Dichotomously switched phase flows

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Abstract:

The general formalism for periodic dichotomous noise on nonpotential flows is considered. This uncorrelated noise process switches suddenly at integer values of period [Formula Presented]. The effect of additive noise of this kind on the planar FitzHugh-Nagumo ordinary differential equations [R. FitzHugh, Biophys. J. 1, 445 (1961); J. Nagumo, S. Arimoto, and Y. Yoshikawa, Proc. IRE 50, 2061 (1962)] is examined. For large [Formula Presented], quasifractal attractors are observed, whereas for the white-noise limit, where [Formula Presented] is small, a Fokker-Planck equation describes the evolution. The magnitude of [Formula Presented] determines the smoothness of the transient evolution and equilibrium density of the system. Typically the stochastic equations give rise to two regions of high density near the stable fixed points of the underlying autonomous system. The stiffness parameter [Formula Presented] in the differential equations determines the fast variable, its associated nullcline, and the resulting flow structure. For small [Formula Presented] the cubic nullcline controls the motion and transitions between the high-density peaks occur along segments of a noisy limit cycle. For large [Formula Presented] the linear nullcline governs the transitions and the peaks are joined by a single band. The statistical behavior of the oscillatory and direct transitions is examined. © 1997 The American Physical Society.