A Fractional-Order Chaotic System with an Infinite Number of Equilibrium Points

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.

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 [15][16][17], some chaotic systems have two equilibrium points [18], and some chaotic systems have three equilibrium points [1,2,5,6,9,10], so a natural and interesting question is can we construct a chaotic (hyperchaotic) system which has an infinite number of equilibrium points?Moreover, many fractionalorder chaotic and hyperchaotic systems also possess chaotic attractor for its corresponded integer-order system, so the other question is as follows: are the fractional-order chaotic and hyperchaotic systems no-chaotic behavior for its corresponded integer-order system?To the best of our knowledge, few results on the above mentioned two questions are reported.
Motivated by the above discussions, a new 4D fractionalorder 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.

A New 4D Fractional-Order Chaotic System
Now, a new 4D fractional-order chaotic system is constructed, which is described as follows: where  = 0.95 is the fractional-order, and   ( = 1, 2, 3, 4) are real state variables.The real equilibrium points of system (1) is calculated by Obviously, ( 1 ,  2 ,  3 ,  4 ) = (0,  2 , 0, −10 2 ) is the real equilibrium points of system (1), where  2 is a any real numbers, so system (1) has an infinite number of real equilibrium points.To the best of our knowledge, this result is different from all the previous fractional-order chaotic and hyperchaotic systems.It implies that we yield a new 4D fractional-order system, which has an infinite number of real equilibrium points.
The dynamical behaviors of system (1) for its corresponded integer-order system ( = 1) can be characterized by its Lyapunov exponents.The Lyapunov exponents for its corresponded integer-order system are 0, 0, −0.779, and −12.724, respectively.Therefore, the fractional-order system (1) no-chaotic behaviors for  = 1, and which is periodic orbit for its corresponded integer-order system.Figure 1 shows the periodic orbit of fractional-order system (1) for its corresponded integer-order system ( = 1).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 fractionalorder system [6].However, frequency domain approximation may result in wrong consequences [19], so the direct time domain approximation [6] numerical simulation is used to solve the fractional-order system in this paper.Let ℎ = /,   = ℎ ( = 0, 1, 2 . . ., ), and initial condition ( 1 (0),  2 (0),  3 (0),  4 (0)), so the fractional-order chaotic system (1) can be discretized as follows: where Figure 1: The periodic orbit of fractional-order system (1) for its corresponded integer-order system ( = 1).
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 fractionalorder chaotic system is no-chaotic behaviors for its corresponded integer-order system ( = 1).The result in our paper is different from all the previous fractional-order chaotic and hyperchaotic systems.

Chaotic Synchronization for the New 4D Fractional-Order Chaotic System
In this section, the chaotic synchronization for the new 4D fractional-order chaotic system (1) is considered.Based on the stability theory of nonlinear fractional-order systems [20][21][22][23][24], one synchronization scheme is proposed, and some numerical simulations are performed.Now, the response fractional-order chaotic system is considered as where   ( = 1, 2, 3, 4) is the feedback controller, and   ( = 1, 2, 3, 4) are real state variables.Our goal is to choose suitable   ( = 1, 2, 3, 4) such that drive system (1) and response system (8) can be achieved with chaotic synchronization.Definition the synchronization errors are   =   −   ( = 1, 2, 3, 4).The following Theorem 1 is given in order to achieve the chaotic synchronization between the fractionalorder chaotic system (1) and the fractional-order chaotic system (8).

Theorem 1. If the feedback controllers are
then the chaotic synchronization between fractional-order chaotic system (1) and fractional-order chaotic system (8) can be arrived.
Let  be one of the eigenvalues of (, ) and  is the corresponding nonzero eigenvector, so where H is conjugate transpose, and  is the conjugate for eigenvalues .
According to (12), one can obtain Therefore so That is Using the stability theory of nonlinear fractional-order systems, one can yield that the error system (10) is asymptotically stable, so lim  → +∞   = 0 ( = 1, 2, 3, 4) .

Conclusions
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.