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Modified generalized projective synchronization of fractionalorder chaotic Lü systems
Advances in Difference Equations volume 2013, Article number: 374 (2013)
Abstract
This paper addresses new modified generalized projective synchronization (MGPS) of fractionalorder chaotic systems based on the stability theory of fractionalorder systems, where the drive and response systems could be asymptotically synchronized up to a desired transformation matrix, not a diagonal matrix. MGPS between the hyperchaotic Lorenz system and the Lü system of the base order 0.95 is implemented as an example. Numerical simulations show the effectiveness and feasibility of the method.
PACS Codes:05.45.
1 Introduction
It is well known that fractional calculus is a classical mathematical notion, with a history as long as calculus itself. But its applications to physics and engineering are a subject of only recent interest [1, 2]. It was found that many systems in interdisciplinary fields can be elegantly described with the help of fractional derivatives, for instance, viscoelastic systems [3], electromagnetic waves [4], dielectric polarization [5], quantitative finance [6], quantum evolution of complex systems [7], and so forth.
More recently, many investigations were devoted to the chaotic behavior, chaotic control, and synchronization of fractionalorder dynamical systems. For example, it has been shown that Chua’s circuit with an order as low as 2.7 can produce a chaotic attractor [8]. In [9], it was shown that nonautonomous Duffing systems with an order less than 2.0 can still behave in a chaotic manner. In [10], chaotic behavior of the fractionalorder ‘jerk’ model, in which chaotic attractors can be obtained with a system of the order as low as 2.1, was studied. Bifurcations and chaos in the fractionalorder simplified Lorenz system [11], chaotic behavior and its control in the fractionalorder Chen system [12] were reported. In [13], chaotic and hyperchaotic behaviors in fractionalorder Rössler equations were studied. Chaotic dynamics and synchronization of fractionalorder Arneodo systems [14], the Lü system [15], and a unified system [16], synchronization of fractionalorder hyperchaotic modified systems [17] were also reported.
Recently, projective synchronization (PS) has been especially extensively studied because it can be used to obtain faster communication with its proportional feature and the unpredictability of the scaling factor can additionally enhance the security of communication [18–20]. In [21, 22], Wu et al. presented the generalized projective synchronization (GPS) method for fractionalorder Chen hyperchaotic systems, which associates with the projective synchronization and the generalized one, where the drive and response systems could be synchronized up to scaling factors ${\theta}_{i}$. In the practical applications, a more general form synchronization, called modified generalized projective synchronization (MGPS), where the drive and response systems can be synchronized up to a transformation matrix that is not diagonal, will increase the complexity of the synchronization and further increase the diversity and the security of communications. Moreover, to the best of our knowledge, most of the existing papers only consider constant scaling factors which is a diagonal matrix, and the MGPS, which has rarely been explored, will contain GPS with constant scaling factors and extend previous works. Therefore, MGPS of fractionalorder chaotic systems becomes a new meaningful problem.
Motivated by the above discussion, this paper introduces a fractionalorder chaotic Lü system and aims to investigate this new MGPS of fractionalorder chaotic systems with different structure, where the drive and response systems could be asymptotically synchronized up to a desired transformation matrix, not a diagonal matrix. Based on the stability theory of fractionalorder systems, the controllers are designed to make the drive and response systems synchronize up to the desired transformation matrix.
This paper is organized as follows. In Section 2, a brief review of the fractional derivative and numerical algorithm for the fractionalorder system is given. Dynamics of a novel fractionalorder chaotic system is numerically studied and demonstrated by computer simulation. In Section 3, a general method of MGPS for coupled fractionalorder chaotic systems is presented based on the stability theory of fractionalorder systems. MGPS between the fractionalorder hyperchaotic Lorenz system and the Lü system is derived, and numerical simulations show the effectiveness and feasibility of the proposed synchronization scheme. Finally, the conclusions are given in Section 4.
