**Authors:**
James Wright, Michele Bartuccelli, Guido Gentile

**Title:** *
The effects of time-dependent dissipation on the basins of attraction for the pendulum with oscillating support
*

**Abstract: **
We consider a pendulum with vertically oscillating support and time-dependent
damping coefficient which varies until reaching a finite final value. Although
it is the final value which determines which attractors eventually exist,
however the sizes of the corresponding basins of attraction are found
to depend strongly on the full evolution of the dissipation.
In particular we investigate numerically how dissipation monotonically
varying in time changes the sizes of the basins of attraction.
It turns out that, in order to predict the behaviour of the system, it is essential
to understand how the sizes of the basins of attraction
for constant dissipation depend on the damping coefficient.
For values of the parameters where the systems can be considered
as a perturbation of the simple pendulum, which is integrable, we characterise
analytically the conditions under which the attractors exist and study numerically
how the sizes of their basins of attraction depend on the damping coefficient.
Away from the perturbation regime, a numerical study of the attractors
and the corresponding basins of attraction
for different constant values of the damping coefficient produces a
much more involved scenario: changing the magnitude
of the dissipation causes some attractors to disappear either leaving
no trace or producing new attractors by bifurcation,
such as period doubling and saddle-node bifurcation.
Finally we pass to the case of an initially non-constant damping coefficient,
both increasing and decreasing to some finite final value,
and we numerically observe the resulting effects on the sizes
of the basins of attraction: when the damping coefficient varies slowly from
a finite initial value to a different final value, without changing the set of attractors,
the slower the variation the closer the sizes of the basins of attraction
are to those they have for constant damping coefficient fixed at the initial value.
Furthermore, if during the variation of the damping coefficient attractors
appear or disappear, remarkable additional phenomena may occur.
For instance it can happen that, in the limit of very large variation time,
a fixed point asymptotically attracts the entire phase space,
up to a zero measure set, even though no attractor with such a property
exists for any value of the damping coefficient between the extreme values.

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

James A. Wright

Department of Mathematics

University of Surrey

Guildford, GU2 7HX, UK

e-mail: j.wright@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