A method of estimation of all parameters of a class of nonlinear uncertain dynamical systems is considered, based on the modified projective synchronization (MPS). The case of modified Colpitts oscillators is investigated. Through a suitable transformation of the dynamical system, sufficient conditions for achieving synchronization are derived based on Lyapunov stability theory. Global stability and asymptotic robust synchronization of the considered system are investigated. The proposed approach offers a systematic design procedure for robust adaptive synchronization of a large class of chaotic systems. The combined effect of both an additive white Gaussian noise (AWGN) and an artificial perturbation is numerically investigated. Results of numerical simulations confirm the effectiveness of the proposed control strategy.

Synchronization of chaotic systems and their potential applications in wide areas of physics and engineering sciences is currently a field of great interest ([

Adaptive control technique is used when the system parameters are unknown. In adaptive method, control law and a parameter update rule for unknown parameters are designed in such a way that the chaotic response system is controlled by the chaotic drive system. Most of the studies in synchronization involve two identical/nonidentical systems under the hypotheses that all the parameters of the master and slave systems are known

From the literature survey, it is seen that, with the development of nonlinear control theory, nowadays adaptive projective synchronization method has become very much effective to control and synchronize the chaotic and hyperchaotic systems with uncertain parameters and external disturbances. Recently, many authors have studied the adaptive synchronization for the chaotic systems. In [

Most of the adaptive control scheme is based on the dynamic parameter estimation. In their book entitled

Recently, there have been many efforts for the study of dynamical properties of this oscillator introduced by Ababei and Marculescu in [

In this paper, we first transform the original system equations of MCO by rigorous mathematical theory and secondly we will study modified synchronization of uncertain MCO which is presented based on Lyapunov stability theory.

The organization of this paper is as follows. In Section

The simplest configuration of the MCO is shown in Figure

Circuit model: (a) Schematic of the Colpitts oscillator. (b) BJT model in common base configuration.

Let us select the parameters:

Consider the dynamics system of MCO given by

Preliminary insights concerning the existence of attractive sets [

Consider the fact that the system (

It is easily shown that the system (

Let us consider a general class of chaotic systems described by the following differential equation:

The states of the chaotic system described by (

The relation (

Let us consider two 3-dimensional chaotic systems which can be represented in a more generic form as follows:

We assume that the asymptotic convergence of (

In the new space, the coordinates can be written in the following form:

If the master system (

Let us write

An illustration of the modified projective synchronization is now presented. Taking into account the synchronization between two chaotic systems, take the drive system as follows:

Chaos synchronization schemes such as complete synchronization and antisynchronization are special cases of modified projective synchronization when

From the definition of error signal (

Let one consider an uniform continuous function

For the given scaling factor

Construct dynamical Lyapunov function as follows:

The equations of MCO in the new space can be expressed as in equations (

By assumption, the master system operates in the chaotic regime; hence, all master signals are bounded. Furthermore, let us temporarily assume that the trajectory of the slave system in closed loop, that is,

Substituting the control input (

For any nonzero scaling factors, the slave system (

The error system is

It is left to show that the trajectories of the slave system are bounded under the feedback. We invoke the following.

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

It may be shown as in [

The calculations of all equations were carried out using the fourth-order Runge-Kutta algorithm. For this numerical simulation, we assume the initial conditions:

(a) Time histories of MCO, (b) Phase portraits (versus

Time histories of MCO when no control is applied ((a)

Divergence of the flow of MCO in the new space.

Phase portrait of MCO in the new space in the presence of artificial pertubation.

Estimated values for unknown parameters.

Time evolution of adaptive gain.

Synchronization errors: (a)

Time evolution of the dynamic error.

In this paper, based on Lyapunov stability theory, theory of changing space of variables, and Barbalat’s lemma we achieved lag synchronization of two modified Colpitts oscillators. This control strategy of the MCO with uncertainties including the coefficients of nonlinear terms was obtained via adaptive control. It appears that it is possible to introduce a specific controller to attenuate any artificial perturbation on the system. The final remark is that the proposed scheme is applicable to various other dynamical systems to efficiently estimate unknown parameters which could be arguments of some other nonlinear functions.

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