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Original Articles

Noise-Induced Resonance in Bistable Systems Caused by Delay Feedback

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Pages 135-194 | Published online: 15 Feb 2007
 

Abstract

The subject of the present paper is a simplified model for a symmetric bistable system with memory or delay, the reference model, which in the presence of noise exhibits a phenomenon similar to what is known as stochastic resonance. The reference model is given by a one-dimensional parametrized stochastic differential equation with point delay; the basic properties of which we check.

With a view to capturing the effective dynamics and, in particular, the resonance-like behavior of the reference model, we construct a simplified or reduced model, the two-state model, first in discrete time, then in the limit of discrete time tending to continuous time. The main advantage of the reduced model is that it enables us to explicitly calculate the distribution of residence times which in turn can be used to characterize the phenomenon of noise-induced resonance.

Drawing on what has been proposed in the physics literature, we outline a heuristic method for establishing the link between the two-state model and the reference model. The resonance characteristics developed for the reduced model can thus be applied to the original model.

ACKNOWLEDGMENT

This work was partially supported by the DFG research center Matheon (FZT 86) in Berlin and the DFG Sonderforschungsbereich 649 “Economic Risk.”

Notes

1Our notation is slightly different from that of Equation (Equation1) in Tsimring and Pikovsky [Citation12, p. 1]. In particular, their parameter ε, indicating the “strength of the feedback,” corresponds to −β, here.

2Alternatively, we could invoke Theorem 10.2.2 in Stroock and Varadhan [Citation12, p. 1] and the fact that pathwise uniqueness holds.

3Equation (1.3) in Redmonda et al. [Citation12, p. 1] is our standard example with V the quartic potential, where −α corresponds to our parameter β.

4For the probability of a singleton {Z} under a discrete measure ν, we just write ν(Z).

5At the moment, “number of changes of sign” would be a label more precise for 𝒥(Z), but cf. Section 4.

6Recall the tuple notation for elements of S M .

7Under suitable conditions the approximating time series converge in distribution to the (weakly unique) solution of the SDDE.

8Skorokhod convergence in D does not necessarily preserve the number of jumps.

9See Jacod and Shiryaev [Citation27, p. 66] for a definition of the integral process w.r.t. a random measure.

10Cf. also the numerical results in Curtin et al. [Citation12, p. 1].

11The jump height measure corresponds to a measure of resonance proposed by Masoller [Citation12, p. 1].

12Skorokhod spaces for E-valued functions on the infinite interval [0,∞) are defined in Ethier and Kurtz [Citation12, p. 1].

13The sets were defined at the beginning of Section 4.2.

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