FM 03-14 (arXiv:1403.4996)

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