The robust adaptive exponential synchronization problem of stochastic chaotic systems with structural perturbations is investigated in mean square. The stochastic disturbances are assumed to be Brownian motions that act on the slave system and the norm-bounded uncertainties exist in all parameters after decoupling. The stochastic disturbances could reflect more realistic dynamical behaviors of the coupled chaotic system presented within a noisy environment. By using a combination of the Lyapunov functional method, the robust analysis tool, the stochastic analysis techniques, and adaptive control laws, we derive several sufficient conditions that ensure the coupled chaotic systems to be robustly exponentially synchronized in the mean square for all admissible parameter uncertainties. This approach cannot only make the outputs of both master and slave systems reach

Synchronization is a fundamental phenomenon that enables coherent behavior in coupled dynamical systems. Since Pecora and Carroll [

Synchronization of chaotic systems affected by both structural and stochastic disturbances poses new challenges for the understanding of stability, sensitivity and robustness, bifurcations and chaos, and so forth. When analyzing the dynamical behaviors of chaotic systems, stochastic disturbances and modeling errors are probably two of the main sources that result in uncertainties. To overcome these difficulties, various adaptive synchronization schemes have been proposed and investigated (see an excellent text in [

In this paper, we discuss the asymptotical synchronization and almost surely synchronization for two different chaotic systems with the consideration of both structural and stochastic perturbations. With the help from the Lyapunov functional method and adaption method, we employ the robust analysis tool and the stochastic analysis techniques to derive some relevant conditions under which the coupled systems is globally robustly synchronized in the mean square for all admissible parameter uncertainties. These conditions guarantee the robustness of the controller against the effect of exoteric perturbations. The rest of the paper is organized as follows: Section

Let two classes of nonlinear systems be given in the following form:

Considering the effect of structural and exoteric perturbation on system’s parameters, the master system is given by

Define the synchronization error vector as

Systems (

Let us define

Given a scalar

Assume that the noise intensity function matrix

There exists continuous function

Systems (

Suppose two continuous functions

We can choose the controller in this following form:

The control problem with a feedback (

There exist two known positive functions

For bounded

The function

The “robust performant” gain scheduled system is assumed to be bounded in response to certain added inputs. Moreover, the performance level must not degrade discontinuously for arbitrarily small errors in controller (

Let us consider three continuous positive functions

For two integrable functions

For any matrices

We can choose a constant matrix

System (

The closed loop error system with control (

Relation (

Relation (

Using inequality (

From Lemma

Relation (

Since the systems are assumed to operate in chaotic mode without feedback, their trajectories converge to compact invariant set. Let one assume that

It may be shown that the system under control is forward complete; that is, if there exists a set of initial conditions and gains such that, together, they generate solutions that tend to infinity, these solutions may unboundedly grow only in infinite time. From this, it follows that for each

In this section, we will provide simulation results for a system where the master oscillator is the perturbed modified Colpitts oscillators and the observer is a perturbed and stochastic Chua system.

From (

If the parameters are taken as

Time history and phase portraits of modified Colpitts oscillator.

Time history and phase portraits of Chua’s circuit.

For convenience, we select the following matrices:

If we let

Let us select the perturbed matrix simply in the following form:

Figure

The time history of the output error with disturbances when no control input is applied.

The time history of the output errors when the control input is applied in presence of structural perturbation.

Time evolution of the estimated feedback gain

Because the adaptive control has some interesting features such as low sensitivity to external disturbances, robustness to the plant uncertainties, and easy realization, in this paper, we use this method to realize exponential synchronization of two different uncertain chaotic systems in which the slave system is noise perturbed. By employing the Lyapunov functional method and adaptive control, several sufficient conditions have been obtained which ensure the coupled chaotic systems to be exponentially robustly synchronized in the mean square. Furthermore, the theoretical analysis is easily verified by using the standard numerical software. We have selected two perturbed systems consisting of modified Colpitts oscillator and Chua’s oscillator. It was found that the controller maintains robust stable synchronization in the presence of exoteric perturbations, structural uncertainties, and noises.

The authors declare that there is no conflict of interests regarding the publication of this paper.