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Complex modified generalized projective synchronization of fractionalorder complex chaos and real chaos
Advances in Difference Equations volume 2015, Article number: 274 (2015)
Abstract
This paper introduces a type of modified generalized projective synchronization with complex transformation matrix (CMGPS) for fractionalorder complex chaos and real chaos with the same dimension and different structures. The transformation matrix in this type of chaos synchronization is a nondiagonal square matrix, and its elements are complex numbers. Based on the stability theory of fractionalorder systems, necessary and sufficient criteria are established to guarantee CMGPS for the fractionalorder complex chaos and fractionalorder real chaos, and for two fractionalorder complex chaotic systems, respectively. Numerical examples are provided to illustrate the feasibility and effectiveness of our theoretical results.
Introduction
In the last thirty years, with the development of interdisciplinary applications, it was found that many systems in interdisciplinary fields can be elegantly described with the help of fractional derivatives, for instance, viscoelastic systems [1], dielectric polarization [2], quantitative finance [3], quantum evolution of complex systems [4], and so forth. Due to the above wide scope of applications, many researchers devoted much effort to chaotic behaviors, chaotic control, and synchronization of fractionalorder dynamical systems in a real space. For example, Hartley et al. introduced the fractionalorder continuous ChuaHartley’s system [5], Arena et al. considered the fractionalorder cellular neural network [6], Gao and Yu presented the fractional continuous Duffing’s systems [7], Wu et al. addressed discrete chaos and synchronization of the fractional logistic, sine and standard maps recently [8–10]. The projective synchronization (PS) [11] 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. In [12], Wu and Lu presented a modified projective synchronization (MPS) 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 \(\delta_{i}\). Liu et al. [13] introduced modified generalized projective synchronization (MGPS) of fractionalorder chaotic systems with different structure, where the drive and response systems could be asymptotically synchronized up to a desired nondiagonal transformation matrix.
In 2013, several research results were proposed about the dynamic properties and synchronization of fractionalorder chaotic systems in a complex space. The fractionalorder complex Lorenz system was proposed and dynamics of the system was investigated in detail [14]. Luo and Wang introduced the fractionalorder complex Chen system and its application to digital secure communication, the complex variables (doubling the number of variables) increase the content of transmitting information signals and enhance their security further [15].
However, all the scaling factors in the above synchronization are real numbers. That is to say, the drive and response systems evolve in the same or inverse direction simultaneously. In fact, for complex dynamical systems, the scaling factors can be complex [16–19], the drive and response systems may evolve in different directions with a constant intersection angle, for example, \(\zeta=\rho e^{j\gamma} \eta\), where \(\rho e^{j\gamma}=\rho(\cos\gamma +j\sin\gamma)\), ζ and η denote the complex state variables of drive and response systems, respectively, \(\rho>0\) denotes the zoom rate, \(\gamma\in[0, 2\pi)\) denotes the rotate angle. Moreover, as the complex scaling factors are arbitrary and more unpredictable than real scaling factors and the operations of complex numbers are complicated, the possibility that an interceptor extracts the information from the transmitted signal is greatly less than real scaling factors, which will also increase security and variety of communications. However, complex synchronization of fractionalorder complex chaos and fractionalorder real chaos is less. Only in [20], Liu introduced modified hybrid projective synchronization with complex transformation matrix (CMHPS) for different dimensional fractionalorder complex chaos and fractionalorder real hyperchaos. Naturally, a question may be put forth: Does there exist another kind of complex synchronization, where the same dimensional fractionalorder complex chaos and real chaos could be synchronized up to a nondiagonal complex transformation matrix \(\Lambda={\Lambda^{r}} +j{\Lambda^{i}}\)? In the practical applications, there does exist this type of synchronization, called modified generalized projective synchronization with complex transformation matrix (CMGPS). By means of the complex state transformation matrix, every state variable in the response system will be involved in multiple state variables of the drive system, which will increase the complexity of the synchronization and further increase the diversity and the security of communications. Therefore, it is interesting and significant to study CMGPS of two fractionalorder complex chaotic systems and that of fractionalorder complex chaos and real chaos. However, to the best of our knowledge, this type of CMGPS for fractionalorder chaotic systems has rarely been reported.
