Basin stability approach for quantifying responses of multistable systems with parameters mismatch

TL;DR

This paper introduces a modified basin stability method to quantify the likelihood of different solutions in multistable nonlinear systems, enhancing safety analysis by efficiently predicting system behavior across parameter ranges.

Contribution

A new algorithm based on basin stability is proposed to estimate the probability of reaching specific solutions in multistable systems with parameter mismatch.

Findings

Proven effectiveness on mechanical systems like Duffing oscillators and impacting beams.

Allows prediction of system behavior with less computational effort.

Highlights advantages over classical nonlinear analysis methods.

Abstract

In this paper we propose a new method to detect and classify coexisting solutions in nonlinear systems. We focus on mechanical and structural systems where we usually avoid multistability for safety and reliability. We want to be sure that in the given range of parameters and initial conditions the expected solution is the only possible or at least has dominant basin of attraction. We propose an algorithm to estimate the probability of reaching the solution in given (accessible) ranges of initial conditions and parameters. We use a modified method of basin stability (Menck et. al., Nature Physics, 9(2) 2013). In our investigation we examine three different systems: a Duffing oscillator with a tuned mass absorber, a bilinear impacting oscillator and a beam with attached rotating pendula. We present the results that prove the usefulness of the proposed algorithm and highlight its strengths in comparison with classical analysis of nonlinear systems (analytical solutions, path-following, basin of attraction ect.). We show that with relatively small computational effort (comparing to classical analysis) we can predict the behaviour of the system and select the ranges in parameter's space where the system behaves in a presumed way. The method can be used in all types of nonlinear complex systems.