非线性动力系统分岔点邻域内随机共振的特性

研究了叉形分岔系统和FitzHugh Nagumo(FHN)细胞模型两种非线性动力系统分岔点邻域内 随机共振的特性.研究结果表明：这两种系统在分岔发生时具有由一个吸引子变为两个吸引 子或者由两个吸引子变为一个吸引子共同的分岔特性，即在分岔点的邻域内, 系统在分岔点 的两侧有分岔前吸引子和分岔后吸引子存在，在噪声的作用下，系统的运动除了像传统随机 共振的机理那样在分岔点一侧共存的吸引子之间跃迁，还在分岔点两侧三个吸引子（分岔前 一个吸引子和分岔后两个吸引子）之间跃迁，并且这种跃迁单独诱发了随机共振 ；在两种 跃迁都发生的情况下, 在其分岔点的邻域内，由第二种跃迁诱发的随机共振在引起第一种跃 迁噪声的强度很大的范围内变化仍可维持, 而第一种跃迁诱发的随机共振在引起第二种跃迁 噪声的强度很小的范围内变化即迅速消失. 关键词： 随机共振 吸引子 分岔点 跃迁     

Strong perturbations in nonlinear systems

A novel case of probabilistic coupling for hybrid stochastic systems with chaotic components via Markovian switching is presented. We study its stability in the norm, in the sense of Lyapunov and present a quantitative scheme for detection of stochastic stability in the mean. In particular we examine the stability of chaotic dynamical systems in which a representative parameter undergoes a Markovian switching between two values corresponding to two qualitatively different attractors. To this end we employ, as case studies, the behaviour of two representative chaotic systems (the classic Rössler and the Thomas-Rössler models) under the influence of a probabilistic switch which modifies stochastically their parameters. A quantitative measure, based on a Lyapunov function, is proposed which detects regular or irregular motion and regimes of stability. In connection to biologically inspired models (Thomas-Rössler models), where strong fluctuations represent qualitative structural changes, we observe the appearance of stochastic resonance-like phenomena i.e. transitions that lead to orderly behavior when the noise increases. These are attributed to the nonlinear response of the system.     

Noise-induced extinction in Bazykin-Berezovskaya population model

A nonlinear Bazykin-Berezovskaya prey-predator model under the influence of parametric stochastic forcing is considered. Due to Allee effect, this conceptual population model even in the deterministic case demonstrates both local and global bifurcations with the change of predator mortality. It is shown that random noise can transform system dynamics from the regime of coexistence, in equilibrium or periodic modes, to the extinction of both species. Geometry of attractors and separatrices, dividing basins of attraction, plays an important role in understanding the probabilistic mechanisms of these stochastic phenomena. Parametric analysis of noise-induced extinction is carried out on the base of the direct numerical simulation and new analytical stochastic sensitivity functions technique taking into account the arrangement of attractors and separatrices.     

Sensitivity analysis of stochastic attractors and noise-induced transitions for population model with Allee effect

We study a stochastically forced predator-prey model with Allee effect. In the deterministic case, this model exhibits non-trivial stable equilibrium or limit cycle corresponding to the coexistence of both species. Computational methods based on the stochastic sensitivity functions technique are suggested for the analysis of the dispersion of random states in stochastic attractors. Our method allows to construct confidence domains and estimate the threshold value of the intensity for noise generating a transition from the coexistence to the extinction.     

Stochastic sensitivity analysis of the attractors for the randomly forced Ricker model with delay

Stochastically forced regular attractors (equilibria, cycles, closed invariant curves) of the discrete-time nonlinear systems are studied. For the analysis of noisy attractors, a unified approach based on the stochastic sensitivity function technique is suggested and discussed. Potentialities of the elaborated theory are demonstrated in the parametric analysis of the stochastic Ricker model with delay nearby Neimark–Sacker bifurcation.     

