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

Abstract

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.

Capturing the cascade: a transseries approach to delayed bifurcations

“…As a consequence of the preceding analysis, we are able to derive an approximation for the solution to the logistic equation in the 2-periodic and 4-periodic parameter regimes, which we denote as R2,app(x) and R4,app(x) respectively. Combining ( 16), ( 17), ( 19), (21), and (24), we find that in the 2-periodic parameter regime…”

“…These expansions may now be used to equate powers of ε and obtain the expressions given in (19) and (21).…”

“…In addition to establishing specific results about the slowly-varying logistic map, this study established that delayed bifurcations can play an essential role in the behaviour of discrete systems. Similar methods were used to study delayed bifurcations in more general unimodal maps [20], as well as discrete maps with noise [6,21,22].…”

Capturing the cascade: a transseries approach to delayed bifurcations

“…By substituting the power series (40) into the governing equation ( 46), it can be seen at leading order as η → 0 and τ 1 → 0 that S 1,0 = −S 3,0 , providing one initial condition for the system ( 47)- (50). The second initial condition may be chosen arbitrarily, as this choice may be absorbed into the expression for σ 1 , in the same manner as the constant C in (22). For algebraic convenience, and without loss of generality, we select S 1,0 = 1.…”

“…In addition to establishing specific results about the slowly-varying logistic map, this study established that delayed bifurcations can play an essential role in the behaviour of discrete systems. Similar methods were used to study delayed bifurcations in more general unimodal maps [21], as well as discrete maps with noise [7,22,23].…”