**Authors:**
J. Wright, J. Deane, M. Bartuccelli, G. Gentile

**Title:** *
Basins of attraction in forced systems with time-varying dissipation
*

**Abstract: **
We consider periodically forced systems with dissipation depending on time and study how
the sizes of the basins of attraction are modified with respect to the case of constant dissipation.
In particular, we investigate cases in which having information as to how the system behaves
for constant dissipation may be used when dissipation varies over a certain time interval before
settling at a constant final value. First, we consider situations where one is interested in the
basins of attraction for damping coefficient growing linearly between two given values, in
the time interval where it varies, for a large number of choices of the interval: we outline a
method to reduce the computation time required to estimate numerically the relative areas of
the basins for all values of the time interval, 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.

**Keywords:**
Attractors; basins of attraction; forced systems; dissipative systems;
damping coefficient; non-constant dissipation; pendulum with oscillating support.

James A. Wright

Department of Mathematics

University of Surrey

Guildford, GU2 7HX, UK

e-mail: j.wright@surrey.ac.uk

Jonathan H.B. Deane

Department of Mathematics

University of Surrey

Guildford, GU2 7HX, UK

e-mail: j.deane@surrey.ac.uk

Michele V. Bartuccelli

Department of Mathematics

University of Surrey

Guildford, GU2 7HX, UK

e-mail: gentile@mat.uniroma3.it

Guido Gentile

Dipartimento di Matematica e Fisica

Università Roma Tre

Largo San Leonardo Murialdo 1 - 00146 Roma - Italy

e-mail: gentile@mat.uniroma3.it