A New Fractional-Order Chaotic Complex System and Its Antisynchronization

and Applied Analysis 3 According to this algorithm, system (5) for initial condition (m 10 , m 20 , m 30 , m 40 , m 50 ) can be discretized as m 1,n+1 = m 10 + h α 1


Introduction
Chaotic behavior and synchronization of fractional-order dynamical systems have been extensively studied over the last decade.Many fractional-order systems can behave chaotically, such as the fractional-order Chua's system [1], the fractional Rössler system [2], the fractional-order Lorenz system [3], the fractional-order Chen system [4], and the fractional-order Lü system [5].It has been shown that some fractional-order systems have chaotic behavior with orders less than 3.Meanwhile, chaos synchronization of fractionalorder systems has attracted much attention, such as the complete synchronization (CS) [6], projective synchronization (PS) [7], and lag projective synchronization [8].
However, most of the studies about fractional-order systems had been based on real variables, and complex systems are rarely involved.Complex systems provide an excellent instrument to describe a variety of physical phenomena, such as detuned laser systems, amplitudes of electromagnetic fields, and thermal convection of liquid flows [9][10][11].And now complex systems have played an important role in many branches of physics, for example, superconductors, plasma physics, geophysical fluids, modulated optical waves, and electromagnetic fields [12].There are some new kinds of synchronization for complex dynamical systems, for example, complex complete synchronization (CCS) [13], complex projective synchronization (CPS) [14], complex modified projective synchronization (CMPS) [15,16], and so forth.These new kinds of synchronization have been widely studied for applications in secure communication [17], because complex variables (doubling the number of variables) increase the contents and security of the transmitted information.Therefore, the dynamical behavior and synchronization of the fractional-order complex nonlinear systems are worth studying.Recently, Luo and Wang proposed the fractional-order complex Lorenz system [18] and the fractional-order complex Chen system [19] and studied their dynamical properties and chaos synchronization.To our best knowledge, there are few results on fractional-order chaotic complex systems until now.
Motivated by the above discussion, the aim of this paper is to investigate the chaotic phenomena in a newly proposed fractional-order complex Lü system, which may provide potential applications in secure communication.As will be shown below, this new system displays many interesting dynamical behaviors, such as fixed points, periodic motions, and chaotic motions.Besides, when the parameters of the system are fixed, the lowest order for chaos to exist is determined.Furthermore, antisynchronization between the new system and fractional-order complex Lorenz system is studied.More generally, we investigate antisynchronization of different fractional-order chaotic complex systems and give a usual scheme.
The remainder of this paper is organized as follows.In Section 2, The fractional-order complex Lü system is presented and its dynamics is discussed by phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents.In Section 3, the antisynchronization of different fractional-order chaotic complex systems is studied, and the proposed new system can antisynchronize the fractionalorder complex Lorenz system.A concluding remark is given in Section 4.

