A period-doubling bifurcation with slow parametric variation and additive noise

Slow sinusoidal modulation of a control parameter can maintain a low-period orbit into parameter regions where the low-period orbit is locally unstable, and a higher-period orbit would normally occur. Whether or not a bifurcation to higher period becomes evident during the modulation depends on the competing effects of stabilization by the modulation and destabilization by inherent very low level system noise. A transition, often rapid, from a locally unstable period-1 orbit to period-2, for example, can be triggered by noise. The competing effects are examined here for a period-doubling bifurcation of a general unimodal map. A nested set of three matched asymptotic expansions (a triple-deck) is used to describe the combined period-1 and period-2 response. The resulting solution gives estimates of whether and where an apparent period-doubling bifurcation occurs. Typical period-1 stability boundaries are obtained that include the effect of the amplitude and frequency of the variation, the noise level in the system, and the allowable maximum threshold level of period-2 response.