Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators Properties

Abstract

The study of dynamical systems is based on the solution of differential equations that may exhibit various behaviors, such as fixed points, limit cycles, periodic, quasi-periodic attractors, chaotic behavior, and coexistence of attractors, to name a few. In this paper, we present a simple and novel method for predicting the occurrence of tipping points in a family of Piece-Wise Linear systems (PWL) that exhibit a transition from monostability to multistability with the variation of a single parameter, without the need to compute time series, i.e., without solving the differential equations of the system. The linearized system of the model is analyzed, the stable and unstable manifolds are taken to be real vectors in space, and the changes suffered by these vectors as a result of the modification of the parameter are examined using such simple metrics as the magnitude of a vector or the angle between two vectors in space. The results obtained with the linear analysis of the system agree well with those obtained with the numerical resolution of the dynamical system itself. The work presented here is an extension of previous results on this topic and contributes to the understanding of the mechanisms by which a system changes its stability by fragmenting its basin of attraction. This, in turn, enriches the field by providing an alternative to numerical resolution to identify quantitative changes in the dynamics of complex systems without having to solve the differential equation system.

Keywords

• Nonlinear dynamics
• Chaotic system
• Multistability
• PWL system
• Bifurcation
• Tipping point

Supporting Institution

J.L.E.M. thanks CONAHCYT for financial support (CVU-706850), and to CICESE.

Project Number

CVU-706850

Ethical Statement

The authors state that there are no ethical conflicts in publishing this paper since it does not involve experiments with living beings.

Thanks

J.L.E.M. thanks CONAHCYT for financial support (CVU-706850), and to J.A.G. for the opportunity to do a postdoctoral fellowship at CICESE.

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