Basins of attraction in forced systems with time-varying dissipation

TL;DR

This paper studies how the basins of attraction in forced dissipative systems change when dissipation varies over time, introducing methods to efficiently estimate these basins and explaining abrupt changes through phase space analysis.

Contribution

It presents a new method to estimate basins of attraction under time-varying dissipation and explains large variations in basins with phase space comparisons.

Findings

Efficient computation of basin areas for varying dissipation.

Abrupt basin changes linked to phase space contraction.

Application to a pendulum with oscillating support.

Abstract

We consider dissipative periodically forced systems and investigate cases in which having information as to how the system behaves for constant dissipation may be used when dissipation varies in time before settling at a constant final value. First, we consider situations where one is interested in the basins of attraction for damping coefficients varying linearly between two given values over many different time intervals: we outline a method to reduce the computation time required to estimate numerically the relative areas of the basins and discuss its range of applicability. Second, we observe that sometimes very slight changes in the time interval may produce abrupt large variations in the relative areas of the basins of attraction of the surviving attractors: we show how comparing the contracted phase space at a time after the final value of dissipation has been reached with the basins of attraction corresponding to that value of constant dissipation can explain the presence of such variations. Both procedures are illustrated by application to a pendulum with periodically oscillating support.