TRUNCATED FRACTAL BASIN BOUNDARIES IN THE PENDULUM WITH NONPERIODIC FORCING

Citation
I. Dobson et Df. Delchamps, TRUNCATED FRACTAL BASIN BOUNDARIES IN THE PENDULUM WITH NONPERIODIC FORCING, Journal of nonlinear science, 4(4), 1994, pp. 315-328
Citations number
18
Categorie Soggetti
Mathematics,"Mathematical Method, Physical Science",Mathematics,Mechanics
ISSN journal
09388974
Volume
4
Issue
4
Year of publication
1994
Pages
315 - 328
Database
ISI
SICI code
0938-8974(1994)4:4<315:TFBBIT>2.0.ZU;2-Z
Abstract
It is well known that oscillators such as the pendulum can have fracta l basin boundaries when they are periodically forced with the conseque nce that the long term behavior of the system may be unpredictable. In engineering and physical applications, the forcing is often nonperiod ic and eventually decays to zero, and simulation of the pendulum with decaying forcing (M. Varghese, J. S. Thorp, Physical Review Letters, v ol. 60, no. 8, pp. 665-668, Feb. 1988) exhibits truncated fractal basi n boundaries which also limit the system predictability. We develop a coordinate change for the pendulum with decaying forcing that allows u s to apply standard qualitative methods to study the basin boundaries. We prove that the basin boundaries cannot be fractal and show by exam ple how the extreme stretching and folding leading to a truncated frac tal basin boundary may arise.