2 A novel fractionalorder chaotic system
2.1 Fractional derivative and its approximation method
There are many definitions of fractional differential operators [1]. The definition of the RiemannLiouville derivative is given as follows:
where $m=\lceil \alpha \rceil $, i.e., m is the first integer which is not less than α, ${J}^{\beta}$ is the βorder RiemannLiouville integral operator as described by
where $\mathrm{\Gamma}(\u2022)$ denotes the gamma function $\mathrm{\Gamma}(x)={\int}_{0}^{\mathrm{\infty}}{t}^{x1}{e}^{t}\phantom{\rule{0.2em}{0ex}}dt$.
Here and throughout, the following Caputo definition is applied:
where $m=\lceil \alpha \rceil $.
Here we choose the Caputo version and use an improved predictorcorrector algorithm, i.e., the AdamsBashforthMoulton predictorcorrectors scheme for fractional differential equations [23–27], where the numerical approximation is a timedomain approach that is more accurate, and the computational cost is greatly reduced.
The improved fractional predictorcorrector algorithm is based on the analytical property of the following differential equation:
which is equivalent to the Volterra integral equation
Now, set $h=T/N$, ${t}_{n}=nh$ ($n=0,1,2,\dots ,N\in {Z}^{+}$). Equation (2.5) can be written as
where the predicted value ${x}_{h}^{\theta}({t}_{n+1})$ is determined by
and
The estimation error in this method is
2.2 Dynamic analysis of a novel fractionalorder system
2.2.1 Novel fractionalorder system model
In 2006, Wang et al. introduced a new modified hyperchaotic Lü system [28], further investigated its dynamical behaviors, and physically implemented the system which is described by
where $(x,y,z,w)\in {R}^{4}$, and a, b, c, d are real constant parameters. The hyperchaotic attractors of system (2.10) for $a=70$, $b=15$, $c=12$, $d=5$ are plotted in Figure 1.
In 2011, Mahmoud et al. studied modified hyperchaotic complex Lü systems [29] given by
where $(x,y)\in {C}^{2}$ are complex variables, $(z,w)\in {R}^{2}$ are real variables, and a, b, c and d are real (or complex) positive constant parameters. The hyperchaotic attractors of (2.11) using the same choice of the parameters and initial conditions as in the real system (2.10) are plotted in Figure 2. More detailed complex dynamics of the modified hyperchaotic complex Lü system can be seen in [29].
In the present paper, based on the above descriptions, we modify the derivative operator in Eq. (2.10) to be with respect to the fractional order α ($0<\alpha \le 1$). Thus the fractional version of the hyperchaotic Lü system is given by
where a, b, c and d are real positive constant parameters. When $\alpha =1$, system (2.12) reduces to the classical integerorder modified hyperchaotic Lü system (2.10).
2.2.2 Dynamic analysis of the fractionalorder Lü system
According to the numerical algorithm for fractional differential systems in Section 2.1, system (2.12) for initial condition $({x}_{0},{y}_{0},{z}_{0},{w}_{0})$ can be written as
where
In the following simulations, the system parameters are always chosen as $a=70$, $b=15$, $c=12$, $d=5$. The simulation results demonstrate that chaos indeed exists in the fractionalorder system (2.12) with order less than 4. For example, when $\alpha =0.95$, a chaotic attractor is found, and the phase portraits of the system are displayed in Figure 3. Equation (2.12) has no solution when $\alpha <0.8320$. A chaotic attractor is found for $\alpha =0.8321$, and its phase portrait is displayed in Figure 4. Thus the lower limit of fractional order for this system to be chaotic is between $\alpha =0.8320$ and $\alpha =0.8321$, and the lowest order we found for this system to yield chaotic is 3.3284.
The largest Lyapunov exponent of the simulation time series [30, 31] is positive only when $\alpha \ge 0.8321$, for example, $\alpha =0.8321$, ${\lambda}_{1}=19.78383$. Obviously, the fractionalorder system (2.12) is chaotic.
3 MGPS between the different fractionalorder systems
3.1 A general method for MGPS of fractionalorder systems
In this section we apply stability analysis to fractionalorder systems. Fractionalorder differential equations are at least as stable as their integerorder counterparts because systems with memory are typically more stable than those without memory [23, 26].
