Recurrence time statistics for chaos analysis in dynamical systems

Abstract

We propose a method of delimiting the existence domains of periodic, quasi-periodic and chaotic solutions of dynamical systems in the parameter space. The method is based on an analysis of the sampling of the times of passing a phase point from the previous intersection of the Poincaré plane to the next one (or from one local maximum to the next). An algorithm for generating a sample of recurrence times with subsequent analysis of the histogram of the obtained sample is constructed. A simple measure of histogram filling allows us to separate periodic and chaotic modes, as well as to estimate the degree of chaoticity of intermediate modes. On simple model signals it is shown that the distribution of recurrence times gives information not contained in the spectral densities of the signal. Then, on the example of the classical Lorentz system, it is shown how a simple measure of filling the histogram of recurrence times allows us to obtain a visual map of modes. The paper presents the results of a comparative analysis of power spectral density and histograms of recurrence times for different modes realized in the Lorentz system at different values of the control parameter (Rayleigh number).