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Cuimei Jiang 1 and Shutang Liu 1 and Chao Luo 2,3
Academic Editor:Felix Sadyrbaev
1, College of Control Science and Engineering, Shandong University, Jinan 250061, China
2, School of Information Science and Engineering, Shandong Normal University, Jinan 250014, China
3, Shandong Provincial Key Laboratory for Novel Distributed Computer Software Technology, Jinan 250014, China
Received 28 March 2014; Revised 28 July 2014; Accepted 28 August 2014; 20 October 2014
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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: [figure omitted; refer to PDF] Here n is the first integer which is not less than α and α > 0 , x ( n ) is the n -order derivative in the usual sense, and J β ( β > 0 ) is the β -order Riemann-Liouville integral operator with expression [figure omitted; refer to PDF] Here Γ stands for Gamma function, and the operator D * α is generally called α -order Caputo differential operator.
In 2007, the complex Lü system was proposed by Mahmoud et al. [22], which can be described as [figure omitted; refer to PDF] where x = ( x 1 , x 2 , x 3 ) T is the vector of state variables, x 1 = m 1 + j m 2 and x 2 = m 3 + j m 4 are complex variables, and x 3 = m 5 is a real variable. j = - 1 , a i > 0 ( i = 1,2 , 3 ) is the system real parameter. When the parameters are chosen as a 1 = 40 , a 2 = 22 , a 3 = 5 , the system (3) is chaotic as shown in Figure 1.
Figure 1: Chaotic attractor of complex Lü system ( a 1 , a 2 , a 3 ) = ( 40,22,5 ) .
[figure omitted; refer to PDF]
In this paper, we modify the derivative operator in (3) to be with respect to the fractional order α ( 0 < α ...4; 1 ) . Thus, the fractional-order complex Lü system can be expressed by [figure omitted; refer to PDF] when the derivative order α i = 1 ( i = 1,2 , 3 ) , system (4) will be the common integer-order complex Lü system. The Caputo differential operator is a linear operator [20]; that is, D * α ( a y ( t ) + b z ( t ) ) = a D * α ( y ( t ) ) + b D * α ( z ( t ) ) , for arbitrary constants a and b . Therefore, D * α i ( x i ) = D * α i ( m 2 i - 1 + j m 2 i ) = D * α i m 2 i - 1 + j D * α i m 2 i , ( i = 1,2 , 3 ) .
Separating the real and imaginary parts of system (4), we can obtain the following system: [figure omitted; refer to PDF]
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 ( m 10 , m 20 , m 30 , m 40 , m 50 ) can be discretized as [figure omitted; refer to PDF]
where [figure omitted; refer to PDF]
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 m 5 -axis, which means it is invariant for the coordinate transformation of ( m 1 , m 2 , m 3 , m 4 , m 5 ) [arrow right] ( - m 1 , - m 2 , - m 3 , - m 4 , m 5 ) .
2.3.2. Equilibria and Stability
The equilibria of system (5) can be calculated by solving the equations D * α j m i = 0 ( j = 1,2 , 3 ; i ... = ... 1,2 , ... , 5 ) , and this system has an isolated equilibria E 0 = ( 0,0 , 0,0 , 0 ) and nontrivial equilibria E θ = ( r cos ... θ , r sin θ , r cos ... θ , r sin θ , a 2 ) , where r = a 2 a 3 , θ ∈ [ 0,2 π ] .
As to the equilibrium E 0 , it is stable when a 2 < 0 and unstable when a 2 > 0 . For E θ , the characteristic polynomial of Jacobian matrix is λ ( λ + a 1 - a 2 ) ( λ 3 + ( a 1 + a 3 - a 2 ) λ 2 + a 1 a 3 λ + 2 a 1 a 2 a 3 ) = 0 when a 2 > 0 . According to the fractional-order Routh-Hurwitz conditions [24], when ( a 1 + a 3 - a 2 ) · ( a 1 a 3 ) > 2 a 1 a 2 a 3 , E θ 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 ( a 1 , a 2 , a 3 ) = ( 42,22,5 ) , and an initial value is ( x 1 , x 2 , x 3 ) T = ( 1 + 2 j , 3 + 4 j , 5 ) T . When varying the fractional derivative order α i ( i = 1,2 , 3 ) , 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 α 1 = α 2 = α 3 = α . The bifurcation diagram is calculated numerically against α ∈ [ 0.9,1 ] , while the incremental value of α is 0.0002. From the bifurcation diagram shown in Figure 2, it is found that chaotic range is α ∈ [ 0.928,1 ] . To identify the route to chaos, the time history of m 5 is shown in Figures 3(a)-3(d). It is clearly shown that the state variables are stable at the fixed point at α = 0.922 , which can be seen in Figure 3(a). When α increases, intermittent dynamical behavior is observed in Figures 3(b)-3(c). As α is further increased, the motion become chaotic as shown for α = 0.928 , where the largest Lyapunov exponent is λ = 0.0154 . In Figures 4(a)-4(b), phase portraits are shown at α = 0.927 and 0.928 , respectively. Numerical evidence displays that the lowest order to yield chaos is 4.64, where α = 0.928 .
