Global stability of synchronization between delay-differential systems with generalized diffusive coupling

Synchronization between general delay feedback systems coupled by generalized diffusive coupling with delay is studied. It is shown that the generalized diffusive coupling synchronizes the units much more effectively than the simple diffusive coupling. Sufficient conditions for the global stability of synchronization for systems of a quite general form are obtained.     

Diagnostics of the degree of noise influence on a nonlinear system using relative metric entropy

In this paper we summarize and substantiate the relative metric entropy approach introduced in our previous papers [1, 2]. Using this approach we study the mixing influence of noise on both regular and chaotic systems. We show that the synchronization phenomenon as well as stochastic resonance decrease, the degree of mixing is caused by white Gaussian noise.     

Chaotic synchronization of nearest-neighbor diffusive coupling Hindmarsh–Rose neural networks in noisy environments

The chaotic synchronization of Hindmarsh–Rose neural networks linked by a nonlinear coupling function is discussed. The HR neural networks with nearest-neighbor diffusive coupling form are treated as numerical examples. By the construction of a special nonlinear-coupled term, the chaotic system is coupled symmetrically. For three and four neurons network, a certain region of coupling strength corresponding to full synchronization is given, and the effect of network structure and noise position are analyzed. For five and more neurons network, the full synchronization is very difficult to realize. All the results have been proved by the calculation of the maximum conditional Lyapunov exponent.     

Random attractors of stochastic partial differential equations: A smooth approximation approach

We use the method of smooth approximation to examine the random attractor for two classes of stochastic partial differential equations (SPDEs). Roughly speaking, we perturb the SPDEs by a Wong-Zakai scheme using smooth colored noise approximation rather than the usual polygonal approximation. After establishing the existence of the random attractor of the perturbed system, we prove that when the colored noise tends to the white noise, the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.     

Stochastic Stability of Some Mechanical Systems

We discuss the behavior, for large values of time, of two linear stochastic mechanical systems. The systems are similar mathematically in that they contain a white noise in their parameters. The initial data may be random as well but are independent of white noise. The expected energy is calculated in both cases. It is well known that for free nonstochastic mechanical systems with viscous damping, the energy approaches zero as time increases. We check that this behavior takes place for the stochastic systems under consideration in the case when the initial data are random but the parameters are not. When the parameters contain a random noise the expected energy may be infinite, approach zero, remain bounded, or increase with no bound. This regime is similar to but more interesting than the known regime for the solutions of differential equations with time dependent periodic coefficients that describes the behavior of a mechanical system with characteristics that are periodic functions of time. We give necessary and sufficient conditions for stability of both systems in terms of the structure of the set of roots of an auxiliary equation.     

Stochastic population dynamics under regime switching II

This is a continuation of our paper [Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl. 334 (2007) 69-84] on stochastic population dynamics under regime switching. In this paper we still take both white and color environmental noise into account. We show that a sufficient large white noise may make the underlying population extinct while for a relatively small noise we give both asymptotically upper and lower bound for the underlying population. In some special but important situations we precisely describe the limit of the average in time of the population.     

Efficient synchronization of structurally adaptive coupled Hindmarsh–Rose neurons

The use of spikes to carry information between brain areas implies complete or partial synchronization of the neurons involved. The degree of synchronization reached by two coupled systems and the energy cost of maintaining their synchronized behavior is highly dependent on the nature of the systems. For non-identical systems the maintenance of a synchronized regime is energetically a costly process. In this work, we study conditions under which two non-identical electrically coupled neurons can reach an efficient regime of synchronization at low energy cost. We show that the energy consumption required to keep the synchronized regime can be spontaneously reduced if the receiving neuron has adaptive mechanisms able to bring its biological parameters closer in value to the corresponding ones in the sending neuron.     