Stochastic dynamical analysis of a kind of vibro-impact system under multiple harmonic and random excitations

There have been many contributions concerned with non-smooth dynamics. The purpose of this study is focused on the global stochastic dynamics of a kind of vibro-impact oscillator under the multiple harmonic and bounded noisy excitations. The well-known cell-to-cell mapping method is firstly developed to investigate the incursive fractal boundaries between the attracting domains of different random attractors, and a specific Poincaré map is then set up to explore the noise-contaminated dynamical transitions in the system. Lastly, the leading Lyapunov exponents and the surrogate tests are used to identify the noise-contaminated dynamics. It is shown that several random attractors will coexist in the phase space of the randomly driven system by adjusting the parameters’ values, and fractal boundaries may also arise between the attracting domains of different random attractors. Under the joint action of the harmonic excitation and the weak bounded noise excitation, the noisy period-doubling process, similar to a deterministic one, can appear in the Poincaré’s global cross-section by increasing the strength of the bounded noisy excitation. Moreover, the noisy periodic, the noisy chaotic, and the random-dominant dynamics are also distinguished from the noise-contaminated signals.     

Gibbsian Dynamics and Invariant Measures for Stochastic Dissipative PDEs

We present a general strategy for proving ergodicity for stochastically forced nonlinear dissipative PDEs. It consists of two main steps. The first step is the reduction to a finite dimensional Gibbsian dynamics of the low modes. The second step is to prove the equivalence between measures induced by different past histories using Girsanov theorem. As applications, we prove ergodicity for Ginzburg–Landau, Kuramoto–Sivashinsky and Cahn–Hilliard equations with stochastic forcing.     

Generalizations of SRB Measures to Nonautonomous,Random, and Infinite Dimensional Systems

We review some developments that are direct outgrowths of, or closely related to, the idea of SRB measures as introduced by Sinai, Ruelle and Bowen in the 1970s. These new directions of research include the emergence of strange attractors in periodically forced dynamical systems, random attractors in systems defined by stochastic differential equations, SRB measures for infinite dimensional systems including those defined by large classes of dissipative PDEs, quasi-static distributions for slowly varying time-dependent systems, and surviving distributions in leaky dynamical systems.     

Unstable attractors induce perpetual synchronization and desynchronization

Common experience suggests that attracting invariant sets in nonlinear dynamical systems are generally stable. Contrary to this intuition, we present a dynamical system, a network of pulse-coupled oscillators, in which unstable attractors arise naturally. From random initial conditions, groups of synchronized oscillators (clusters) are formed that send pulses alternately, resulting in a periodic dynamics of the network. Under the influence of arbitrarily weak noise, this synchronization is followed by a desynchronization of clusters, a phenomenon induced by attractors that are unstable. Perpetual synchronization and desynchronization lead to a switching among attractors. This is explained by the geometrical fact, that these unstable attractors are surrounded by basins of attraction of other attractors, whereas the full measure of their own basin is located remote from the attractor. Unstable attractors do not only exist in these systems, but moreover dominate the dynamics for large networks and a wide range of parameters.     

Stochastic Attractors for Shell Phenomenological Models of Turbulence

Recently, it has been proposed that the Navier–Stokes equations and a relevant linear advection model have the same long-time statistical properties, in particular, they have the same scaling exponents of their structure functions. This assertion has been investigate rigorously in the context of certain nonlinear deterministic phenomenological shell model, the Sabra shell model, of turbulence and its corresponding linear advection counterpart model. This relationship has been established through a “homotopy-like” coefficient λ which bridges continuously between the two systems. That is, for λ=1 one obtains the full nonlinear model, and the corresponding linear advection model is achieved for λ=0. In this paper, we investigate the validity of this assertion for certain stochastic phenomenological shell models of turbulence driven by an additive noise. We prove the continuous dependence of the solutions with respect to the parameter λ. Moreover, we show the existence of a finite-dimensional random attractor for each value of λ and establish the upper semicontinuity property of this random attractors, with respect to the parameter λ. This property is proved by a pathwise argument. Our study aims toward the development of basic results and techniques that may contribute to the understanding of the relation between the long-time statistical properties of the nonlinear and linear models.     

