FM 2004-04 (mp_arc 04-123)

Author: Michele Bartuccelli, Alberto Berretti, Jonathan Deane, Guido Gentile and Stephen Gourley

Title: Periodic orbits and scaling laws for a driven damped quartic oscillator

Abstract: In this paper we investigate the conditions under which periodic solutions of a certain nonlinear oscillator persist when the differential equation is perturbed by adding a driving periodic force and a dissipative term. We conjecture that for any periodic orbit, characterized by its frequency, there exists a threshold for the damping coefficient, above which the orbit disappears, and that this threshold is infinitesimal in the perturbation parameter, with integer order depending on the frequency. Some rigorous analytical results toward the proof of these conjectures are provided. Moreover the relative size and shape of the basins of attraction of the existing stable periodic orbits are investigated numerically, giving further support to the validity of the conjectures.

Keywords: Periodic motions; Asymptotic stability; Basins of attraction; Elliptic functions; Scaling; Dissipative Systems; Resonances; Perturbation theory.

Michele Bartuccelli
Department of Mathematics and Statistics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: m.bartuccelli@surrey.ac.uk

Alberto Berretti
Dipartimento di Matematica
II Università di Roma (Tor Vergata)
Via della Ricerca Scientifica, 00133 Roma, Italy
e-mail: berretti@mat.uniroma2.it

Jonathan Deane
Department of Mathematics and Statistics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: j.deane@surrey.ac.uk

Guido Gentile
Dipartimento di Matematica
Universitaà di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
e-mail: gentile@mat.uniroma3.it

Stephen Gourley
Department of Mathematics and Statistics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: s.gourley@surrey.ac.uk