New bifurcation diagrams based on hypothesis testing: Pseudo-periodic surrogates with correlation dimension as discriminating statistic

Abstract

This paper discusses a possibility of using a new type of bifurcation diagrams that can be computed for time series being either solutions of nonlinear oscillatory dynamical systems or measured and recorded in a laboratory when no mathematical model is known. The obtained or measured noisy time-series in mechanical (and in other areas of engineering) dynamical systems are analyzed in a certain range of a slowly varying parameter and the added noise is uncorrelated with the dynamics of the system. In such a scenario with noise, typical well-known bifurcation diagrams cannot be used, so a new method of computing bifurcation diagrams via hypothesis testing from statistics is proposed - a method based on pseudo-periodic surrogates and correlation dimension as the discriminating statistics. An illustrative example based on the chaotic Rössler system is provided to show promising results of this new approach.