A priori tests of a stochastic mode reduction strategy

Several a priori tests of a systematic stochastic mode reduction procedure recently devised by the authors [Proc. Natl. Acad. Sci. 96 (1999) 14687; Commun. Pure Appl. Math. 54 (2001) 891] are developed here. In this procedure, reduced stochastic equations for a smaller collections of resolved variables are derived systematically for complex nonlinear systems with many degrees of freedom and a large collection of unresolved variables. While the above approach is mathematically rigorous in the limit when the ratio of correlation times between the resolved and the unresolved variables is arbitrary small, it is shown here on a systematic hierarchy of models that this ratio can be surprisingly big. Typically, the systematic reduced stochastic modeling yields quantitatively realistic dynamics for ratios as large as 1/2. The examples studied here vary from instructive stochastic triad models to prototype complex systems with many degrees of freedom utilizing the truncated Burgers–Hopf equations as a nonlinear heat bath. Systematic quantitative tests for the stochastic modeling procedure are developed here which involve the stationary distribution and the two-time correlations for the second and fourth moments including the resolved variables and the energy in the resolved variables. In an important illustrative example presented here, the nonlinear original system involves 102 degrees of freedom and the reduced stochastic model predicted by the theory for two resolved variables involves both nonlinear interaction and multiplicative noises. Even for large value of the correlation time ratio of the order of 1/2, the reduced stochastic model with two degrees of freedom captures the essentially nonlinear and non-Gaussian statistics of the original nonlinear systems with 102 modes extremely well. Furthermore, it is shown here that the standard regression fitting of the second-order correlations alone fails to reproduce the nonlinear stochastic dynamics in this example.     

Physical models of cognition

This paper presents and discusses physical models for simulating some aspects of neural intelligence, and, in particular, the process of cognition. The main departure from the classical approach here is in utilization of a terminal version of classical dynamics introduced by the author earlier. Based upon violations of the Lipschitz condition at equilibrium points, terminal dynamics attains two new fundamental properties: it is spontaneous and nondeterministic. Special attention is focused on terminal neurodynamics as a particular architecture of terminal dynamics which is suitable for modeling of information flows. Terminal neurodynamics possesses a well-organized probabilistic structure which can be analytically predicted, prescribed, and controlled, and therefore which presents a powerful tool for modeling real-life uncertainties. Two basic phenomena associated with random behavior of neurodynamic solutions are exploited. The first one is a stochastic attractor—a stable stationary stochastic process to which random solutions of a closed system converge. As a model of the cognition process, a stochastic attractor can be viewed as a universal tool for generalization and formation of classes of patterns. The concept of stochastic attractor is applied to model a collective brain paradigm explaining coordination between simple units of intelligence which perform a collective task without direct exchange of information. The second fundamental phenomenon discussed is terminal chaos which occurs in open systems. Applications of terminal chaos to information fusion as well as to explanation and modeling of coordination among neurons in biological systems are discussed. It should be emphasized that all the models of terminal neurodynamics are implementable in analog devices, which means that all the cognition processes discussed in the paper are reducible to the laws of Newtonian mechanics.     

永磁同步风力发电机随机分岔现象的全局分析

永磁同步风力发电机在运行过程中不可避免地会受到风能的随机干扰,本文建立了在输入机械转矩存在随机干扰情况下永磁同步风力发电机的数学模型,采用胞映射方法分析了随机干扰强度变化时系统全局结构的演化行为,并通过数值模拟对理论分析进行验证.研究结果表明,随着随机干扰强度的增大,系统中会出现随机内部激变和随机边界激变,即由于随机吸引子与其吸引域内的随机鞍发生碰撞而产生的随机分岔现象和由于随机吸引子与其吸引域边界发生碰撞而产生的随机分岔现象.研究结果揭示了随机干扰对永磁同步风力发电机运行性能影响的机理,为永磁同步风力发电系统的运行和设计提供了理论依据.     

