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This work investigates modified function projective synchronization between two different hyperchaotic dynamical systems, namely, hyperchaotic Lorenz system and hyperchaotic Chen system with fully unknown parameters. Based on Lyapunov stability theory, the adaptive control law and the parameter update law are derived to achieve modified function projective synchronized between two diffierent hyperchaotic dynamical systems. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.

During the last three decades, synchronization of chaotic systems has attracted increasing attention from scientists and engineers and has been explored intensively both theoretically and experimentally. Since Pecora and Carrol [

A focused problem in the study of chaos synchronization is how to design a physically available and simple controller to guarantee the realization of high-quality synchronization in coupled chaotic systems. Linear feedback is of course a practical technique, but the shortcoming is that it needs to find the suitable feedback constant. Recently, Huang proposed a simple adaptive feedback control method, which neednot to estimate or find feedback constant, to effectively synchronize two almost arbitrary identical chaotic systems in his series paper [

In this work, we investigate modified function projective synchronization (MFPS) between hyperchaotic Lorenz system and hyperchaotic Chen system with fully unknown parameters. This work is organized as follows. In Section

Consider the following master and slave system:

The system (

It is easy to see that the definition of modified function projective synchronization encompasses function projective synchronization when the scaling matrix

The hyperchaotic Lorenz system is described as follows [

The attractor of hyperchaotic Lorenz dynamical system at

Hyperchaotic Chen system is described as [

The attractor of hyperchaotic Chen dynamical system at

In order to achieve the synchronization behavior between hyperchaotic Lorenz system and hyperchaotic Chen system, we assume that hyperchaotic Lorenz system is the drive system whose four variables are denoted by subscript 1 and hyperchaotic Chen system is the response system whose variables are denoted by subscript 2. The drive and response systems are described by the following equations, respectively,

We have the following error dynamical system:

Substitution of (

Our aim is to find control laws

For given constant scaling matrix

Define a Lyapunov function,

The time derivative of the Lyapunov function along the trajectory of error system (

Inserting (

Since

Therefore, the drive system (

Note that complete synchronization and antisynchronization between two different hyperchaotic dynamical systems are special cases of MFPS with the scaling function

Note that function projective synchronization (FPS) between two different hyperchaotic dynamical systems is special case of MFPS with the scaling factors

Note that generalized projective synchronization (GPS) and modified generalized projective synchronization (MGPS) between two different hyperchaotic dynamical systems are special cases of MFPS with the scaling function

By suitable choosing for

In this section, numerical examples are used to demonstrate the effectiveness of the proposed method. By using Maple 12 to solve the systems of differential equations (

Let the scaling function be

The behavior of the trajectories

Let the scaling function be

The behavior of the trajectories

The estimates

The estimates

Let the scaling function be

(a) The behavior of the trajectories

Let the scaling function be

(a) The behaviour of the trajectories

Let the scaling function be

(a) The behavior of the trajectories

Let the scaling function be

(a) The behavior of the trajectories

Let the scaling function be

(a) The behaviour of the trajectories

This work investigated modified function projective synchronization between the hyperchaotic Lorenz system and hyperchaotic Chen system with fully unknown parameters. Based on Lyapunov stability theory, we design adaptive synchronization controllers with corresponding parameter update laws to synchronize the two systems. The MFPS includes complete synchronization, antisynchronization, function projective synchronization (FPS), generalized projective synchronization (GPS), and modified generalized projective synchronization (MGPS). All the theoretical results are verified by numerical simulations to demonstrate the effectiveness of the proposed synchronization schemes. Thus, our synchronization method is successful for some systems with two positive Lyapunov exponents.

The authors would like to thank the editor and the anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper. The first author acknowledges with thanks the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia for his support this article.