Slow Sweep Through a Period-Doubling Cascade: An Example of a Noisy Parametric Bifurcation

Abstract

The response of a nonlinear system when a parameter is slowly varied through a bifurcation point may be influenced strongly by very-low-level noise. Noise, which is inherent in any system, both in experiments and in numerical simulations, is actually necessary in some cases to trigger the change in response from the initial form to the bifurcated form. This is the case for example when the initial form of the solution continues to exist even though in an unstable state after the bifurcation point. We analyse two such systems involving two different types of bifurcations, one involving a continuous system and the other a discrete map, and show the critical effect of noise in each case.

• (i) A ship model with coupled pitch and roll modes exhibits a saturation phenomenon of the pitch mode, with a corresponding large amplitude increase of the roll mode as the wave excitation amplitude is slowly increased. This is a form of transcritical bifurcation. During a slow sinusoidal variation of the excitation amplitude, we show that noise determines whether and when the possibly catastrophic increase in roll amplitude occurs.

• (ii) Quadratic maps exhibit cascades of period-doubling bifurcations. We use a renormalisation scheme to describe high-period orbits. Low-period orbits can be stabilised by slow sinusoidal sweep of a control parameter. Increasing the rate of sweep has a stabilising effect, but increasing the noise level destabilises the orbits by triggering period-doubling.

We analyse all these systems using matched asymptotic expansions in terms of the small rate of variation of the parameter. A nested set of three expansions is needed to describe the jump phenomenon from one type of behaviour to another, that is, from the now unstable original form to the stable bifurcated form. The innermost expansion in each case describes how noise is necessary to trigger exponential growth for example of period-2 response away from the locally unstable period-1 response; the mean-square period-2 response is calculated. This expansion matches to an inner expansion that describes the rapid change from one form of response behaviour to the other, and this in turn matches to the outer expansion that describes the bifurcated response. Comparisons of the analytic estimates with numerical simulations of the describing equations are excellent in all cases.