Additive Temporal Coloured Noise Induced Eckhaus Instability in Complex Ginzburg-Landau Equation System

The effect of additive coloured noises, which are correlated in time, on one-dimensional travelling waves in the complex Ginzburg-Landau equation is studied by numerical simulations. We found that a small coloured noise with temporal correlation could considerably influence the stability of one-dimensional wave trains. There exists an optimal temporal correlation of noise where travelling waves are the most vulnerable. To elucidate the phenomena, we statistically calculated the convective velocities Vg of the wave packets, and found that the coloured noise with an appropriate temporal correlation can decrease Vg, making the system convectively more unstable.     

Role of the driving frequency in a randomly perturbed Hodgkin-Huxley neuron with suprathreshold forcing

The present paper examines the influence of the forcing frequency on the response of a randomly perturbed Hodgkin-Huxley system in the realm of suprathreshold amplitudes. Our results show that, in the presence of noise, the choice of driving frequency can seriously affect the precision of the external information transmission. At the same level of noise the precision can either decrease or increase depending on the driving frequency. We demonstrate that the destructive influence of noise on the interspike interval can be effectively reduced. That is, with driving signals in certain frequency ranges, the system can show stable periodic spiking even for relatively large noise intensities. Here, the most accurate transmission of an external signal occurs. Outside these frequency ranges, noise of the same intensity destroys the regularity of the spike trains by suppressing the generation of some spikes. On the other hand, we show that noise can have a reconstructive role for certain driving frequencies. Here, increasing noise intensity enhances the coherence of the neuronal response.     

Behavior of a Logistic Map Driven by White Noise

In the real world, every nonlinear system is inevitably affected by noise. As an example, a logistic map driven by white noise is studied. Unlike previous studies which focused on the behavior under local parameters to find analytical results, we investigate the whole driven logistic map. For a white noise driven logistic map, its nondivergent interval decreases with increasing white noise. The white noise does not change the equilibrium point and two-cycle intervals in statistics, if the driven logistic map is kept non-divergent. In particular, chaos can be excited by white noise only after the four-cycle bifurcation begins. The latest result is a necessary condition which has not been given in the literature [Int. J. Bifur. Chaos 18 （2008） 509], and it can be deduced from Sharkovsky＇s theorem. Numerical simulations prove these analytical results.     

Secure communication based on a four-wing chaotic system subject to disturbance inputs

In this paper, a two-input two-output secure communication scheme based on a four-wing four-dimensional chaotic system with disturbance inputs is discussed. Based on parameter modulation theory and Lyapunov stability theory, synchronization and secure communication between transmitter and receiver are achieved and two message signals are recovered via a convenient robust high-order sliding mode adaptative controller. In addition, the gains of the receiver system can be adjusted continually, the unknown parameters can be identified precisely and the disturbance inputs can be suppressed simultaneously by the proposed adaptative controller. Synchronization under the effect of noise is also considered. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient.     

Rhythm Synchronization of Coupled Neurons with Temporal Coding Scheme

Encoding information by firing patterns is one of the basic neural functions, and synchronization is important collective behaviour of a group of coupled neurons. Taking account of two schemes for encoding information （that is, rate coding and temporal coding）, rhythm synchronization of coupled neurons is studied. There are two types of rhythm synchronization of neurons： spike and burst synchronizations. Firstly, it is shown that the spike synchronization is equivalent to the phase synchronization for coupled neurons. Secondly, the similarity function of the slow variables of neurons, which have relevant to the bursting process, is proposed to judge the burst synchronization. It is also found that the burst synchronization can be achieved more easily than the spike synchronization, whatever the firing patterns of the neurons are. Hence the temporal encoding scheme, which is closely related to both the spike and burst synchronizations, is more comprehensive than the rate coding scheme in essence.     

Adaptive synchronization of weighted complex dynamical networks with coupling time-varying delays

This Letter considers the problem of controlling a weighted complex dynamical network with coupling time-varying delay toward an assigned evolution. Adaptive controllers have been designed for nodes of the controlled network. Analytical results show that the states of the weighted dynamical network can globally asymptotically synchronize onto a desired orbit under the designed controllers. In comparison with the common linear feedback controllers, the adaptive controllers have strong robustness against asymmetric coupling matrix, time-varying weights, delays, and noise. Numerical simulations illustrated by a nearest-neighbor coupling network verify the effectiveness of the proposed controllers.     