时变耦合变时滞非自治神经网络模型的全局同步性分析

利用Lyapunov泛函方法,对一类时变线性耦合神经网络模型的全局同步性进行了研究．在去掉耦合矩阵的对称性、不可约性和扩散耦合限制的基础上,得到了确保耦合时滞神经网络模型全局同步的充分性条件．所得结果仅依赖于系统中的参数,条件易于验证且不必求矩阵的特征值．     

Antiphase synchronization of chaos by noncontinuous coupling: two impacting oscillators

We show that two identical chaotic oscillators can evolve in antiphase synchronization regime when noncontinuous coupling between them is introduced. As an example, we consider dynamics of two mechanical oscillators coupled by impacts.     

Positive role of multiplication noise in attaining complete synchronization on large complex networks of dynamical systems

In this paper, the role of multiplicative noise in attaining complete synchronization on large complex networks of dynamical systems is investigated by theoretical analysis and numerical simulations. Based on the stability theory of stochastic differential equation, we prove that the multiplicative noise plays a positive role in attaining synchronization if the complex networks of dynamical systems are bounded. Moreover, the theoretical result shows that smaller second eigenvalue of coupling matrix is of benefit in attaining complete synchronization. To demonstrate the correctness of theoretical results, the coupled Lorenz systems, Hindmarsh–Rose neuronal systems and Rössler-like systems are performed as numerical examples.     

Stability,coherent spiking and synchronization in noisy excitable systems with coupling and internal delays

We study the onset and the adjustment of different oscillatory modes in a system of excitable units subjected to two forms of noise and delays cast as external or internal according to whether they are associated with inter- or intra-unit activity. Conditions for stability of a single unit are derived in case of the linearized perturbed system, whereas the interplay of noise and internal delay in shaping the oscillatory motion is analyzed by the method of statistical linearization. It is demonstrated that the internal delay, as well as its coaction with external noise, drive the unit away from the bifurcation controlled by the excitability parameter. For the pair of interacting units, it is shown that the external/internal character of noise primarily influences frequency synchronization and the competition between the noise-induced and delay-driven oscillatory modes, while coherence of firing and phase synchronization substantially depend on internal delay. Some of the important effects include: (i) loss of frequency synchronization under external noise; (ii) existence of characteristic regimes of entrainment, where under variation of coupling delay, the optimized unit (noise intensity fixed at resonant value) may be controlled by the adjustable unit (variable noise) and vice versa, or both units may become adjusted to coupling delay; (iii) phase synchronization achieved both for noise-induced and delay-driven modes.     

Signal-to-noise ratio gain by stochastic resonance in a bistable system

It was shown recently that the signal-to-noise ratio (SNR) could be improved by stochastic resonance (SR) in certain monostable systems and certain systems with monotonous nonlinearity working in the nonlinear response (NLR) regime. Here we demonstrate that a simple bistable system driven by periodic pulse train and band-limited Gaussian white noise can also provide an SNR gain. We show the reasons why SNR gain had not been found in double well potential systems, despite strong efforts made during the last 15 years.     

Stochastic SIRS model under regime switching

We propose a new stochastic SIRS model under regime switching in which both white and color environmental noises are taken into account. We show that white noise suppresses explosions in the model and the disease-free equilibrium of the model is stochastically asymptotically stable under certain conditions. Moreover, we show that the model is stochastically ultimately bounded and the moment average in time of the model solution is bounded. An example illustrates the boundedness of the moment average in time of the model solution.     

Generalized synchronization in discrete maps. New point of view on weak and strong synchronization

In the present Letter we show that the concept of the generalized synchronization regime in discrete maps needs refining in the same way as it has been done for the flow systems Koronovskii et al. [Koronovskii AA, Moskalenko OI, Hramov AE. Nearest neighbors, phase tubes, and generalized synchronization. Phys Rev E 2011;84:037201]. We have shown that, in the general case, when the relationship between state vectors of the interacting chaotic maps are considered, the prehistory must be taken into account. We extend the phase tube approach to the systems with a discrete time coupled both unidirectionally and mutually and analyze the essence of the generalized synchronization by means of this technique. Obtained results show that the division of the generalized synchronization into the weak and the strong ones also must be reconsidered. Unidirectionally coupled logistic maps and Hénon maps coupled mutually are used as sample systems.     

