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A new 4D fractional-order chaotic system, which has an infinite number of equilibrium points, is introduced. There is no-chaotic behavior for its corresponded integer-order system. We obtain that the largest Lyapunov exponent of this 4D fractional-order chaotic system is 0.8939 and yield the chaotic attractor. A chaotic synchronization scheme is presented for this 4D fractional-order chaotic system. Numerical simulations is verified the effectiveness of the proposed scheme.

Because the chaotic (hyperchaotic) signal can be used in electrical engineering, telecommunications, information processing, material engineering, and so forth much attention has been paid to effectively generating chaotic and hyperchaotic systems. Many chaotic (hyperchaotic) systems and fractional-order chaotic (hyperchaotic) systems are reported in recent years [

However, for all the previous integer-order and fractional-order chaotic (hyperchaotic), many systems have a finite number of equilibrium points. For example, some chaotic systems have one equilibrium point [

Motivated by the above discussions, a new 4D fractional-order chaotic system is presented in this paper. This new 4D fractional-order chaotic system has an infinite number of equilibrium points, and no-chaotic behavior for its corresponded integer-order system. The largest Lyapunov exponent and chaotic attractor are yielded for the new 4D fractional-order chaotic system. A chaotic synchronization scheme is presented for this new 4D fractional-order chaotic system.

Now, a new 4D fractional-order chaotic system is constructed, which is described as follows:

The real equilibrium points of system (

Obviously, (

The Jacobian

The dynamical behaviors of system (

The periodic orbit of fractional-order system (

Now, we discuss the numerical solution of fractional differential equations. It is well known that there are direct time domain approximation (the improved version of Adams-Bashforth-Moulton numerical algorithm) and frequency domain approximation for nonlinear fractional-order system [

The error of this approximation is described as follows:

The dynamical behaviors of 4D fractional-order system (

The chaotic attractor of 4D fractional-order system (

According to the above mentioned, we obtain a new 4D fractional-order chaotic system, which has an infinite number of real equilibrium points. Moreover, the 4D fractional-order chaotic system is no-chaotic behaviors for its corresponded integer-order system

In this section, the chaotic synchronization for the new 4D fractional-order chaotic system (

Now, the response fractional-order chaotic system is considered as

Definition the synchronization errors are

If the feedback controllers are

Combining the fractional-order chaotic system (

Let

According to (

Using the stability theory of nonlinear fractional-order systems, one can yield that the error system (

Now, numerical simulations are considered. The numerical results are shown as Figure

The synchronization errors between the drive systems (

In this paper, we obtain a new 4D fractional-order chaotic system, which has an infinite number of equilibrium points and no-chaotic behavior for its corresponded integer-order system. We yield the largest Lyapunov exponent of the new 4D fractional-order system and the Lyapunov exponents for its corresponded integer-order system. The chaotic attractor for the new 4D fractional-order chaotic system and the periodic orbit for its corresponded integer-order system are given. Finally, we realize the chaotic synchronization for the new 4D fractional-order chaotic system, and some numerical simulations are performed.

This work was supported by the Foundation of Science and Technology Project of Chongqing Education Commission (KJ110525, KJ100513) and by the National Natural Science Foundation of China (61104150).