Noise-Induced Resonance in Bistable Systems Caused by Delay Feedback

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.

Keywords:

• Delay differential equation
• Effective dynamics
• Markov chain
• Stationary process
• Stochastic differential equation
• Stochastic resonance
• Stochastic synchronization

Mathematics Subject Classification:

• Primary 34K50, 60H10
• Secondary 60G17, 34K11, 34K13, 34K18, 60G10

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 (1) in Tsimring and Pikovsky [12, 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 [12, p. 1] and the fact that pathwise uniqueness holds.

³Equation (1.3) in Redmonda et al. [12, 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 [27, p. 66] for a definition of the integral process w.r.t. a random measure.

¹⁰Cf. also the numerical results in Curtin et al. [12, p. 1].

¹¹The jump height measure corresponds to a measure of resonance proposed by Masoller [12, p. 1].

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

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