Robust approximation algorithms for the detection of attraction basins in dynamical systems

TL;DR

This paper introduces robust algorithms for detecting and reconstructing attraction basins in dynamical systems with multiple stable equilibria, using RBF-based methods and extensive numerical validation.

Contribution

It presents novel algorithms for identifying basin boundary points and reconstructing attraction manifolds in multi-stable dynamical systems, supported by a Matlab package.

Findings

Algorithms effectively detect basin boundary points.

Reconstruction of attraction manifolds is accurate and robust.

Numerical tests validate the methods' reliability.

Abstract

In dynamical systems saddle points partition the domain into basins of attractions of the remaining locally stable equilibria. This problem is rather common especially in population dynamics models. Precisely, a particular solution of a dynamical system is completely determined by its initial condition and by the parameters involved in the model. Furthermore, when the omega limit set reduces to a point, the trajectory of the solution evolves towards the steady state. But, in case of multi-stability it is possible that several steady states originate from the same parameter set. Thus, in these cases the importance of accurately reconstruct the attraction basins follows. In this paper we focus on dynamical systems of ordinary differential equations presenting three stable equilibia and we design algorithms for the detection of the points lying on the manifolds determining the basins of attraction and for the reconstruction of such manifolds. The latter are reconstructed by means of the implicit partition of unity method which makes use of radial basis functions (RBFs) as local approximants. Extensive numerical test, carried out with a Matlab package made available to the scientific community, support our findings.