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A hybrid stability checking method is proposed to verify the establishment of synchronization between two hyperchaotic systems. During the design stage of a synchronization scheme for chaotic fractional-order systems, a problem is sometimes encountered. In order to ensure the stability of the error signal between two fractional-order systems, the arguments of all eigenvalues of the Jacobian matrix of the erroneous system should be within a region defined in Matignon’s theorem. Sometimes, the arguments depend on the state variables of the driving system, which makes it difficult to prove the stability. We propose a new and efficient hybrid method to verify the stability in this situation. The passivity-based control scheme for synchronization of two hyperchaotic fractional-order Chen-Lee systems is provided as an example. Theoretical analysis of the proposed method is validated by numerical simulation in time domain and examined in frequency domain via electronic circuits.

Nonlinear systems may exhibit dynamical chaotic behavior. The study of chaos synchronization has received increasing attention due to its predicted potentials in technological applications in recent years. In 1983, Fujisaka and Yamada [

Fractional calculus has been studied in a speedy pace during the recent years. It has been implemented in various engineering fields such as control [^{
m}E method based on the Adams-Bashforth-Moulton algorithm [

In 2004, Chen and Lee [

The concept of passive control theory [

In this paper, the synchronization between two hyperchaotic Chen-Lee systems with different initial conditions was established via the passive control technique first. The influence of the controller parameters was discussed for enhancing the efficiency of synchronization. Next, the proposed controller was also applied to fractional-order hyperchaotic Chen-Lee systems. Synchronization was ensured based on the stability theorem by Matignon [

Matignon’s theorem and the passivity theory were reviewed in this section. The former was used for analyzing the stability of fractional differential equations and the latter for designing the synchronization scheme between two systems.

Consider an error dynamical system

The trivial solution of the system

The response system is asymptotically synchronized with the driving system if and only if

Consider the following nonlinear system [

For all

If a system is passive, a suitable controller can stabilize asymptotically the equilibrium point

If the system (

The system (

Chen et al. [

The aim is to design the controller

Let

Because

If the ordinary differential operators in (

System (

To gain the eigenvalues at the origin,

Obviously, the eigenvalues depend on the system states of (

An electronic circuit was constructed for implementing the proposed scheme. Figure

Schematic diagram of the synchronization system.

Electronic circuits of (a) the driving system and (b) the response system.

Electronic circuit of the controllers: (a)

The above circuit can be converted to a fractional-order one by simply replacing the capacitors on the feedback path of the integrators by chain fractance [

To approximately realize the fractional-order operator with

It can be realized by a chain fractance of order 3 (Figure

Chain fractance of

In this section, the Runge-Kutta method of order 4 was used to solve the differential equations in the systems (

First of all, the value of

Time histories of the synchronization errors (

Time histories of the synchronization errors (

The influence of the controller parameter

The order of the systems was chosen to be

Phase diagrams of the hyperchaotic systems: (a) numerical solution in time domain; (b) circuit simulation in frequency domain.

Time histories of

Time histories of the error dynamical system (

The simulation was done on Multisim package. The initial conditions were set to

Time histories of the circuit outputs of the driving system (red) and the response system (blue) with integer order.

For simulating systems with fractional order, the capacitors were replaced by chain fractance. The value of the resistor

Time histories of the circuit outputs of the driving system (red) and the response system (blue) with fractional order.

A novel and efficient strategy was proposed to examine the stability of the error dynamical systems with fractional order. The chaos synchronization between two hyperchaotic Chen-Lee systems with fractional order was achieved via feedback passive control technique. The passive controller was first designed for the integer-order system. The control scheme was proved based on the stability theorem for fractional calculus. Numerical simulation was given to validate the proposed approaches. Chain fractance was designed for approximating the fractional order

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

The authors would like to thank the referees for their valuable suggestions and comments. This research was partially supported by a Grant (NSC98-2212-M-164-001-MY3) from the National Science Council,

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