Stochastic Resonance: A Comparative Study of Two-State Models

Abstract

Abstract. We consider a dynamical system describing the motion of a particle in a double well potential with a periodic perturbation of very small frequency, and an additive stochastic perturbation of amplitude ε. It is in stochastic resonance if the solution trajectories amplify the small periodic perturbation in a 'best possible way'. Systems of this type first appeared in simple energy balance models designed for a qualitative explanation of global glacial cycles. Large deviations theory provides a lower bound for the proportion of the amplitude and the logarithm of the period above which quasi-deterministic periodic behavior can be observed. However, to obtain optimality, one has to measure periodicity with a measure of quality of tuning such as spectral power amplification favored in the physical literature. In a situation where the potential switches discontinuously between two spatially antisymmetric double well states we encounter a surprise. Contrary to physical intuition, the stochastic resonance pattern is not correctly given by the reduced dynamics described by a two state Markov chain with periodic hopping rates between the potential minima which mimic the large (spatial) scale motion of the diffusion. Only if small scale fluctuations inside the potential wells where the diffusion spends most of its time are carefully eliminated, the Markov chain gives the correct picture.

A comparative study of stochastic resonance for a model with two pathways by escape times, linear response, invariant measures and the conditional Kolmogorov-Smirnov Test

“…As mentioned above in leading order the transitions of the diffusion process X t between the wells can be approximated by a two state Markov Chain Y t = ±1 which have been studied [46,47,48,42]. Further comparative studies of the stochastic resonance for the diffusion case X t versus the Markov Chain Y t case were done by Hermann, Imkeller, Pavlyukevich and Peithmann in [49,50,51,52]. A collection of papers on comparative studies between stochastic resonance in diffusion and Markov Chains can be found in the monograph [42].…”

Stochastic Resonance for a Model with Two Pathways

“…As mentioned above in leading order the transitions of the diffusion process X t between the wells can be approximated by a two state Markov Chain Y t = ±1 which have been studied [45,46,47,41]. Further comparative studies of the stochastic resonance for the diffusion case X t versus the Markov Chain Y t case were done by Hermann, Imkeller, Pavlyukevich and Peithmann in [48,49,50,51]. A collection of papers on comparative studies between stochastic resonance in diffusion and Markov Chains can be found in the monograph [41].…”

Equilibrium trading of climate and weather risk and numerical simulation in a Markovian framework

“…The more realistic second one is given by a conceptual bi-stable diffusion model driven by a Brownian motion with a time-periodic potential function which has two minima the depths of which fluctuate periodically. The noise is implemented with intensities at which the solution trajectories show some random periodicity which can be measured by means of quality of periodic tuning notions (see Herrmann and Imkeller 2003;Herrmann et al 2003;Imkeller and Pavlyukevich 2002).…”