Motivated by the above discussion, CMGPS is addressed for fractionalorder complex chaos and real chaos with the same dimension and different structures based on the stability theory of fractionalorder systems. In addition, CMGPS will contain MPS with real constant scaling matrix and MGPS with real transformation matrix and extend previous works.
The rest of this paper is organized as follows. In Section 2, a brief review of the fractional derivative and numerical algorithm and the stability theory of the fractionalorder system is given. General methods of CMGPS for the fractionalorder real chaotic drive system and the complex chaotic response system, for the fractionalorder complex chaotic drive system and the real chaotic response system, and for two fractionalorder complex chaotic systems are presented in Section 3, Section 4 and Section 5, respectively. Three numerical examples are provided in Section 6. Finally, some conclusions are drawn in Section 7.
Notations
\(\mathbb{R}^{n}\) stands for ndimensional real vector space, \(\mathbb{C}^{n}\) stands for ndimensional complex vector space. If \(z\in\mathbb{C}^{n}\) is a complex vector, then \(z=z^{r}+jz^{i}\), \(j=\sqrt{1}\) is the imaginary unit, superscripts r and i stand for the real and imaginary parts of z, respectively, \(z^{\mathrm{T}}\) are the transpose of z, respectively, and \(\z\\) implies the 2norm of z. If z is a complex scalar, z̄ is the conjugate of z.
Assume \(\alpha>0\), then \(\lceil\alpha \rceil\) is just the value α rounded up to the nearest integer, \(J^{\alpha}\) denotes RiemannLiouville type fractional integral of order α, \(D^{\alpha}\) denotes RiemannLiouville type fractional derivative of order α, \(D_{*}^{\alpha}\)denotes Caputo type fractional derivative of order α, \(\Gamma(\cdot)\) denotes the gamma function \(\Gamma(x) = \int_{0}^{\infty}{{t^{x  1}}{e^{  t}}\,dt}\), \(x > 0 \).
Preliminaries
The definition of fractional derivative
There are many definitions of fractional derivative [21]. The definition of the RiemannLiouville derivative is given as
where \(\alpha>0\), \(m: = \lceil\alpha \rceil\), \(J^{\beta}\) is the βorder RiemannLiouville integral operator as described by
where \(0 < \beta \le1\).
The Caputo fractional derivative is defined as
where \(\alpha>0\), \(m: = \lceil\alpha \rceil\).
Generally speaking, there are two numerical methods suitable for chaos synchronization of fractional differential systems. One is the frequencydomain method [22], another is the timedomain method. Here, the Caputo version and an improved predictorcorrector algorithm, i.e., the AdamsBashforthMoulton predictorcorrectors scheme are chosen for fractional differential equations, where the numerical approximation is a timedomain approach that is more accurate, and the computational cost is greatly reduced [23, 24].
Numerical algorithms
The fractional predictorcorrector algorithm [23] is based on the analytical property of the following fractional differential equation:
which is equivalent to the Volterra integral equation
Now, set \(h = {T/N}\), \({t_{n}} = nh\) (\(n = 0,1,2, \ldots,N \in {Z^{+} }\)). Equation (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
where \(\theta = \min(2,1 + \alpha)\).
The stability of fractionalorder systems
For a given fractionalorder linear timeinvariant system
with \(x(0) = {x_{0}}\), where \(0<\alpha<1\) and \(x \in{R^{n}}\), M is a constant matrix.
Lemma 1
([25])
System (11) is

(i)
asymptotically stable if and only if
$$ \bigl\vert {\arg\bigl({\lambda_{\ell}}(M)\bigr)} \bigr\vert > \frac{{\alpha\pi }}{2}\quad (\ell = 1,2, \ldots,n), $$(12)where \(\arg({\lambda_{\ell}}(M))\) denotes the argument of the eigenvalue \({\lambda_{\ell}}\) of M. In this case, each component of the states decays toward 0 like \({t^{  \alpha}}\).

