STABILISATION OF PERIOD-1 ORBITS OF THE HENON MAP; PARAMETRIC VARIATION AND ADDITIVE NOISE

摘要

Slow sinusoidal modulation of a control parameter can maintain a low period orbit of a nonlinear dynamic system into parameter regions where the low period orbit is unstable, and a higher period orbit would normally occur. We examine this case for a period-doubling bifurcation of the Henon map. Whether or not effective stabilisation of the period-1 orbit takes place depends on both the frequency of the variation and on the noise. Increasing the frequency tends to stabilise the orbit; increasing the noise level is destabilising. We use matched asymptotic expansions to describe the response, and find explicit expressions for the response which can estimate whether and where an apparent period doubling occurs.