Chaotically spiking canards in an excitable system with 2D inertial fast manifolds

We introduce a new class of excitable systems with two-dimensional fast dynamics that includes inertia. A novel transition from excitability to relaxation oscillations is discovered where the usual Hopf bifurcation is followed by a cascade of period doubled and chaotic small excitable attractors and, as they grow, by a new type of canard explosion where a small chaotic background erratically but deterministically triggers excitable spikes. This scenario is also found in a model for a nonlinear Fabry-Perot cavity with one pendular mirror.     

Ergodicity for the Dissipative Boussinesq Equations with Random Forcing

We study the stationary measure for the two-dimensional Boussinesq equation with random forcing. We prove the ergodicity for the two-dimensional stochastically forced Boussinesq equation. We also study the Galerkin truncations of the three-dimensional Boussinesq equations under degenerate stochastic forcing. We follow closely the previous results on the stochastically forced Navier–Stokes equations.     

Attractor selection in chaotic dynamics

For different settings of a control parameter, a chaotic system can go from a region with two separate stable attractors (generalized bistability) to a crisis where a chaotic attractor expands, colliding with an unstable orbit. In the bistable regime jumps between independent attractors are mediated by external perturbations; above the crisis, the dynamics includes visits to regions formerly belonging to the unstable orbits and this appears as random bursts of amplitude jumps. We introduce a control method which suppresses the jumps in both cases by filtering the specific frequency content of one of the two dynamical objects. The method is tested both in a model and in a real experiment with a CO2 laser.     

Complex behavior of simple maps with fluctuating delay times

Delay systems used to model retarded actions are relevant in many fields such as optics, mechanical machining, biology or physiology. A frequently encountered situation is that the length of the delay time changes with time. In this study, we use a simple map system to investigate the influence of the fluctuating delay time on the system dynamics. For simplicity, we start from a case with the delay time taking only the value of zero or one discrete time steps, where the system dynamics reduces to one- and two-dimensional map, respectively. We study two situations, periodic or random variation of the delay. Rich dynamics including coexisting multiple attractors, strange nonchaotic attractors and on-off intermittency are observed. For a special case we can show analytically the existence of a dense set of singularities of the Lyapunov exponent. Finally we present results for higher dimensional generalizations to show the relevance of our findings to more general situations.     

Vulnerability to paroxysmal oscillations in delayed neural networks: a basis for nocturnal frontal lobe epilepsy?

Resonance can occur in bistable dynamical systems due to the interplay between noise and delay (τ) in the absence of a periodic input. We investigate resonance in a two-neuron model with mutual time-delayed inhibitory feedback. For appropriate choices of the parameters and inputs three fixed-point attractors co-exist: two are stable and one is unstable. In the absence of noise, delay-induced transient oscillations (referred to herein as DITOs) arise whenever the initial function is tuned sufficiently close to the unstable fixed-point. In the presence of noisy perturbations, DITOs arise spontaneously. Since the correlation time for the stationary dynamics is ～τ, we approximated a higher order Markov process by a three-state Markov chain model by rescaling time as t?→?2sτ, identifying the states based on whether the sub-intervals were completely confined to one basin of attraction (the two stable attractors) or straddled the separatrix, and then determining the transition probability matrix empirically. The resultant Markov chain model captured the switching behaviors including the statistical properties of the DITOs. Our observations indicate that time-delayed and noisy bistable dynamical systems are prone to generate DITOs as switches between the two attractors occur. Bistable systems arise transiently in situations when one attractor is gradually replaced by another. This may explain, for example, why seizures in certain epileptic syndromes tend to occur as sleep stages change.     

Self-organization of temporal structures in a multiequilibrium self-excited oscillator system with frequency control

An analysis is made of the nonlinear dynamics of a model of a self-excited oscillator system with automatic fine tuning of the frequency and possessing more than one equilibrium state. It is shown that, depending on the delay parameters of the control loop and the initial frequency detuning, various periodic and stochastic temporal structures may be formed, accompanied by the generation of various limit cycles and chaotic attractors in phase space. Reasons for the onset of self-modulation are set forth, and the position of the different oscillation regions is established. The main bifurcations are studied together with scenarios for the transformation of the oscillator self-modulation modes as a function of the parameters. Zh. Tekh. Fiz. 67, 1–8 (March 1997)