Saratov State University 83, Astrakhanskaya str., 410012, Saratov, Russia Tatjana E. Vadivasova Institute of Nonlinear Dynamics, Department of Physics, Saratov State University 83, Astrakhanskaya str., 410012, Saratov, Russia Galina I. Strelkova Institute of Nonlinear Dynamics, Department of Physics, Saratov State University 83, Astrakhanskaya str., 410012, Saratov, Russia George A. Okrokvertskhov Institute of Nonlinear Dynamics, Department of Physics, Saratov State University 83, Astrakhanskaya str., 410012, Saratov, Russia (Communicated by Stefano Boccaletti) Abstract. This study presents a survey of the results obtained by the au- thors on statistical description of dynamical chaos and the effect of noise on dynamical regimes. We deal with nearly hyperbolic and nonhyperbolic chaotic attractors and discuss methods of diagnosing the type of an attractor. We consider regularities of the relaxation to an invariant probability measure for different types of attractors. We explore peculiarities of autocorrelation de- cay and of power spectrum shape and their interconnection with Lyapunov exponents, instantaneous phase diffusion and the intensity of external noise. Numeric results are compared with experimental data. 1. Introduction. Dynamical chaos, like a random process, requires a statistical description. When chaotic systems are studied in computer or physical experiments, probability characteristics (such as a stationary probability distribution on an at- tractor, correlation functions, power spectra and others) are usually calculated or measured. Chaotic oscillations that correspond to different types of chaotic attrac- tors in the phase space of dynamical systems are characterized by various statistical properties as well as by a different degree of sensitivity of the statistical character- istics to the influence of noise. In the rigorous theory, hyperbolic chaos is often called “true” chaos and is char- acterized by a homogeneous and topologically stable structure [1, 2, 3, 4]. However, strange chaotic attractors of dynamical systems are not, as a rule, robust hyper- bolic sets. Rather they are referred to as nearly hyperbolic attractors; for example, the Lorenz attractor. Nearly hyperbolic (quasihyperbolic) attractors include some 2000 Mathematics Subject Classification. 92D30. Key words and phrases. nonhyperbolic attractors, spiral and funnel chaos, autocorrelation function, instantaneous phase, phase variance, effective diffusion coefficient. 161