BSTAB: an open-source software for computing the basin stability of multi-stable dynamical systems

Norbert Hoffmann,Merten Stender

doi:10.1007/s11071-021-06786-5

Norbert Hoffmann, Merten Stender

Open Access

https://doi.org/10.1007/s11071-021-06786-5

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Abstract

The pervasiveness of multi-stability in nonlinear dynamical systems calls for novel concepts of stability and a consistent quantification of long-term behavior. The basin stability is a global stability metric that builds on estimating the basin of attraction volumes by Monte Carlo sampling. The computation involves extensive numerical time integrations, attractor characterization, and clustering of trajectories. We introduce bSTAB, an open-source software project that aims at enabling researchers to efficiently compute the basin stability of their dynamical systems with minimal efforts and in a highly automated manner. The source code, available at https://github.com/TUHH-DYN/bSTAB/, is available for the programming language Matlab featuring parallelization for distributed computing, automated sensitivity and bifurcation analysis as well as plotting functionalities. We illustrate the versatility and robustness of bSTAB for four canonical dynamical systems from several fields of nonlinear dynamics featuring periodic and chaotic dynamics, complicated multi-stability, non-smooth dynamics, and fractal basins of attraction. The bSTAB projects aims at fostering interdisciplinary scientific collaborations in the field of nonlinear dynamics and is driven by the interaction and contribution of the community to the software package.

Highlights

Stability analysis plays a central role in nonlinear dynamics research, whether for structural dynamics, fluid flows, chemical reactions, or others [1]
The basin stability proposed by Menck et al [2,3] is a global stability concept based on the estimated volume of the basins of attraction in the system’s phase space, which can overcome some limitations of classical local stability concepts
This stability concept is of relevance especially for designing practical application systems which are always subject to uncertainty of initial conditions or instantaneous perturbations

Summary

Introduction

Stability analysis plays a central role in nonlinear dynamics research, whether for structural dynamics, fluid flows, chemical reactions, or others [1]. Linear and local stability concepts are standard methods for the characterization of point attractors and (quasi-) periodic orbits These stability concepts are limited to small perturbations when assessing the longterm system behavior and stability against perturbations. The basin stability indicates the probability of the system to arrive on one of the multiple solutions when perturbing the current state of the system. This stability concept is of relevance especially for designing practical application systems which are always subject to uncertainty of initial conditions or instantaneous perturbations. Albeit being a rather simplistic theoretical concept, the practical computation of the basin stability involves extensive Monte Carlo simulations and several technical challenges for researchers who want to study the basin stability of their dynamical systems

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