A Fractional-Order System with Coexisting Chaotic Attractors and Control Chaos via a Single State Variable Linear Controller

A 3D fractional-order nonlinear system with coexisting chaotic attractors is proposed in this paper. The necessary condition of the existence chaos is q ≥ 0.8477. The fractional-order system exhibits chaotic attractors with the order as low as 2.5431. The largest Lyapunov exponent varying as fractional order q is given. Furthermore, there are the coexisting “positive attractor” and “negative attractor” in this fractional-order chaotic system, and the necessary condition for “positive attractor” and “negative attractor” is obtained. Meanwhile, a control scheme for the stabilization of the unstable equilibrium is suggested via a single state variable linear controller. Numerical results show that the control scheme is valid.


Introduction
Chaotic behaviors in nonlinear is a very interesting phenomenon. The high irregularity, unpredictability, and complexity in chaotic systems [1,2] have been widely used in the field of engineering and technology such as secure communications, image steganography, authenticated encryption, motor control, and power system protection. Recently, coexisting chaotic attractors have been found in chaotic systems [3][4][5][6][7][8][9]. For example, the coexisting chaotic attractors in a 3D no-equilibrium system were reported by Pham et al. [3], the coexisting multiple attractors in Hopfield neural network were found by Bao et al. [4], the coexisting chaotic attractors in a hyperchaotic hyperjerk system were given by Wang et al. [5], the coexisting "positive attractor" and "negative attractor" in a 3D autonomous continuous chaotic system were found by Zhou and Ke [6], and so on [7][8][9]. Therefore, more and more attention has been focused on the coexisting chaotic attractors in nonlinear chaotic systems.
On the other hand, the fractional-order differential equations [10][11][12] can be accurately described in the real-world physical systems such as viscoelasticity, dielectric polarization, electrode-electrolyte polarization, electromagnetic waves, heat conduction, diffusion-wave, and superdiffusion. Chaotic behaviors have been found in many real-world physical fractional-order nonlinear systems, for example, the fractional-order chaotic brushless DC motor [13], the fractional-order electronic circuits [14], the fractional-order microelectromechanical system [15], and the fractional-order gyroscopes [16]. Therefore, more and more attention has been paid to the chaotic behaviors in fractional-order nonlinear systems.
Motivated by the above discussions, based on a 3D autonomous continuous chaotic system reported by Zhou and Ke [6], we suggested a 3D autonomous continuous fractional-order system. We have shown that the chaotic system reported by Zhou and Ke [6] can be extended to its fractional-order version where the coexisting "positive attractor" and "negative attractor" can be observed. We obtained that the fractional-order system with the order as low as 2.5431 exhibits chaotic attractors. Moreover, we obtained the largest Lyapunov exponent varying as fractional order. Finally, for the stabilization of the unstable equilibrium, one control scheme is proposed via a single state variable linear controller.

Complexity
The outline of this paper is organized as follows. In Section 2, based on a 3D autonomous continuous chaotic system reported by Zhou and Ke [6], the fractional-order version nonlinear system is given, and some basic dynamical properties of this fractional-order version nonlinear system are obtained including the necessary condition of the existence chaos, the largest Lyapunov exponent varying as fractional order, and the coexisting "positive attractor" and "negative attractor." In Section 3, by a single state variable, stabilization of the unstable equilibrium points of the fractional-order chaotic system is discussed. Finally, the conclusions are given in Section 4.

System Model and Basic Characteristics
In this paper, the Caputo definition of the fractional derivative will be used in next. The Caputo definition of the fractional derivative is described as where 0 is the Caputo operator, is the first integer which is not less than , and ( ) ( ) is the -order derivative in usual sense of ( ).
Next, based on the 3D autonomous continuous chaotic system reported by Zhou and Ke [6], a fractional-order system is addressed as where fractional order 0 < < 1. Fractional-order system (2) has five equilibrium points. They are 0 = (0, 0, 0),  Tavazoei and Haeri [17] have obtained that a necessary condition for a fractional-order nonlinear system to exist chaotic is where is the eigenvalues of saddle equilibrium point of index two in fractional-order nonlinear system. Now, we can obtain the necessary condition of the existence chaos in fractional-order system (2). According to (3), we have the following: Thus, the necessary condition of existence chaos in fractionalorder system (2) is ≥ 0.8477. This result indicates that fractional-order system (2) with the order as low as 2.5431 can exhibit chaotic attractors.
According to Figure 1, we can obtain that the largest Lyapunov exponent is larger zero for 0.8477 ≤ ≤ 1. This result indicates that the chaotic attractor is emerged in fractional-order system (2) for 0.8477 ≤ ≤ 1. For example, let = 0.8477, the largest Lyapunov exponent is 0.9853, and the chaotic attractor in fractional-order system (2) with initial condition (−5, 2, 5) is shown as Figure 2.
Next, some numerical simulations are given for = 0.9. Here, the largest Lyapunov exponent is 0.763 for = 0.9.
According to Figures 3 and 4, the coexisting chaotic attractors are found in fractional-order chaotic system (2). These results indicate that the chaotic system reported by Zhou and Ke [6] can be extended to its fractional-order version where the coexisting "positive attractor" and "negative attractor" can be observed.
Lemma 2 (for more details, see [19]). Fractional-order nonlinear system (9) is said to be asymptotically stable if the next two conditions hold: where ( ) and ‖ ‖ denote the eigenvalues and the 2 -norm with respect to matrix , respectively.
Next, we discuss how to stable the unstable equilibrium point in fractional-order chaotic system (2) via single state variable linear controller. Now, let the unstable equilibrium points in system (2) be ( 1 , 2 , 3 ), and the controlled fractional-order system is shown as follows: where 0.8477 ≤ ≤ 1 and ( 2 − 2 ) ( = 1, 2, 3) is a linear controller determined by a single state variable 2 .
Therefore, according to Lemma 2, the origin of system (12) is asymptotically stable. That is, the unstable equilibrium points ( 1 , 2 , 3 ) of fractional-order chaotic system (2) in controlled system (10) are asymptotically stable. The proof is finished.
Remark 4. In our control scheme, the linear controller is only determined by one single state variable, so our control scheme is different from many previous control schemes.

Numerical Simulations Results
Next, in order to show the effectiveness of the proposed control approach, the numerical simulations are performed for = 0.9.
The simulative results in Figures 5-9 show the effectiveness of our control scheme.

Conclusions
In this paper, a fractional-order chaotic system is proposed. The necessary condition of existence chaos in this fractionalorder chaotic system is ≥ 0.8477. The largest Lyapunov exponent varying as fractional order is given. The coexisting "positive attractor" and "negative attractor" can be observed, and the necessary conditions for "positive attractor" and "negative attractor" are obtained. Meanwhile, by a single state variable, a linear controller is used for the stabilization of the unstable equilibrium points of the fractional-order chaotic system. The numerical results show that the control approach is effective.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare no conflicts of interest.