The Proposal of the Fractional-Order Complex Lü System.
There are many definitions of fractional derivatives [20,21], such as Riemann-Liouville, Grünwald-Letnikov, and Caputo definitions.In this paper, we use the Caputo definition which is defined as follows: Here  is the first integer which is not less than  and  > 0,  () is the -order derivative in the usual sense, and   ( > 0) is the -order Riemann-Liouville integral operator with expression Here Γ stands for Gamma function, and the operator   * is generally called -order Caputo differential operator.
(1) Commensurate Order  1 =  2 =  3 = .The bifurcation diagram is calculated numerically against  ∈ [0.9, 1], while the incremental value of  is 0.0002.From the bifurcation diagram shown in Figure 2, it is found that chaotic range is  ∈ [0.928, 1].To identify the route to chaos, the time history of  5 is shown in Figures 3(a)-3(d).It is clearly shown that the state variables are stable at the fixed point at  = 0.922, which can be seen in Figure 3(a).When  increases, intermittent dynamical behavior is observed in Figures 3(b)-3(c).As  is further increased, the motion become chaotic as shown for  = 0.928, where the largest Lyapunov exponent is  = 0.0154.In Figures 4(a)-4(b), phase portraits are shown at  = 0.927 and 0.928, respectively.Numerical evidence displays that the lowest order to yield chaos is 4.64, where  = 0.928.
(2)  2 =  3 = 1, and Let  1 Vary.The bifurcation diagram is calculated numerically against  1 ∈ [0.78, 1], while the incremental value of  1 is 0.0002.Figure 5 shows that chaotic motions exist in the range  1 ∈ [0.812, 1].To identify the route to chaos, the time history of  5 is shown in Figures 6(a)-6(d).At  1 = 0.807, the state variables are stable at the fixed point as depicted in Figure 6(a).When  1 increases, intermittent dynamical behavior is observed in Figures 6(b)-6(c).As  1 is further increased, the motion become chaotic as shown for  1 = 0.812, where the largest Lyapunov exponent is  = 0.0899.In this case, the lowest order for system (5) to be chaotic is 4.624, where  1 = 0.812.
To observe the dynamical behavior of system, the region of  2 ∈ [0.75, 0.85] is expanded in step size of 0.0002 as shown in Figure 7(b).The period-doubling bifurcations can be seen in Figure 7(b).Phase diagrams shown in Figures 8(a)-8(d) exhibit period-1, period-2, period-4, and chaotic behaviors.Thus, Figure 8 identifies a period doubling route to chaos.In this case, the lowest order for system (5) to be chaotic is 4.61, where  2 = 0.805 and the largest Lyapunov exponent is  = 0.0461.At  1 = 0.820, the state variables are stable at the fixed point as depicted in Figure 10(a).When  increases, intermittent dynamical behavior is observed in Figures 10(b)-10(c).As  is further increased, the motion become chaotic as shown for  = 0.831, where the largest Lyapunov exponent is  = 0.0516.Numerical evidence displays that the lowest order for system (5) to be chaotic is 4.831, where  3 = 0.831.

Antisynchronization between Different Fractional-Order Complex Systems
In this section, we give a general method to achieve antisynchronization of different fractional-order complex systems firstly.Consequently, antisynchronization between fractional-order complex Lü and Lorenz system can be achieved.Without loss of generality, we assume that the derivative order is  ( < 1) in both master system and slave system.
Remark 1.Some fractional-order chaotic complex systems can be described by (8), such as the fractional-order complex Lorenz, Lü, and Chen systems.Now we give the stability results for linear fractionalorder systems.
Lemma 2 (see [28]).Autonomous linear system of the fractional-order   *  = , with (0) =  0 is asymptotically stable if and only if | arg(  ())| > /2, ( = 1, 2, 3 . ..).In this case, the component of the state decay towards 0 like  − .Also, this system is stable if and only if either it is asymptotically stable or those critical eigenvalues which satisfy | arg(  ())| = /2 have geometric multiplicity one, where arg(  ()) denotes the argument of the eigenvalue   of .Proof.From the definition of antisynchronization, we obtain the error vector between (8) and ( 9) as follows: The derivative of the error vector ( 12) can be expressed as  Substituting (10) into (13), the error dynamical system (13) can be written as Since | arg(  ( − ))| > /2, according to Lemma 2, the error vector () asymptotically converges to zero as  → ∞.So antisynchronization between different fractional-order complex systems is achieved by using the controller (10).This completes the proof.

Antisynchronization between Fractional-Order Complex
Lü and Lorenz System.In this section, the antisynchronization behavior between the fractional-order complex Lü and Lorenz systems is made.It is assumed that the fractionalorder complex Lü system drives the fractional-order complex Lorenz system [18].Thus the master system is described by where  15) and ( 17) (time/s).

Conclusions
In this paper, a new fractional-order chaotic complex system is proposed.By means of phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents, we investigate chaotic behavior of this new system.Our results show that the new system displays many interesting dynamical behaviors, such as fixed points, periodic motions, and chaotic motions.Two typical routes to chaosperiod doubling and intermittency-are found in this system.Besides, when the parameters of the system are fixed, the lowest order for chaos to exist is determined.Moreover, antisynchronization of different fractional-order chaotic complex systems has been studied.Meanwhile, the new system and the fractional-order complex Lorenz system can achieve antisynchronization.