Stochastic population dynamics under regime switching

In this paper we will develop a new stochastic population model under regime switching. Our model takes both white and color environmental noises into account. We will show that the white noise suppresses explosions in population dynamics. Moreover, from the point of population dynamics, our new model has more desired properties than some existing stochastic population models. In particular, we show that our model is stochastically ultimately bounded.     

Cooperation of deterministic and stochastic mechanisms resulting in the intermittent behavior

Intermittent behavior near the boundary of chaotic phase synchronization in the presence of noise (when deterministic and stochastic mechanisms resulting in intermittency take place simultaneously) is studied. The noise of small intensity is shown to do not affect on the characteristics of intermittency whereas the noise of large amplitude induces new effects near the boundary of the synchronous regime. In the first case the eyelet intermittency takes place near the boundary of the synchronous regime, in the second one the ring intermittency or coexistence of both types of intermittency is realized. Main results are illustrated using the example of two unidirectionally coupled Rössler systems. Similar effects are shown to be observed in coupled spatially distributed Pierce beam–plasma systems.     

Sampled-data nonlinear observer design for chaos synchronization: A Lyapunov-based approach

This paper considers sampled-data based chaos synchronization using observers in the presence of measurement noise for a large class of chaotic systems. We study discretized model of chaotic systems which are perturbed by white noise and employ Lyapunov-like theorems to come up with a simple yet effective observer design. For the choice of observer gain, a suboptimal criterion is obtained in terms of LMI. We present semiglobal as well as global results. The proposed scheme can also be extended for discrete-time chaotic systems. Numerical simulations have been carried out to verify the effectiveness of theoretical results.     

Random dynamics and limiting behaviors for 3D globally modified Navier–Stokes equations driven by colored noise

This paper is mainly concerned with the long-term random dynamics for the nonautonomous 3D globally modified Navier–Stokes equations with nonlinear colored noise. We first prove the existence of random attractors of the nonautonomous random dynamical system generated by the solution operators of such equations. Then we establish the existence of invariant measures supported on the random attractors of the underlying system. Random Liouville-type theorem is also derived for such invariant measures. Moreover, we further investigate the limiting relationship of invariant measures between the above equations and the corresponding limiting equations when the noise intensity approaches to zero. In addition, we show the invariant measures of such equations with additive white noise can be approximated by those of the corresponding equations with additive colored noise as the correlation time of the colored noise goes to zero.     

Characterization of spatial patterns produced by a Turing instability in coupled dynamical systems

We quantify the degree of spatial order of patterns at fixed time generated by lattices of coupled dynamical systems, using correlation-based and recurrence-based numerical diagnostics. These patterns are obtained through numerical integration of differential equations describing the interplay between activator and inhibitor species generating Turing patterns. We consider different types of coupling: linear (diffusive) interaction with nearest-neighbors, global (all-to-all) coupling and intermediate (nonlocal) coupling. Numerical simulations are performed in one and two spatial dimensions. The effects of noise are briefly discussed. We introduce a recurrence-based quantity (recurrence-rate matrix) to characterize two-dimensional spatial patterns.     

Synchronization of tubular pressure oscillations in interacting nephrons

The pressure and flow regulation in the individual functional unit of the kidney (the nephron) tends to operate in an unstable regime. For normal rats, the regulation displays regular self-sustained oscillations, but for rats with high blood pressure the oscillations become chaotic. We explain the mechanisms responsible for this behavior and discuss the involved bifurcations. Experimental data show that neighboring nephrons adjust their pressure and flow regulation in accordance with one another. For rats with normal blood pressure, in-phase as well as anti-phase synchronization can be observed. For spontaneously hypertensive rats, indications of chaotic phase synchronization are found. Accounting for a hermodynamics as well as for a vascular coupling between nephrons that share a common interlobular artery, we present a model of the interaction of the pressure and flow regulations between adjacent nephrons. It is shown that this model, with physiologically realistic parameter values, can reproduce the different types of experimentally observed synchronization, including multistability and partial phase synchronization with respect to the slow and fast dynamics.