Predicting the Emergence of Multistability in a Monoparametric PWL System

Abstract

One of the main problems in the study of dynamical systems is to explore the asymptotic behavior of the model when a parameter varies continuously. When these variations lead to the appearance of coexisting states, the study of the global properties of the system becomes an even more complex task, since it is almost impossible to predict the stability change. In this paper, we present a simple method for characterizing qualitative changes in the dynamics of a family of Piece-Wise Linear (PWL) chaotic systems, that transit from monostable to multistable behavior by a single bifurcation parameter. By characterizing the magnitude of the stable and unstable manifolds associated with the eigendirections, it is possible to analytically find tipping points in the linear model that are consistent with the occurrence of coexisting states in the dynamics. The results show agreement between the bifurcation diagrams of the linear operator, the bifurcation diagrams of the PWL system, and the multistability phenomenon validation in analog electronics. The presented work makes it possible to know the mechanism by which the system exhibits the break of its stability and the corresponding basin of attraction. This introduces a new methodology for the analysis of dynamical systems in search of dynamical changes such as coexisting attractors.