Correlation times in stochastic equations with delayed feedback and multiplicative noise

Abstract

We obtain the characteristic correlation time associated with a model stochastic differential equation that includes the normal form of a pitchfork bifurcation and delayed feedback. In particular, the validity of the common assumption of statistical independence between the state at time t and that at t − τ , where τ is the delay time, is examined. We find that the correlation time diverges at the model's bifurcation line, thus signaling a sharp bifurcation threshold, and the failure of statistical independence near threshold. We determine the correlation time both by numerical integration of the governing equation, and analytically in the limit of small τ . The correlation time T diverges as T ∼ a −1 , where a is the control parameter so that a = 0 is the bifurcation threshold. The small-τ expansion correctly predicts the location of the bifurcation threshold, but there are systematic deviations in the magnitude of the correlation time.

Activation process in excitable systems with multiple noise sources: Large number of units

“…At some point, the increase of noise induces more or less coherent oscillations of units which are easy to synchronize. This conforms to the onset of the collective mode according to the scenario of stochastic bifurcation [22][23][24][25]. In the supercritical state, the global variables follow a limit cycle attractor whose profile is similar to that of relaxation oscillations of individual units.…”

The transition between strong and weak chaos in delay systems: Stochastic modeling approach

“…For the stochastic delay system there is no corresponding theory to the well-known Fokker-Planck formalism. Existing works on that field either focus on the small-delay limit or provide techniques which do not retrace the phenomena in which we are interested [12][13][14][15]. One can regard Eq.…”

“…we study the chaotic scaling laws systematically. The maximum LE and the sub-LE of the map (14) are calculated as described in the previous section. Figure 1 shows the maximum LE and the sub-LE as a function of the feedback strength k. For k = 0 the two exponents coincide, λ = λ 0 = ln (2).…”

Activation process in excitable systems with multiple noise sources: One and two interacting units

“…What we postulate is that the transition between the domains (i) and (ii) may qualitatively be accounted for by the fact that the excitable unit undergoes stochastic bifurcation induced by D 1 and D 2 . Note that the phenomenological stochastic bifurcation [32][33][34][35] we refer to corresponds to the noise-induced transition from the stochastically stable fixed point (stationary probability distribution P (x, y) focused around the fixed point) to the stochastically stable limit cycle (stationary probability distribution P (x, y) showing non-negligible contribution for (x, y) values along the spiking and the refractory branches of the xnullcline). Intuitively, one understands that the fixed point can be considered stochastically stable if the amplitude of fluctuations around the fixed point is of the order of noise intensity.…”