Abstract
We propose a new fractional-order chaotic complex system and study its dynamical properties including symmetry, equilibria and their stability, and chaotic attractors. Chaotic behavior is verified with phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents. And we find that chaos exists in this system with orders less than 5 by numerical simulation. Additionally, antisynchronization of different fractional-order chaotic complex systems is considered based on the stability theory of fractional-order systems. This new system and the fractional-order complex Lorenz system can achieve antisynchronization. Corresponding numerical simulations show the effectiveness and feasibility of the scheme.
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 fractional-order 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 [9–11]. 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 fractional-order complex Lorenz system. A concluding remark is given in Section 4.
2. The Fractional-Order Complex Lü System
2.1. 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 , is the -order derivative in the usual sense, and is the -order Riemann-Liouville integral operator with expression Here stands for Gamma function, and the operator is generally called -order Caputo differential operator.
In 2007, the complex Lü system was proposed by Mahmoud et al. [22], which can be described as where is the vector of state variables, and are complex variables, and is a real variable., is the system real parameter. When the parameters are chosen as , , , the system (3) is chaotic as shown in Figure 1.
In this paper, we modify the derivative operator in (3) to be with respect to the fractional order . Thus, the fractional-order complex Lü system can be expressed by when the derivative order , system (4) will be the common integer-order complex Lü system. The Caputo differential operator is a linear operator [20]; that is, , for arbitrary constants and . Therefore, , .
Separating the real and imaginary parts of system (4), we can obtain the following system:
2.2. Numerical Algorithm for the Fractional-Order Complex Lü System
In 2002, Diethelm et al. proposed the Adams-Bashforth-Moulton predictor-corrector scheme [23], which is numerically stable and can be used to both linear and nonlinear fractional differential equations. According to this algorithm, system (5) for initial condition can be discretized as
where
2.3. Dynamics of the Fractional-Order Complex Lü System
2.3.1. Symmetry and Invariance
Note that the symmetry of system (5) is symmetric about -axis, which means it is invariant for the coordinate transformation of .
2.3.2. Equilibria and Stability
The equilibria of system (5) can be calculated by solving the equations , and this system has an isolated equilibria and nontrivial equilibria , where .
As to the equilibrium , it is stable when and unstable when . For , the characteristic polynomial of Jacobian matrix is when . According to the fractional-order Routh-Hurwitz conditions [24], when , will be stable.
2.3.3. Chaotic Behavior and Attractors
Using the above discretization scheme (6), we find that chaotic behaviors exist in the fractional-order complex Lü system. In the numerical simulations, the system parameters are chosen as , and an initial value is . When varying the fractional derivative order , system (5) will display diverse motions. The existence of chaos is demonstrated with the time histories, phase diagrams, bifurcation diagrams, and the largest Lyapunov exponents. It is well known that there are many effective algorithms for the calculation of the Lyapunov exponents [25–27]. In this paper, the largest Lyapunov exponents are calculated by Wolf algorithm [25].
(1) Commensurate Order . The bifurcation diagram is calculated numerically against , while the incremental value of is 0.0002. From the bifurcation diagram shown in Figure 2, it is found that chaotic range is . To identify the route to chaos, the time history of is shown in Figures 3(a)–3(d). It is clearly shown that the state variables are stable at the fixed point at , 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 , where the largest Lyapunov exponent is . In Figures 4(a)-4(b), phase portraits are shown at and , respectively. Numerical evidence displays that the lowest order to yield chaos is 4.64, where .
(a)
(b)
(c)
(d)
(a)
(b)
(2) , and Let Vary. The bifurcation diagram is calculated numerically against , while the incremental value of is 0.0002. Figure 5 shows that chaotic motions exist in the range . To identify the route to chaos, the time history of is shown in Figures 6(a)–6(d). At , the state variables are stable at the fixed point as depicted in Figure 6(a). When increases, intermittent dynamical behavior is observed in Figures 6(b)-6(c). As is further increased, the motion become chaotic as shown for , where the largest Lyapunov exponent is . In this case, the lowest order for system (5) to be chaotic is 4.624, where .
(a)
(b)
(c)
(d)
(3) , and Let Vary. The bifurcation diagram is calculated numerically against , while the incremental value of is 0.0005. Figure 7(a) shows that the chaotic zone covers most of the range of . To observe the dynamical behavior of system, the region of 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 and the largest Lyapunov exponent is .
(a)
(b)
(a)
(b)
(c)
(d)
(4) , and Let Vary. The system is calculated numerically against with the step size of 0.001. Figures 9(a)–9(d) displays the phase portraits at and , respectively. Results show that chaos exists in the range . To identify the route to chaos, the time history of is shown in Figures 10(a)–10(d). At , 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 , where the largest Lyapunov exponent is . Numerical evidence displays that the lowest order for system (5) to be chaotic is 4.831, where .
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
3. 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 () in both master system and slave system.
3.1. A General Method for Antisynchronization of Fractional-Order Complex Systems
Consider the following fractional-order complex system: where is the state complex vector, , and define , . is a vector of nonlinear complex functions and is the matrix of system parameters. Superscripts and stand for the real and imaginary parts of the state complex vector. System (8) is considered as the master system and the slave system is given by where is the state complex vector, , and define , . is designed controller, where , .
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 fractional-order systems.
Lemma 2 (see [28]). Autonomous linear system of the fractional-order , with is asymptotically stable if and only if , . In this case, the component of the state decay towards like . Also, this system is stable if and only if either it is asymptotically stable or those critical eigenvalues which satisfy have geometric multiplicity one, where denotes the argument of the eigenvalue of .
Theorem 3. Antisynchronization between (8) and (9) will be achieved, if the controller is designed as follows: The real and imaginary parts of (10) are here is the control gain matrix, which satisfies for all the eigenvalues 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 , 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.
Remark 4. If and , systems (8) and (9) become identical. Therefore, our scheme is also applicable to achieve antisynchronization of two identical fractional-order chaotic complex systems.
3.2. 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 fractional-order complex Lü system drives the fractional-order complex Lorenz system [18]. Thus the master system is described by where The slave system is where In the numerical simulations, the initial values of the master and slave systems are and , respectively. Choose the parameters of the master and slave system as , , . In order to satisfy , we choose the gain control matrix as follows:
According to Theorem 3, the controller is designed as
The errors of antisynchronization converge asymptotically to zero in a quite short period as depicted in Figure 11. Figure 12 shows state variables of drive system and response system, in which the state complex variables are demonstrated by real and imaginary part, respectively. The above results verify that antisynchronization between fractional-order complex Lü system and Lorenz system has been achieved.
4. 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 chaos—period 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The research is supported by the National Nature Science Foundation of China (nos. 61273088 and 10971120) and the Nature Science Foundation of Shandong province (no. ZR2010FM010). The authors would like to thank the editors and anonymous referees for their constructive comments and suggestions.