Estimating model parameters in nonautonomous chaotic systems using synchronization

In this Letter, a technique is addressed for estimating unknown model parameters of multivariate, in particular, nonautonomous chaotic systems from time series of state variables. This technique uses an adaptive strategy for tracking unknown parameters in addition to a linear feedback coupling for synchronizing systems, and then some general conditions, by means of the periodic version of the LaSalle invariance principle for differential equations, are analytically derived to ensure precise evaluation of unknown parameters and identical synchronization between the concerned experimental system and its corresponding receiver one. Exemplifies are presented by employing a parametrically excited 4D new oscillator and an additionally excited Ueda oscillator. The results of computer simulations reveal that the technique not only can quickly track the desired parameter values but also can rapidly respond to changes in operating parameters. In addition, the technique can be favorably robust against the effect of noise when the experimental system is corrupted by bounded disturbance and the normalized absolute error of parameter estimation grows almost linearly with the cutoff value of noise strength in simulation.     

Pattern Synchronization in a Two-Layer Neuronal Network

Pattern synchronization in a two-layer neuronal network is studied. For a single-layer network of Rulkov map neurons, there are three kinds of patterns induced by noise. Additive noise can induce ordered patterns at some intermediate noise intensities in a resonant way; however, for small and large noise intensities there exist excitable patterns and disordered patterns, respectively. For a neuronal network coupled by two single-layer networks with noise intensity differences between layers, we find that the two-layer network can achieve synchrony as the interlayer coupling strength increases. The synchronous states strongly depend on the interlayer coupling strength and the noise intensity difference between layers.     

Stochastic synchronization of coupled neural networks with intermittent control

In this Letter, we study the exponential stochastic synchronization problem for coupled neural networks with stochastic noise perturbations. Based on Lyapunov stability theory, inequality techniques, the properties of Weiner process, and adding different intermittent controllers, several sufficient conditions are obtained to ensure exponential stochastic synchronization of coupled neural networks with or without coupling delays under stochastic perturbations. These stochastic synchronization criteria are expressed in terms of several lower-dimensional linear matrix inequalities (LMIs) and can be easily verified. Moreover, the results of this Letter are applicable to both directed and undirected weighted networks. A numerical example and its simulations are offered to show the effectiveness of our new results.     

Impulsive control of a class of vibro-impact systems

In this Letter, the impulsive control method is developed to stabilize the chaotic motions in a class of vibro-impact systems. The strategy of the control is to implement the pulses just when the impact occurs. As applications of this method, we present the numerical simulations of two impact oscillators. Our numerical results indicate that the method used here could suppress chaos into periodic orbits which embedded in the chaotic attractor effectively, and also show that the method is robust even for high levels of multiplicative noise or additive noise.     

Stationary response of multi-degree-of-freedom vibro-impact systems to Poisson white noises

The stationary response of multi-degree-of-freedom (MDOF) vibro-impact (VI) systems to random pulse trains is studied. The system is formulated as a stochastically excited and dissipated Hamiltonian system. The constraints are modeled as non-linear springs according to the Hertz contact law. The random pulse trains are modeled as Poisson white noises. The approximate stationary probability density function (PDF) for the response of MDOF dissipated Hamiltonian systems to Poisson white noises is obtained by solving the fourth-order generalized Fokker-Planck-Kolmogorov (FPK) equation using perturbation approach. As examples, two-degree-of-freedom (2DOF) VI systems under external and parametric Poisson white noise excitations, respectively, are investigated. The validity of the proposed approach is confirmed by using the results obtained from Monte Carlo simulation. It is shown that the non-Gaussian behaviour depends on the product of the mean arrival rate of the impulses and the relaxation time of the oscillator.     

Stochastic Simulation of Turing Patterns

We investigate the effects of intrinsic noise on Turing pattern formation near the onset of bifurcation from the homogeneous state to Turing pattern in the reaction-diffusion Brusselator. By performing stochastic simulations of the master equation and using Gillespie＇s algorithm, we check the spatiotemporal behaviour influenced by internal noises. We demonstrate that the patterns of occurrence frequency for the reaction and diffusion pro- cesses are also spatially ordered and temporally stable. Turing patterns are found to be robust against intrinsic fluctuations. Sfochastic simulations also reveal that under the influence of intrinsic noises, the onset of Turing instability is advanced in comparison to that predicted deterministically.     

Exponential decay and scaling laws in noisy chaotic scattering

In this Letter we present a numerical study of the effect of noise on a chaotic scattering problem in open Hamiltonian systems. We use the second order Heun method for stochastic differential equations in order to integrate the equations of motion of a two-dimensional flow with additive white Gaussian noise. We use as a prototype model the paradigmatic Hénon-Heiles Hamiltonian with weak dissipation which is a well-known example of a system with escapes. We study the behavior of the scattering particles in the scattering region, finding an abrupt change of the decay law from algebraic to exponential due to the effects of noise. Moreover, we find a linear scaling law between the coefficient of the exponential law and the intensity of noise. These results are of a general nature in the sense that the same behavior appears when we choose as a model a two-dimensional discrete map with uniform noise (bounded in a particular interval and zero otherwise), showing the validity of the algorithm used. We believe the results of this work be useful for a better understanding of chaotic scattering in more realistic situations, where noise is presented.     

