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 higherperiod 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.