OFFPRINT Additive noise may change the stability of nonlinear systems

The present work studies the effect of additive noise on two high-dimensional systems. The first system under study is two-dimensional, evolves close to the deterministic stability threshold and exhibits an additive noise-induced shift of the control parameter when driving one variable by uncorrelated Gaussian noise. After a detailed analytical and numerical study of this effect, the work further focusses on the extended Swift-Hohenberg equation subjected to global noise, i.e. noise constant in space and uncorrelated in time. This spatial system generalizes the two-dimensional system and thus reveals phase transitions induced by additive global noise. Numerical studies confirm this effect. Further closer investigations reveal that the occurence of the noise-induced shift is subjected to the model nonlinearity and the shifts sign depends on the sign of the nonlinearity prefactors. Copyright c © EPLA, 2008 The study of random fluctuations on real-world systems has attracted much attention in the last decades. Effects of such fluctuations are manifold, e.g. the noise-induced negative mobility in particle systems [1], the improvement of the signal transmission in driven systems (stochastic resonance) [2,3], highly coherent oscillations in noisy excitable systems [4] or the loss of the systems stability induced by noise [5,6]. The interaction of the random fluctuations with the system is specific to the corresponding effect. For instance stochastic resonance involves the direct noisy driving of the systems activity (additive noise) [7,8], while the noise-induced state transitions may occur when the random fluctuations affect the system parameters (multiplicative noise) [9]. Interestingly to our best knowledge no noise-induced change of stability has been found by additive noise, though it may affect the systems stability at the presence of multiplicative noise [10]. The study of additive noise is important from both the theoretical and the experimental point of view. Typically physical systems are subjected to external fluctuations and may be viewed as being embedded into a heat bath of a specific temperature. If both systems operate in a thermodynamical equilibrium, the heat bath represents an external random force to the systems elements, i.e. it represents additive random fluctuations, and the fluctuation-dissipation theorem relates the resulting fluctuation variance of the system to the bath (a)E-mail: [email protected] temperature. Several previous studies have considered such fluctuations observed experimentally in theoretical models by an additive noise term, e.g. in hydrodynamic instabilities [11]. From an experimental point of view, additive random fluctuations may represent the external stimulation applied experimentally to the system similar to external periodic forcing, cf. [12–14]. The current work shows that additive noise may change the stability of a system if the system exhibits specific nonlinear interactions. The noise may advance the instability, i.e. shifts the systems control parameter to larger values, or it may delay it. The major focus of this letter is the noise-advanced instability, while we discuss briefly the delayed case found in a previous work [15]. This letter illustrates the de-stabilization by additive noise for two different systems. At first the study of a twodimensional dynamical system reveals in detail the basic mechanism and the conditions for which the effect occurs. In a second step a high-dimensional dynamical system reflecting the spatio-temporal dynamics of a spatial system also shows the de-stabilizing effect by an additive noiseinduced phase transition. To further clarify the effect in both systems, the two-dimensional system is chosen as an approximation limit of the high-dimensional system. To discuss the proposed mechanism in detail, at first let us study the specific dynamical system dx= (αcx+ γc(xy +x)−μcx )dt, (1) dy= (α0y+ γ0x y)dt+ ηdW (t), (2)