Delay induced transitions in an asymmetry bistable system and stochastic resonance

Abstract

The delay Fokker-Planck equation is given for an asymmetry bistable system with correlated Gaussian white noises. The small delay approximation based on the probability density approach is used and the approximate stationary probability density function is obtained. The phenomenon of delay induced transitions is found. When a weak periodic signal is added, the phenomenon of stochastic resonance is investigated. Expression of the signal-to-noise ratio (SNR) is obtained by using the two-state theory. It is shown that the time delay can suppress or promote the stochastic resonance phenomenon. stochastic resonance, time-delayed asymmetry bistable system, correlated noises PACS: 02.50.-r, 05.10.-a, 05.40.-a Effects of time delay on deterministic systems have been a subject of many studies [1-5]. Many helpful analytical and computational methods have been developed for deterministic systems. Recently, stochastic systems with time-delayed feedback have attracted a great deal of attention. The delay Fokker-Planck equation, as an important theoretical tool, was first introduced by Guilouzic et al. [6,7], and the small delay approximation of stationary distributions based on the stochastic delay differential equation approach was studied. Then Frank [8,9] shown that the delay Fokker-Planck equations can also been derived by using Novikov's theorem like the ordinary Fokker-Planck equations and the small delay approximation of stationary distributions based on probability density approach was introduced. The impact of time delays on noise-induced transitions for a Hongler model was studied in [10,11].The phenomenon of stochastic resonance (SR) has been investigated widely since it was first proposed by Benzi [12]. There are many theoretical development for conventional symmetric bistable systems [13][14][15][16][17]. Recently, stochastic resonance in time-delayed bistable systems driven by weak periodic signal was investigated [18]. The effects of time delays on the transient properties of a time-delayed metastable *Corresponding author (email: huiqingzhang@nwpu.edu.cn) system subjected to cross-correlated noises were studied by means of a stochastic simulation method [19].The effects of correlations between additive and multiplicative noise for stochastic systems have been studied by many authors [20][21][22][23][24][25][26]. Most of the previous studies focus on the symmetric case. However, the introduction of asymmetry is necessary [27][28][29]. Wio [30] studied the role of potential asymmetry for stochastic resonance in bistable systems. Li [31] investigated the effects of asymmetry for stochastic resonance induced by multiplicative noise. Stochastic resonance in an asymmetric bistable system driven by correlated noises was studied in [32].In our previous work, we extended the method presented in [8,9] to a class of stochastic time-delayed systems driven by correlated Gaussian white noises [33]. In this paper, the delay Fokker-Planck equations are applied for a time-delayed asymmetry bistable system. Then, approximate st...

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