Analysis of noise-induced transitions from regular to chaotic oscillations in the Chen system

Abstract

The stochastically perturbed Chen system is studied within the parameter region which permits both regular and chaotic oscillations. As noise intensity increases and passes some threshold value, noiseinduced hopping between close portions of the stochastic cycle can be observed. Through these transitions, the stochastic cycle is deformed to be a stochastic attractor that looks like chaotic. In this paper for investigation of these transitions, a constructive method based on the stochastic sensitivity function technique with confidence ellipses is suggested and discussed in detail. Analyzing a mutual arrangement of these ellipses, we estimate the threshold noise intensity corresponding to chaotization of the stochastic attractor. Capabilities of this geometric method for detailed analysis of the noiseinduced hopping which generates chaos are demonstrated on the stochastic Chen system. Noise-induced chaos in stochastically perturbed nonlinear deterministic systems with regular attractors has been studied extensively. This phenomenon has received a great deal of interest among researchers from various fields of physics, life sciences, and engineering. Nonlinearity, even in a non-chaotic regime, can imply a non-uniformity of phase portraits, diverse forms of coexicting regular attractors with a complicated geometry of basins of attraction. Under random disturbances, a phase trajectory can cross separatices and induce stochastic hopping between both coexisting regular attractors and their different but close portions. As a consequence of this hopping, the random trajectories with a high probability fall within zones of divergency (local instability), and the dynamics of the perturbed system as a whole becomes chaotic. The positivity of the largest Lyapunov exponent can be used as a measure that this new noise-induced regime is chaotic.In this paper, for the constructive prediction of the noise-induced transitions from regular to chaotic oscillations in a dynamical system, a new stochastic sensitivity function technique is developed and applied. This technique enables to find dispersion ellipses of the random states for the stochastically perturbed dynamical system around deterministic attractors. The sizes of these ellipses are enlarged as noise intensity increases. The constructive method of the dispersion ellipses allows to estimate the threshold of the noise intensity corresponding to the deformation of a regular stochastic attractor to a chaotic one. The effectiveness of this approach is demonstrated by the stochastically perturbed Chen system.

Sea Ice Dynamics Induced by External Stochastic Fluctuations

“…The freezing process from below occurs, in particular, under ice shelves, where freshwater glacial run-off at a temperature very near 0°C accumulates behind the ice shelf until the fresh water flows out beneath the shelf. Another interesting example is that the East Antarctic ice sheet grows, in particular, by freezing from the base and in some places, up to half of the ice thickness freezes from below (BELL et al, 2011). Our model results, in particular, show that these ice growth processes from below will be decelerated by natural fluctuations of the oceanic velocity.…”

“…As it is known in physics, there is a broad variety of natural phenomena and processes connected with different type of noises. So, for example, the stochastic resonance and bifurcations, noise-induced transitions and chaos may be mentioned among others (HORSTHEMKE and LEFEVER, 1984;ANISHCHENKO et al, 2007;BASHKIRTSEVA and RYASHKO, 2009;BASHKIRTSEVA et al, 2012). We demonstrate below how stochastic fluctuations in the main external parameters influence on the sea ice dynamics.…”

Stochastically driven transitions between climate attractors

“…Stochastic sensitivity function technique was successfully applied to the analysis of noise-induced chaos (Bashkirtseva et al, 2012) stochastic bifurcations (Bashkirtseva et al, 2010) and stochastic excitability (Bashkirtseva et al, 2013).…”

Analysis of stochastic model for nonlinear volcanic dynamics

“…It is well-known, that an interplay between nonlinearity and noise can generate various probabilistic phenomena such as noise-induced transitions (Horsthemke and Lefever, 1984), stochastic resonance (McDonnell et al, 2008;Pikovsky and Kurths, 1997;Arathi, 2013), and noiseinduced chaos (Lai and Tél, 2011;Bashkirtseva et al, 2012). Stochastic effects in nonlinear models are the subjects of intensive investigations in various research domains (Horsthemke and Lefever, 1984;Lindner et al, 2004;Alexandrov et al, 2013).…”

“…Without better modeling forecasts of these dynamic processes, the highly important questions of where, when and how volcanic eruptions occur will remain substantially empirical. Nowadays, an elaboration of the adequate mathematical models for volcanic dynamics is a challenging problem (Melnik and Sparks, 1999;Barmin et al, 2002;Nakanishi and Koyaguchi, 2008;Costa et al, 2012).…”