**Author:**
Michele Bartuccelli, Jonathan Deane and Guido Gentile

**Title:** *
Attractiveness of periodic orbits in parametrically forced systems with time-increasing friction
*

**Abstract: **
We consider dissipative one-dimensional systems subject to a periodic force
and study numerically how a time-varying friction affects the dynamics.
As a model system, particularly suited for numerical analysis,
we investigate the driven cubic oscillator in the presence of friction.
We find that, if the damping coefficient increases in time up to a final
constant value, then the basins of attraction of the
leading resonances are larger than they would have been if the coefficient
had been fixed at that value since the beginning.
From a quantitative point of view, the scenario depends both on
the final value and the growth rate of the damping coefficient.
The relevance of the results for the spin-orbit model are discussed in some detail.

**Keywords:**
Dissipative systems;
Time-dependent friction;
Periodic motions;
Periodically forced systems;
Periodic attractors;
Basins of attraction;
Bifurcations;
Melnikov problem.

Michele V. Bartuccelli

Department of Mathematics

Surrey University

Guildford, GU2 7HX, UK

e-mail: m.bartuccelli@surrey.ac.uk

Jonathan H.B. Deane

Department of Mathematics

Surrey University

Guildford, GU2 7HX, UK

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

Guido Gentile

Dipartimento di Matematica

Università di Roma Tre

Largo San Leonardo Murialdo 1, 00146 Roma, Italy

e-mail: gentile@mat.uniroma3.it