Adaptive synchronization of weighted complex dynamical networks through pinning

This paper considers the problem of controlling weighted complex dynamical networks by applying adaptive control to a fraction of network nodes. We investigate the local and global synchronization of the controlled dynamical network through the construction of a master stability function and a Lyapunov function. Analytical results show that a certain number of nodes can be controlled by using adaptive pinning to ensure the synchronization of the entire network. We present numerical simulations to verify the effectiveness of the proposed scheme. In comparison with feedback pinning, the proposed pinning control scheme is robust when tested by noise, different weighting and coupling structures, and time delays.     

Chaotic Dynamics of a Periodically Modulated Josephson Junction

We study the chaotic dynamics of a periodically modulated Josephson junction with damping. The general solution of the first-order perturbed equation is constructed by using the direct perturbation technique. It is theoretically found that the boundedness conditions of the general solution contain the Melnikov chaotic criterion. When the perturbation conditions cannot be satisfied, numerical simulations demonstrate that the system can step into chaos through a period doubling route with the increase of the amplitude of the modulating term. Regulating specific parameters can effectively suppress the chaos.     

Different Types of Synchronization in Time-Delayed Systems

We investigate different types of synchronization between two unidirectionally nonlinearly coupled identical delay- differential systems related to optical bistable or hybrid optical bistable devices. This system can represent some kinds of delay-differential models, i.e. Ikeda model, Vall~e model, sine-square model, Mackey Glass model, and so on. We find existence and sufficient stability conditions by theoretical analysis and test the correctness by＂ numerical simulations. Lag, complete and anticipating synchronization are observed, respectively. It is found that the time-delay system can be divided into two parts~ one is the instant term and the other is the delay term. Synchronization between two identical chaotic systems can be derived by adding a coupled term to the delay term in the driven system.     

Relation between single neuron and population spiking statistics and effects on network activity

To simplify theoretical analyses of neural networks, individual neurons are often modeled as Poisson processes. An implicit assumption is that even if the spiking activity of each neuron is non-Poissonian, the composite activity obtained by summing many spike trains limits to a Poisson process. Here, we show analytically and through simulations that this assumption is invalid. Moreover, we show with Fokker-Planck equations that the behavior of feedforward networks is reproduced accurately only if the tendency of neurons to fire periodically is incorporated by using colored noise whose autocorrelation has a negative component.     

Detection of Mechanism of Noise-Induced Synchronization between Two Identical Uncoupled Neurons

We investigate the noise-induced synchronization between two identical uncoupled Hodgkin-Huxley neurons with sinusoidal stimulations. The numerical results confirm that the value of critical noise intensity for synchronizing two systems is much less than the magnitude of mean size of the attractor in the original system, and the deterministic feature of the attractor in the original system remains unchanged. This finding is significantly different from the previous work [Phys. Rev. E 67 （2003） 027201] in which the value of the critical noise intensity for synchronizing two systems was found to be roughly equal to the magnitude of mean size of the attractor in the original system, and at this intensity, the noise swamps the qualitative structure of the attractor in the original deterministic systems to synchronize to their stochastic dynamics. Further investigation shows that the critical noise intensity for synchronizing two neurons induced by noise may be related to the structure of interspike intervals of the original systems.     

General methods for modified projective synchronization of hyperchaotic systems with known or unknown parameters

This work is concerned with the general methods for modified projective synchronization of hyperchaotic systems. A systematic method of active control is developed to synchronize two hyperchaotic systems with known parameters. Moreover, by combining the adaptive control and linear feedback methods, general sufficient conditions for the modified projective synchronization of identical or different chaotic systems with fully unknown or partially unknown parameters are presented. Meanwhile, the speed of parameters identification can be regulated by adjusting adaptive gain matrix. Numerical simulations verify the effectiveness of the proposed methods.     

Synchronization of Complex Network with Drivingly Coupled Scheme

We present a network model with a new coupled scheme which is the generalization of drive-response systems called a drivingly coupled network. The synchronization of the network is investigated by numerical simulations based on Lorenz systems. By calculating the largest transversal Lyapunov exponents of such network, the stable and unstable regions of synchronous state for eigenvalues in such network can be obtained and many kinds of drivingly coupled arrays based on Lorenz systems such as all-to-all, star-shape, ring-shape and chain-shape networks are considered.