Consider the fractionalorder chaotic drive and response systems as
and
where the state vectors $X,Y\in {R}^{n}$, $f,g:{R}^{n}\to {R}^{n}$ are continuous vector functions and $U(X,Y):{R}^{2n}\to {R}^{n}$ is a controller to be determined later.
Decompose the fractionalorder systems (3.1) and (3.2) as
and
where $A,B:{R}^{n\times n}\to {R}^{n\times n}$ is the Jacobian matrix of the system at the origin and $F,G:{R}^{n}\to {R}^{n}$ are the nonlinear parts.
Remark 1 Many fractionalorder chaotic systems belong to the class individualized by (3.3). Examples include the fractionalorder hyperchaotic modified Rössler system [13, 21], the Chen system [19], and the Lorenz systems [20, 22].
Definition 1 The drive system (3.3) and response system (3.4) are said to achieve MGPS if there exists a controller U such that ${lim}_{t\to \mathrm{\infty}}\parallel YCX\parallel =0$, where $C:{R}^{n\times n}\to {R}^{n\times n}$ is a transformation matrix of the drive system (3.3).
Remark 2 In particular, if the transformation matrix is diagonal and all diagonal elements are the same, the MGPS is simplified to the GPS.
Hence, the error system is defined as
which means that system (3.4) synchronizes with the projection of system (3.3). If ${lim}_{t\to \mathrm{\infty}}\parallel e\parallel =0$, systems (3.3) and (3.4) substituted into (3.5) give an error system that can be expressed as
Then the discussions of MGPS between the two coupled systems (3.3) and (3.4) can be translated into the analysis of the asymptotic stability of the zero solution of the error system (3.6). Next, a suitable controller is provided to ensure the asymptotic stability of the zero solution of the error system (3.6) based on the stability theorem of linear fractionalorder systems.
For a given autonomous fractionalorder linear system
with $z(0)={z}_{0}$, where $0<\alpha <1$ and $z\in {R}^{n}$, M is a constant matrix.
Lemma 1 [32]
System (3.1) is

(i)
asymptotically stable if and only if
$$arg({\lambda}_{i}(M))>\alpha \pi /2\phantom{\rule{1em}{0ex}}(i=1,2,\dots ,n),$$(3.8)
where $arg({\lambda}_{i}(M))$ denotes the argument of the eigenvalue ${\lambda}_{i}$ of M. In this case, each component of the states decays toward 0 like ${t}^{\alpha}$,

(ii)
stable if and only if $arg({\lambda}_{i}(M))\ge \alpha \pi /2$ ($i=1,2,\dots ,n$), and those critical eigenvalues ${\lambda}_{i}$ that satisfy $arg({\lambda}_{i}(M))=\alpha \pi /2$ ($i=1,2,\dots ,n$) have geometric multiplicity one.
Now, due to Lemma 1, the following results can be obtained.
Theorem 1 Given a fractionalorder drive system (3.3) and a response system (3.4), there exists a suitable controller
where $K\in {R}^{n\times n}$ is a gain matrix. MGPS between systems (3.3) and (3.4) can be achieved if and only if all the eigenvalues of $B+K$ satisfy $arg({\lambda}_{i}(B+K))>\alpha \pi /2$ ($i=1,2,\dots ,n$).
Proof Substituting controller (3.9) into system (3.6), the error system (3.6) can be rewritten as
Due to Lemma 1, we arrive at the conclusion that system (3.10) is asymptotically stable if and only if all the eigenvalues ${\lambda}_{i}(B+K)$ satisfy $arg({\lambda}_{i}(B+K))>\alpha \pi /2$ ($i=1,2,\dots ,n$). That is, ${lim}_{t\to \mathrm{\infty}}\parallel e\parallel =0$, or systems (3.3) and (3.4) realize MGPS. This completes the proof. □
3.2 MGPS between the fractionalorder hyperchaotic Lorenz system and the Lü system
In this section the MGPS behavior between two different fractionalorder systems, the hyperchaotic Lorenz system and the modified Lü system, is made. It is assumed that the fractionalorder Lü system drives the hyperchaotic Lorenz system. Therefore, we define the Lü system as a master system and the hyperchaotic Lorenz system as a slave system as follows. The master system described through Eq. (2.12) is
The slave system described in [22] is
where k, m, n, p are parameters and $x,y,z,w\in R$ are variables, four functions ${u}_{i}(t)$ ($i=1,2,3,4$) are the controller to be determined later so that the drive and response systems can be synchronized in the sense of MGPS.