Figure 2: Bifurcation diagram of system (5) with α 1 = α 2 = α 3 = α ∈ [ 0.9,1 ] .
[figure omitted; refer to PDF]
Time histories showing the rout to chaos via intermittency for system (5) at α 1 = α 2 = α 3 = α .
(a) α = 0.922
[figure omitted; refer to PDF]
(b) α = 0.926
[figure omitted; refer to PDF]
(c) α = 0.927
[figure omitted; refer to PDF]
(d) α = 0.928
[figure omitted; refer to PDF]
Phase portraits of system (5) at α 1 = α 2 = α 3 = α .
(a) α = 0.927
[figure omitted; refer to PDF]
(b) α = 0.928
[figure omitted; refer to PDF]
( 2) α 2 = α 3 = 1 , and Let α 1 Vary . The bifurcation diagram is calculated numerically against α 1 ∈ [ 0.78,1 ] , while the incremental value of α 1 is 0.0002. Figure 5 shows that chaotic motions exist in the range α 1 ∈ [ 0.812,1 ] . To identify the route to chaos, the time history of m 5 is shown in Figures 6(a)-6(d). At α 1 = 0.807 , the state variables are stable at the fixed point as depicted in Figure 6(a). When α 1 increases, intermittent dynamical behavior is observed in Figures 6(b)-6(c). As α 1 is further increased, the motion become chaotic as shown for α 1 = 0.812 , where the largest Lyapunov exponent is λ = 0.0899 . In this case, the lowest order for system (5) to be chaotic is 4.624, where α 1 = 0.812 .
Figure 5: Bifurcation diagram of system (5) with α 1 ∈ [ 0.78,1 ] .
[figure omitted; refer to PDF]
Time histories showing the rout to chaos via intermittency for system (5) at α 2 = α 3 = 1 with α 1 varying.
(a) α 1 = 0.807
[figure omitted; refer to PDF]
(b) α 1 = 0.810
[figure omitted; refer to PDF]
(c) α 1 = 0.811
[figure omitted; refer to PDF]
(d) α 1 = 0.812
[figure omitted; refer to PDF]
( 3) α 1 = α 3 = 1 , and Let α 2 Vary . The bifurcation diagram is calculated numerically against α 2 ∈ [ 0.75,1 ] , while the incremental value of α is 0.0005. Figure 7(a) shows that the chaotic zone covers most of the range of α 2 ∈ [ 0.805,1 ] . To observe the dynamical behavior of system, the region of α 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 α 2 = 0.805 and the largest Lyapunov exponent is λ = 0.0461 .
Bifurcation diagram of system (5) at α 1 = α 3 = 1 with α 2 varying.
(a) α 2 ∈ ( 0.75,1 )
[figure omitted; refer to PDF]
(b) α 2 ∈ ( 0.75,0.85 )
[figure omitted; refer to PDF]
Phase portraits of system (6) at α 1 = α 3 = 1 with α 2 varying.
(a) α 2 = 0.801
[figure omitted; refer to PDF]
(b) α 2 = 0.802
[figure omitted; refer to PDF]
(c) α 2 = 0.804
[figure omitted; refer to PDF]
(d) α 2 = 0.805
[figure omitted; refer to PDF]
( 4) α 1 = α 2 = 1 , and Let α 3 Vary . The system is calculated numerically against α 3 ∈ [ 0.8,1 ] with the step size of 0.001. Figures 9(a)-9(d) displays the phase portraits at α 3 = 0.830,0.831,0.90 and 0.950 , respectively. Results show that chaos exists in the range α 3 ∈ [ 0.831,1 ] . To identify the route to chaos, the time history of m 5 is shown in Figures 10(a)-10(d). At α 1 = 0.820 , the state variables are stable at the fixed point as depicted in Figure 10(a). When α increases, intermittent dynamical behavior is observed in Figures 10(b)-10(c). As α is further increased, the motion become chaotic as shown for α = 0.831 , where the largest Lyapunov exponent is λ = 0.0516 . Numerical evidence displays that the lowest order for system (5) to be chaotic is 4.831, where α 3 = 0.831 .
