Additive noise in noise-induced nonequilibrium transitions

Noisy-flow-induced instability in a reaction-diffusion system

We consider a generic reaction-diffusion-advection system where the flow velocity of the advection term is subjected to dichotomous noise with zero mean and Ornstein-Zernike correlation. A general condition for noisy-flow-induced instability is derived in the flow velocity-correlation rate parameter plane. Full numerical simulations on Gierer-Meinhardt model with activator-inhibitor kinetics have been performed to show how noisy differential flow can lead to symmetry breaking of a homogeneous stable state in the presence of noise resulting in traveling waves.

Towards a critical transition theory under different temporal scales and noise strengths

The mechanism of critical phenomena or critical transitions has been recently studied from various aspects, in particular considering slow parameter change and small noise. In this article, we systematically classify critical transitions into three types based on temporal scales and noise strengths of dynamical systems. Specifically, the classification is made by comparing three important time scales τ(λ), τ(tran), and τ(ergo), where τ(λ) is the time scale of parameter change (e.g., the change of environment), τ(tran) is the time scale when a particle or state transits from a metastable state into another, and τ(ergo) is the time scale when the system becomes ergodic. According to the time scales, we classify the critical transition behaviors as three types, i.e., state transition, basin transition, and distribution transition. Moreover, for each type of transition, there are two cases, i.e., single-trajectory transition and multitrajectory ensemble transition, which correspond to the transition of individual behavior and population behavior, respectively. We also define the critical point for each type of critical transition, derive several properties, and further propose the indicators for predicting critical transitions with numerical simulations. In addition, we show that the noise-to-signal ratio is effective to make the classification of critical transitions for real systems.

Dichotomous-noise-induced pattern formation in a reaction-diffusion system

We consider a generic reaction-diffusion system in which one of the parameters is subjected to dichotomous noise by controlling the flow of one of the reacting species in a continuous-flow-stirred-tank reactor (CSTR) -membrane reactor. The linear stability analysis in an extended phase space is carried out by invoking Furutzu-Novikov procedure for exponentially correlated multiplicative noise to derive the instability condition in the plane of the noise parameters (correlation time and strength of the noise). We demonstrate that depending on the correlation time an optimal strength of noise governs the self-organization. Our theoretical analysis is corroborated by numerical simulations on pattern formation in a chlorine-dioxide-iodine-malonic acid reaction-diffusion system.

Diffusion process with two reflecting barriers in a time-dependent potential

We consider a Brownian particle which is driven by a harmonically oscillating force, the motion of which is restricted by two reflecting boundaries. We solve the Fokker-Planck equation using the finite-element method and focus on the dynamics of the mean position of the particle in the time-asymptotic regime. As a function of the strength of the external force, the response of the system, i.e., the amplitude of the mean position and the dynamical shift, in the stationary limit shows a resonancelike behavior as a function of the diffusion coefficient for certain parameter regimes. We explain these numerical results heuristically and give some qualitative estimates.

Resonant-pattern formation induced by additive noise in periodically forced reaction-diffusion systems

We report frequency-locked resonant patterns induced by additive noise in periodically forced reaction-diffusion Brusselator model. In the regime of 2:1 frequency-locking and homogeneous oscillation, the introduction of additive noise, which is colored in time and white in space, generates and sustains resonant patterns of hexagons, stripes, and labyrinths which oscillate at half of the forcing frequency. Both the noise strength and the correlation time control the pattern formation. The system transits from homogeneous to hexagons, stripes, and to labyrinths successively as the noise strength is adjusted. Good frequency-locked patterns are only sustained by the colored noise and a finite time correlation is necessary. At the limit of white noise with zero temporal correlation, irregular patterns which are only nearly resonant come out as the noise strength is adjusted. The phenomenon induced by colored noise in the forced reaction-diffusion system is demonstrated to correspond to noise-induced Turing instability in the corresponding forced complex Ginzburg-Landau equation.

Fokker-Planck analysis of stochastic coherence in models of an excitable neuron with noise in both fast and slow dynamics

We provide a detailed and quantitative Fokker-Planck analysis of noise-induced periodicity (stochastic coherence, also known as coherence resonance) in both a discrete-time model and a continuous-time model of excitable neurons. In particular, we show that one-dimensional models can explain why the effects of noise added to the fast and slow dynamics of the models are dramatically different. We argue that such effects should occur in any excitable system with two or more distinct time scales and need to be taken into account in experiments investigating stochastic coherence.

Noise-induced instability: an approach based on higher-order moments

Noise-induced transitions in the organization of systems far from equilibrium have been of vital interest. Although the effects of additive and multiplicative noise have been widely studied, it is only the multiplicative noise that can be dealt with within the scope of a linear analysis of first moments of the spatiotemporal perturbations, by the application of Novikov's theorem. For the case of additive noise, the corresponding straightforward linear analysis of the first moment throws no light on the effect of the noise on stability conditions. We propose here a simple approach based on higher-order moments to show how additive noise can give rise to noise-induced instability in spatially extended systems, at times leading to pattern formation. Our theoretical analysis is corroborated by numerical simulations on two simple one-component reaction-diffusion systems in two dimensions.

Nonequilibrium transitions induced by the cross-correlation of white noises

We study the role that the cross-correlation of noises plays in the statistical behavior of systems driven by two multiplicative Gaussian white noises. The temporal evolution of the system is described by a Langevin equation, for which we adopt a general interpretation that includes the Ito as well as the Stratonovich interpretation. We derive the stochastically equivalent Fokker-Planck equation by means of the two-stage averaging of a state-dependent function. Analyzing the stationary solution of the Fokker-Planck equation for specific examples, we show explicitly that the cross-correlation of white noises can induce nonequilibrium transitions.

Limit cycle induced by multiplicative noise in a system of coupled Brownian motors

We study a model consisting of N nonlinear oscillators with global periodic coupling, and local multiplicative and additive noises. The model was shown to undergo a nonequilibrium phase transition towards a broken-symmetry phase exhibiting noise-induced "ratchet" behavior. A previous study [H. S. Wio, S. Mangioni, and R. Deza, Physica D 168-169, 184 (2002)] focused on the relationship between the character of the hysteresis loop, the number of "homogeneous" mean-field solutions, and the shape of the stationary mean-field probability distribution function. Here, we show-as suggested by the absence of stable solutions when the load force is beyond a critical value-the existence of a limit cycle induced by both multiplicative noise and global periodic coupling.