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
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.