تجزیه و تحلیل حساسیت تصادفی از اینترمتنسی ناشی از نویز و گذار به آشوب در سیستم های زمان گسسته تک بعدی

We study a phenomenon of noise-induced intermittency for the stochastically forced one-dimensional discrete-time system near tangent bifurcation. In a subcritical zone, where the deterministic system has a single stable equilibrium, even small noises generate large-amplitude chaotic oscillations and intermittency. We show that this phenomenon can be explained by a high stochastic sensitivity of this equilibrium. For the analysis of this system, we suggest a constructive method based on stochastic sensitivity functions and confidence intervals technique. An explicit formula for the value of the noise intensity threshold corresponding to the onset of noise-induced intermittency is found. On the basis of our approach, a parametrical diagram of different stochastic regimes of intermittency and asymptotics are given.

Due to the interaction between nonlinearity and stochasticity, noise can induce a number of interesting unexpected phenomena in dynamical systems, such as noise-induced transitions [1] and [2], noise-induced resonance [3], [4] and [5], noise-induced excitement [6], noise-induced order [7] and [8] and chaos [9] and [10]. The transition to chaos is a fundamental and widely studied problem in deterministic nonlinear dynamics. Among the possible routes to chaos is an intermittency route. The system demonstrating intermittent behavior remains for a long duration in some regular regime (laminar state) and at unpredictable moments begins to exhibit chaotic oscillations (turbulent state) before returning to the laminar state. Pomeau and Manneville [11] and [12] have proposed a simple deterministic one-dimensional model and classified three different types of intermittency. These types (I, II and III) correspond to a tangent bifurcation, a subcritical Hopf bifurcation, or an inverse period-doubling bifurcation. A renormalization group approach to analyze type-I intermittency has been used in Refs. [13] and [14]. In this paper, we focus on the study of the noise-induced type-I intermittency phenomenon. An influence of noise on the intermittent behavior of nonlinear dynamical systems has been widely studied [15], [16], [17], [18], [19], [20] and [21]. Frequently, noise-induced intermittency is caused by the multistability of the initial nonlinear deterministic system. Indeed, let the system have coexisting regular (equilibrium or limit cycle) and chaotic attractors. Due to random disturbances, a phase trajectory can cross a separatrix between basins of the attraction and exhibit a new dynamical regime which has no analog in the deterministic case. Random trajectories hopping between basins of coexisting deterministic attractors form a new stochastic attractor. This stochastic attractor joins together two types of dynamics. Trajectories in this attractor exhibit the alternation of phases of noisy regular and noisy chaotic dynamics near initial deterministic attractors and define corresponding type of noise-induced intermittency. However, the multistability is not an obligatory condition of the noise-induced intermittency. The phenomenon of noise-induced intermittency can be observed in the specific dynamical systems with a single stable equilibrium only. For these systems, a basin of attraction of equilibrium can be separated on two zones. If the initial point belongs to the first zone localized near the equilibrium, the system quickly relaxes back into the stable equilibrium. Once the initial point lies in second zone, a large excursion of the trajectory is observed. In this case, the system demonstrates high-amplitude oscillations until the trajectory returns to the first zone. Under the small random disturbances, trajectory of this type system leaves a stable equilibrium and forms some probabilistic distribution around it. This noisy equilibrium is localized in the first zone. Once the noise intensity exceeds a certain threshold, the random trajectory hits at second zone and exhibits long-time noisy oscillations until return to first zone and so on. In such a way the stochastically forced system with super-threshold noise demonstrates noise-induced intermittency. Under the random disturbances, this system is transformed from order to chaos. The standard model with this type noise-induced intermittency is a one-dimensional map in a zone of tangent bifurcation. Similar phenomena when small noises generate large-amplitude oscillations can also be observed in continuous-time systems with a single stable equilibrium. The FitzHugh–Nagumo model is a well known example of such noise-induced excitement [22] and [6]. A probabilistic analysis of the noise-induced phenomena is based on the investigation of corresponding stochastic attractors. A detailed description of stochastic attractors for continuous-time systems is given by the Kolmogorov–Fokker–Planck equation. For discrete-time systems, this description is given by the corresponding integral equation with Frobenius–Perron operator. However, a direct usage of these equations is very difficult even for the simplest cases. To avoid this complexity, various asymptotics and approximations can be considered [23] and [24]. A stochastic sensitivity function (SSF) method has been used for the constructive probabilistic description of stochastic attractors for both continuous [25] and discrete-time [26] systems. The aim of our work is to demonstrate how the SSF technique can be applied to the parametrical analysis of the noise-induced intermittency for discrete-time nonlinear systems. Our general approach is illustrated on the example of the simple one-dimensional model. In Section 2, we introduce this model and discuss phenomena of noise-induced intermittency and noise-induced chaotization in a subcritical zone near the tangent bifurcation. The main results of our paper are shown in Section 3. In Section 3.1, we present a brief theoretical background of the general SSF technique for stochastic equilibria of discrete-time dynamical systems. A constructive description of the dispersion of random states in the stochastic equilibria is given by confidence intervals. The size of the confidence interval is defined by the noise intensity, value of stochastic sensitivity and fiducial probability. In Section 3.2, this technique is applied to the detailed parametrical analysis of noise-induced intermittency for the one-dimensional model introduced in Section 2. Through this study, we find an explicit formula for the value of noise intensity threshold corresponding the onset of noise-induced intermittency and construct a parametrical diagram of different stochastic regimes. In Section 3.3, constructive abilities of our approach for the asymptotic analysis of the noise-induced intermittency in a tangent bifurcation zone for the general one-dimensional systems are demonstrated.

We study an intermittency phenomenon for nonlinear systems forced by the random disturbances. Our paper is focused on the noise-induced intermittency and chaotization observed near tangent bifurcation. Through the study of a simple one-dimensional stochastic system, we present the main probabilistic phenomena and methods of their analysis. The remarkable feature of the dynamics of the model considered here is that small noises generate large-amplitude chaotic oscillations even in the subcritical zone where the deterministic system has a single stable equilibrium. We show that this phenomenon can be explained by the high stochastic sensitivity of this equilibrium. In this paper, for the probabilistic distribution of random states in the stochastic attractor, we use the constructive approximation based on the stochastic sensitivity function technique. On the basis of the SSF technique, we find confidence intervals for these random states and use them in the parametrical analysis of noise-induced intermittency. For a sufficiently small noise, the confidence intervals are localized near the stable equilibrium. As the noise intensity increases, the confidence intervals expand and begin to occupy the zone of large-amplitude oscillations. This occupation means that random trajectories of the forced system with a high probability can exceed the bounds of the unexcited regime and go on a large excursion generating chaotic oscillations. The noise intensity that corresponds to the beginning of this occupation can be used as the estimation of the threshold value. In the present work, we have found an explicit formula for the value of noise intensity threshold corresponding to the onset of noise-induced intermittency and constructed a parametrical diagram of different stochastic regimes. Our method enables to determine the asymptotic of the critical noise intensity as a function of parameters of tangent bifurcation in a general case.