Phase portraits of system (5) at α 1 = α 2 = 1 with α 3 varying.
(a) α 3 = 0.830
[figure omitted; refer to PDF]
(b) α 3 = 0.831
[figure omitted; refer to PDF]
(c) α 3 = 0.90
[figure omitted; refer to PDF]
(d) α 3 = 0.950
[figure omitted; refer to PDF]
Time histories showing the rout to chaos via intermittency for system (5) at α 1 = α 2 = 1 with α 3 varying.
(a) α 3 = 0.820
[figure omitted; refer to PDF]
(b) α 3 = 0.825
[figure omitted; refer to PDF]
(c) α 3 = 0.830
[figure omitted; refer to PDF]
(d) α 3 = 0.831
[figure omitted; refer to PDF]
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 α ( α < 1 ) 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: [figure omitted; refer to PDF] where x = ( x 1 , x 2 , ... , x n ) T is the state complex vector, x = x r + j x i , and define x 1 = m 1 + j m 2 , x 2 = m 3 + j m 4 , ... , x n = m 2 n - 1 + j m 2 n . f = ( f 1 , f 2 , ... , f n ) T is a vector of nonlinear complex functions and A ∈ R n × n is the matrix of system parameters. Superscripts r and i 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 [figure omitted; refer to PDF] where y = ( y 1 , y 2 , ... , y n ) T is the state complex vector, y = y r + j y i , and define y 1 = s 1 + j s 2 , y 2 = s 3 + j s 4 , ... , y n = s 2 n - 1 + j s 2 n . u = u r + j u i is designed controller, where u r = ( u 1 , u 3 , ... , u 2 n - 1 ) T , u i = ( u 2 , u 4 , ... , u 2 n ) T .
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 D * α x = A x , with x ( 0 ) = x 0 is asymptotically stable if and only if | arg ( λ i ( A ) ) | > α π / 2 , ( i = 1,2 , 3 ... ) . In this case, the component of the state decay towards 0 like t - α . Also, this system is stable if and only if either it is asymptotically stable or those critical eigenvalues which satisfy | arg ( λ i ( A ) ) | = α π / 2 have geometric multiplicity one, where arg ( λ i ( A ) ) denotes the argument of the eigenvalue λ i of A .
Theorem 3.
Antisynchronization between (8) and (9) will be achieved, if the controller is designed as follows: [figure omitted; refer to PDF] The real and imaginary parts of (10) are [figure omitted; refer to PDF] here K is the control gain matrix, which satisfies | arg ( λ i ( B - K ) ) | > α π / 2 for all the eigenvalues of B - K .
Proof.
From the definition of antisynchronization, we obtain the error vector between (8) and (9) as follows: [figure omitted; refer to PDF] The derivative of the error vector (12) can be expressed as [figure omitted; refer to PDF] Substituting (10) into (13), the error dynamical system (13) can be written as [figure omitted; refer to PDF] Since | arg ( λ i ( B - K ) ) | > α π / 2 , according to Lemma 2, the error vector e ( t ) asymptotically converges to zero as t [arrow right] ∞ . So antisynchronization between different fractional-order complex systems is achieved by using the controller (10). This completes the proof.
Remark 4.
If A = B and f ( · ) = g ( · ) , 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 [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The slave system is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] In the numerical simulations, the initial values of the master and slave systems are ( x 1 m , x 2 m , x 3 m ) T = ( 1 + 2 j , 3 + 4 j , 5 ) T and ( x 1 s , x 2 s , x 3 s ) T = ( - 1 + 9 j , 8 - 5 j , 1 ) T , respectively. Choose the parameters of the master and slave system as ( a 1 , a 2 , a 3 ) = ( 42,22,5 ) , ( b 1 , b 2 , b 3 ) = ( 10,180,1 ) , α 1 = α 2 = α 3 = 0.95 . In order to satisfy | ... arg ( λ i ( B - K ) ) | > α π / 2 , we choose the gain control matrix K as follows: [figure omitted; refer to PDF]
According to Theorem 3, the controller is designed as [figure omitted; refer to PDF]
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.
Figure 11: The time evolution of the synchronization errors between systems (15) and (17) (time/s).
[figure omitted; refer to PDF]
Figure 12: State variables of master system (15) and slave (17) (time/s).
[figure omitted; refer to PDF]
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.
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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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.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer