In this paper, a general method of synchronizing noise-perturbed chaotic systems with unknown parameters is proposed. Based on the LaSalle-type invariance principle for stochastic differential equations and by employing a combination of feedback control and adaptive control, some sufficient conditions of chaos synchronization between these noise-perturbed systems with unknown parameters are established. The model used in the research is the chaotic system, but the method is also applicable to the hyperchaotic systems. Unified system and noise-perturbed RSssler system, hyperchaotic Chen system and nolse-perturbed hyperchaotic RSssler system are taken for illustrative examples to demonstrate this technique.
With both additive and multiplicative noise excitations, the effect on the chaotic behaviour of the dynamical system is investigated in this paper. The random Melnikov theorem with the mean-square criterion that applies to a type of dynamical systems is analysed in order to obtain the conditions for the possible occurrence of chaos. As an example, for the Duffing system, we deduce its concrete expression for the threshold of multiplicative noise amplitude for the rising of chaos, and by combining figures, we discuss the influences of the amplitude, intensity and frequency of both bounded noises on the dynamical behaviour of the Duffing system separately. Finally, numerical simulations are illustrated to verify the theoretical analysis according to the largest Lyapunov exponent and Poincaré map.
It is shown how the cross-correlation time and strength of coloured cross-correlated white noises can set an upper bound for the time derivative of entropy in a nonequilibrium system. The value of upper bound can be calculated directly based on the Schwartz inequality principle and the Fokker-Planck equation of the dynamical system driven by coloured cross-correlated white noises. The present calculations can be used to interpret the interplay of the dissipative constant and cross-correlation time and strength of coloured cross-correlated white noises on the upper bound.
This paper deals with the problem of chaos control and synchronization of the Chen-Liao system. From rigorous mathematic justification, the chaotic trajectories of the Chen-Liao system are led to a type of points whose fourdimensional coordinates have a particular functional relation among them. Meanwhile, a new synchronization manner, reduced-order generalized synchronization （RGS）, is proposed which has the characteristic of having a functional relation between the slave and the partial master systems. It is shown that this new synchronization phenomenon can be realized by a novel technique. Numerical simulations have verified the effectiveness of the proposed scheme.
In this paper, we apply a simple adaptive feedback control scheme to synchronize two bi-directionally coupled chaotic systems. Based on the invariance principle of differential equations, sufficient conditions for the global asymptotic synchronization between two bi-directionally coupled chaotic systems via an adaptive feedback controller are given. Unlike other control schemes for bi-directionally coupled systems, this scheme is very simple to implement in practice and need not consider coupling terms. As examples, the autonomous hyperchaotic Chen systems and the new nonautonomous 4D systems are illustrated. Numerical simulations show that the proposed method is effective and robust against the effect of weak noise.
In this paper, based on the invaxiance principle of differential equations, we propose a simple adaptive control method to synchronize the network with coupling of the general form. Comparing with other control approaches, this scheme only depends on each node＇s state output. So we need not to know the concrete network structure and the solutions of the isolate nodes of the network in advance. In order to demonstrate the effectiveness of the method, a special example is provided and numerical simulations are performed. The numerical results show that our control scheme is very effective and robust against the weak noise.
In this paper the stochastic resonance （SR） is studied in an overdamped linear system driven by multiplicative noise and additive quadratic noise. The exact expressions are obtained for the first two moments and the correlation function by using linear response and the properties of the dichotomous noise. SR phenomenon exhibits in the linear system. There are three different forms of SR： the bona fide SR, the conventional SR and SR in the broad sense. Moreover, the effect of the asymmetry of the multiplicative noise on the signal-to-noise ratio （SNR） is different from that of the additive noise and the effect of multiplicative noise and additive noise on SNR is different.
Investigations of low energy transfer trajectories are important for both celestial mechanics and astronautics. Methodologies using the theories from dynamical systems are developed in recent years. This paper investigates the dynamics of the earth-moon system. Low energy transfer trajectories are solved numerically by employing a hybrid strategy： first, a genetic hide and seek method performs a search in large domain to confine the global minimum f（η） （objective function） region; then, a deterministic Nelder-Mead method is utilized to refine the minimum quickly. Some transfer trajectories of the spacecraft in the earth-moon system are successfully simulated which verify the desired efficiency and robustness of the method of this paper.
How to predict the dynamics of nonlinear chaotic systems is still a challenging subject with important real-life applications. The present paper deals with this important yet difficult problem via a new scheme of anticipating synchronization. A global, robust, analytical and delay-independent sufficient condition is obtained to guarantee the existence of anticipating synchronization manifold theoretically in the framework of the Krasovskii-Lyapunov theory. Different from ＇traditional techniques （or regimes）＇ proposed in the previous literature, the present scheme guarantees that the receiver system can synchronize with the future state of a transmitter system for an arbitrarily long anticipation time, which allows one to predict the dynamics of chaotic transmitter at any point of time if necessary. Also it is simple to implement in practice. A classical chaotic system is employed to demonstrate the application of the proposed scheme to the long-term prediction of chaotic states.
This paper investigates the stochastic resonance （SR） phenomenon in an asymmetric system with coupling between multiplicative and additive noise when the coupling between two noise terms is coloured. The approximate expression of signal-to-noise ratio has been obtained by applying the two-state theory and SR exhibits in the bistable system. Moreover, the potential asymmetry r and cross-correlation strength λ can weaken the SR phenomenon, while the cross-correlation time r can strengthen the SR phenomenon.