Comparing systems (3.11) and (3.12) with systems (3.3) and (3.4), one has
Here, choose a transformation matrix
The error system $e=YCX$ between systems (3.11) and (3.12) is
According to Theorem 1, with a suitable controller $U=CAx+CF(x)BCxG(y)+Ke$, $K\in {R}^{n\times n}$, the error dynamical system can be obtained in the form
Then Theorem 1 assures that there exists a feedback gain matrix K so that systems (3.11) and (3.12) realize the synchronization.
There is not a unique choice for such a matrix K. For example, when $\alpha =0.95$, $(k,m,n,p)=(10,28,8/3,0.1)$, and $(a,b,c,d)=(70,15,12,5)$, we can set $B+K=diag(1,2,1,3)$ with the matrix
so that all the eigenvalues ${\lambda}_{i}=(1,2,1,3)$ of $B+K$ satisfy $arg({\lambda}_{i}(B+K))>0.95\pi /2$ ($i=1,2,\dots ,n$).
In the numerical simulations, the initial values of the drive and response systems are arbitrarily chosen as $({x}_{1}(0),{x}_{2}(0),{x}_{3}(0),{x}_{4}(0))=(1,2,3,4)$, and $({y}_{1}(0),{y}_{2}(0),{y}_{3}(0),{y}_{4}(0))=(0.3,0.6,0.3,1)$. The time evolutions of the states are plotted in Figure 5(a)(d). Figure 5(e) displays the evolution of the MGPS error $e={({e}_{1},{e}_{2},{e}_{3},{e}_{4})}^{T}$ which tends to zero as $t\to \mathrm{\infty}$, which implies that the error system (3.14) between the drive and response systems (3.11)(3.12) is globally and asymptotically stable.
What should be mentioned is that the fractional order and control gains, i.e., the elements of K, play an important role in the synchronization rate.
4 Conclusion
In this paper, MGPS of a novel fractionalorder chaotic system is presented. First, we numerically study the chaotic behavior of a novel fractionalorder fourdimensional Lü system according to the generalized AdamsBashforthMoulton predictorcorrector algorithm. Then, based on the stability theory of fractionalorder systems, the MGPS method for a class of fractionalorder chaotic systems is presented, and a nonlinear controller is given to control the slave system to become a projection of the master system. In MGPS, the drive and response systems could be asymptotically synchronized up to a desired transformation matrix, but not a diagonal matrix.
This synchronization method can be easily extended to other chaotic systems. The classical ‘PC’ complete synchronization, antisynchronization, and generalized projective synchronization can be considered as special cases of our scheme. For verifying the effectiveness and feasibility of the presented synchronization scheme, some numerical simulations were performed.
It is worth mentioning that there are still many interesting problems about chaos synchronization of different fractionalorder systems that warrant investigation in future, such as lag MGPS and adaptive MGPS.
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Acknowledgements
This research is supported by the National Natural Science Foundation of China (Grant No. 61273088), the Natural Science Foundation of Shandong Province, China (Grant Nos. ZR2010FM010, ZR2011AL007 and ZR2010AL016), the Foundation for University Young Key Teacher Program of Shandong Provincial Education Department, China. The authors would like to thank the referees for their helpful comments and suggestions which greatly improved the presentation of the paper.
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Liu, J., Liu, S. & Yuan, C. Modified generalized projective synchronization of fractionalorder chaotic Lü systems. Adv Differ Equ 2013, 374 (2013). https://doi.org/10.1186/168718472013374
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Keywords
 fractionalorder system
 chaotic system
 Lyapunov exponent
 modified generalized projective synchronization