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The problem of matrix projective synchronization (MPS) in discrete-time chaotic systems is investigated, and a new type of discrete chaos synchronization called inverse matrix projective synchronization (IMPS) is introduced. Sufficient conditions are derived for achieving MPS and IMPS between chaotic dynamical systems in discrete-time of different and identical dimensions. Based on new control schemes, Lyapunov stability theory, and stability theory of linear dynamical systems in discrete-time, some synchronization criteria are obtained. Numerical examples and simulations are used to illustrate the use of the proposed schemes.

Over the past few decades, chaos synchronization has become an active research subject in nonlinear science and attracted much attention from many fields due its high-potential applications [

Up to now, many types of synchronization have been found in interesting chaotic systems in discrete-time such as projective synchronization [

In this paper, based on new design control method using Lyapunov stability theory, we would like to present constructive schemes to investigate two new synchronization types: matrix projective synchronization (MPS) and inverse matrix projective synchronization (IMPS) between chaotic dynamical systems in discrete-time. Numerical examples are given to illustrate the effectiveness of the proposed schemes. To be specific, we apply the MPS between discrete-time chaotic systems of different dimensions: the drive 3D Hénon-like map and the controlled 2D Fold map. Furthermore, for the proposed scheme of IMPS, we apply it to 3D generalized Hénon map and the controlled 3D Baier-Klein map.

The rest of this paper is arranged as follows. In Section

In this section, to study the problem of matrix projective synchronization (MPS), the following drive chaotic system is considered:

As a response system, we consider the following chaotic system:

Now, we present the definition of matrix projective synchronization (MPS) between the drive system (

The

Then, the error system between the drive system (

To achieve MPS between systems (

The drive system (

By substituting (

The following result is a corollary of Theorem

If

Although this result is a corollary of Theorem

To this end, consider a Lyapunov function in the form

Thus, from the Lyapunov stability theory, it is immediate that the zero solution of the error system (

In this section, we investigate the problem of IMPS. The drive and the response chaotic systems are in the following forms:

The definition of inverse matrix projective synchronization (IMPS) for the coupled drive-response chaotic systems given in (

The drive system (

The error system between the drive system (

To achieve IMPS between systems (

With that in mind, we formulate the following result.

The drive system (

By substituting the control law (

Thus, by asymptotic stability of autonomous linear discrete-time systems, it is immediate that all solutions of error system (

In this section, to illustrate the applicability of the theoretical synchronization results derived above, two examples are considered.

Here, we consider the Hénon-like map as the drive system and the controlled Fold map as the response system. The Hénon-like map can be described as

Chaotic attractor of the Hénon-like map when

The controlled Fold map can be described as

Chaotic attractor of the Fold map when

In this example, the synchronization criterion presented in Section

According to the general control law given in Section

Using simple calculations, we can show that

The error function evolution is shown in Figure

Time evolution of MPS errors between systems (

In this example, we apply the control scheme proposed in Section

Chaotic attractor of the generalized Hénon map when

The controlled Baier-Klein map [

Chaotic attractor of the Baier-Klein map.

According to our approach presented in Section

In this case, by using the same formula of the control law given by (

It is easy to show that all eigenvalues of

The error function evolution is shown in Figure

Time evolution of IMPS errors between systems (

In this paper, the problems of matrix projective synchronization (MPS) and inverse matrix projective synchronization (IMPS) in different and identical dimensional discrete-time chaotic systems have been analyzed. Based on nonlinear controllers, Lyapunov stability theory, and stability theory of linear dynamical systems, some synchronization criteria have been obtained and new conditions have been derived for achieving MPS and IMPS. Firstly, to achieve MPS behavior between different dimensional systems, the derived control scheme was proposed by controlling the linear part of the response system. Secondly, to achieve IMPS between identical dimensional systems, the presented control method was proposed by controlling the linear part of the drive system. Numerical examples and simulations were used to verify the effectiveness of the proposed approaches.

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