Transitions induced by bounded noise

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Abstract

We study noise-induced transitions when the system is driven separately by a Gaussian colored noise and a non-Gaussian bounded noise with the same mean and stationary correlation. Clear differences in the nature of such transitions for these two cases are observed. These transitions are shown to be enhanced essentially by the bounded noisy driving. The results are obtained through a new approach based on numerical analysis of hyperbolic systems.

Introduction

Traditionally, stochastic dynamical systems used in the physical sciences have involved Gaussian noise. In recent times, however, it has been recognized that the assumption of Gaussianity is not appropriate in some cases (see Ref. [1] for a review). The Gaussian noise is unbounded, i.e., there exists a positive chance of having very large values. Strictly speaking, this fact contradicts the very nature of a real physical quantity which is always bounded [2]. Studies of dynamical systems with non-Gaussian continuous noise are much more complicated, especially analytically. Only recently, some results in this area have been obtained, see Refs. [1], [3], [4], [5], [6].

In this paper we consider one of the simplest nonlinear stochastic flows, viz., the so-called genetic model:x˙=12-x+λx(1-x)+x(1-x)ξ(t),which appears in population genetics or in a chemical reaction modeling [7], [8]. Two forms of the multiplicative noise ξ(t) are considered. The first one is the Gaussian process (Ornstein–Uhlenbeck (OU) noise) ξ1(t) with zero mean and the correlationK(t-s)=ξ1(t)ξ1(s)=Dτ-1exp{-τ-1|t-s|},where τ is the correlation time and D denotes the noise intensity. The second one is the process ξ2(t), ξ2(t)=2Dτ-1sin(2τ-1w(t)),with w(t) standing for the standard Wiener process. It is important to note that the noise ξ2(t) is non-Gaussian because absolute values of all its trajectories are bounded by 2Dτ-1. Let us call it the sine-Wiener (SW) noise. Using the well-known properties of the Wiener process and the Euler representation of the sine function one can easily show that ξ2(t)=0,ξ2(t)ξ2(s)=Dτexp-(t-s)τ1-exp-4sτ,ts.Therefore the noises ξ1(t) and ξ2(t) have the same mean and stationary correlation. Moreover, their odd moments are zero. Our aim is to show that despite the above facts the probability density functions (PDFs) for Eq. (1) with OU and SW noises can differ essentially: the stationary PDF in the SW noise case can be bimodal even if the PDF in the OU noise case is unimodal. The transitions between the unimodal and the bimodal stationary PDF are referred to as the noise-induced transitions [7], [9]. In this paper, we obtain the PDFs for Eq. (1) by a new method based on approximations of these functions via solutions of hyperbolic systems. We believe that this approach represents a new technique for studying colored-noise-driven dynamical systems.

Recently, another type of non-Gaussian noise has been used in the study of transitions for genetic model [6]. This interesting noise is a particular case of Markov diffusion process (see Refs. [1], [3], [6], [10] for details). Its stationary PDF is similar to the generalized thermostatistics proposed by Tsallis [11]. Although that noise is phenomenologically richer than the SW noise, its influence on dynamical systems is more difficult to analyze.

Section snippets

Hyperbolic systems: OU noise

Extending the non-Markovian solutions x(t) of Eq. (1) to the pairs {x(t),ξ1(t)} and {x(t),w(t)} we obtain Markov processes whose joint PDFs satisfy Fokker–Planck equations [12]. For example, the time evolution of the joint PDF P(t,x,y) of the process {x(t),ξ1(t)} is given by the two-dimensional Fokker–Planck equationP(t,x,y)t=-x12-x+λx(1-x)+x(1-x)yP+1τy(yP)+Dτ22Py2,with the condition -P(0,x,y)dx=D2πτexp-τy22D.Our interest lies in the system dynamics of x(t) alone but Eq. (3) contains

Hyperbolic systems: SW noise

Now let us consider Eq. (5) with SW noise. Of course q(t,x) is a functional of the Wiener process w(t), and therefore we shall write q(t,x)=q(t,x;w(s)), t0st. To disentangle the correlation ξ(t)q(t,x) we use the following relation:[exp{iγw(t)}q(t,x;w(s))]=exp-γ2t2q(t,x;w(s)+iγs),which follows from the Cameron–Martin formula for the density of the Wiener measure under translation [19], [20]. In what follows, equality (8) will play a role similar to Eq. (6) in the OU noise case. Since sin2

Probability density functions

Hierarchies (7) and (9) cannot be solved analytically in general case. Nevertheless they can be useful in asymptotic analysis. For example, one can obtain asymptotic expansions for the PDF in powers of the correlation time τ using the approach from Ref. [17]. These expansions have both the regular and the boundary layer parts. If the correlation time is not small we analyze the hyperbolic systems which are obtained after the truncation of the hierarchies. No analytical solutions are known in

Acknowledgements

The authors thank O. Hryniv for a revision of the manuscript and ESF for support.

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