(ii)
stable if and only if
$$ \bigl\vert {\arg\bigl({\lambda_{\ell}}(M)\bigr)} \bigr\vert \geq\frac{{\alpha\pi }}{2}\quad (\ell = 1,2, \ldots,n), $$(13)and those critical eigenvalues \({\lambda_{i}}\) that satisfy \(\vert {\arg({\lambda_{\ell}}(M))} \vert = \alpha\pi/2 \) (\(\ell = 1,2, \ldots,n\)), have geometric multiplicity one.
Fractionalorder differential equations are at least as stable as their integer order counterpart because systems with memory are typically more stable than those without memory [23].
CMGPS scheme of fractionalorder chaotic real drive system and complex response system
Mathematical model and problem descriptions
First, a class of ndimensional fractionalorder chaotic real drive systems is considered as
where \(x=(x_{1},x_{2},\ldots, x_{n})^{\mathrm{T}}\in\mathbb{R}^{n}\) is a real state vector, \(M\in\mathbb{R}^{n\times n}\) is the coefficient matrix of x, while \(h=(h_{1},h_{2},\ldots, h_{n})^{\mathrm{T}}\) is a vector of complex nonlinear function.
The fractionalorder complex chaotic response system with the controller is written as
where \(w=w^{r}+jw^{i}\in\mathbb{C}^{n}\) and \(w^{r}=(w^{r}_{1},w^{r}_{2},\ldots ,w^{r}_{n})^{\mathrm{T}}\in\mathbb{R}^{n}\), \(w^{i}=(w^{i}_{1},w^{i}_{2},\ldots ,w^{i}_{n})^{\mathrm{T}} \in\mathbb{R}^{n}\), \(B\in\mathbb{R}^{n\times n}\) is the coefficient matrix of w, while \(g=(g_{1},g_{2},\ldots, g_{n})^{\mathrm{T}}\) is a vector of complex nonlinear function, \(v=v^{r}+jv^{i}\in\mathbb {C}^{n}\) is the controller to be designed.
Next the definition of CMGPS with complex transformation matrix is introduced for the fractionalorder real chaotic drive system and the complex chaotic response system based on that of MGPS with real transformation matrix for two fractionalorder real chaotic systems [13].
Definition 1
For the fractionalorder real chaotic drive system (14) and the complex chaotic response system (15), it is said to be CMGPS with complex constant matrix \(\Lambda={\Lambda^{r}} +j{\Lambda^{i}}\) between \(x(t)\) and \(w(t)\) if there exists a controller \(v=v^{r}+jv^{i}\in \mathbb{C}^{n}\) such that
i.e.,
and
while the matrix \(\Lambda\in{\mathbb{C}^{n \times n}}\) is defined as a complex transformation matrix of the fractionalorder real chaotic drive system (14).
If the error of CMGPS is defined as
then
the objective of this section is to design a controller v to ensure that synchronization error tends to zero asymptotically, i.e.,
and
Remark 1
Most of the classical fractionalorder real chaotic systems can be formed as system (14), such as fractionalorder real ChuaHartley’s system [5], fractionalorder real chaotic Lorenzlike system [6, 11–13], and most of the classical fractionalorder complex chaotic systems can be formed as system (15), such as fractionalorder complex Lorenz system [14], fractionalorder complex Chen system [15].
Remark 2
Several types of synchronization are special cases of CMGPS, such as complex modified projective synchronization (CMPS), complex projective synchronization (CPS), modified generalized projective synchronization (MGPS), modified projective synchronization (MPS), projective synchronization (PS), antisynchronization (AS), complete synchronization (CS); see Table 1.
Therefore, the CMGPS will contain most existing works and extend previous works.
Remark 3
In particular, if the transformation matrix Λ is zero, the CMGPS problem degenerates to the control problem of the fractionalorder complex chaotic system (15).
General method of CMGPS
Theorem 1
For the given complex transformation matrix \(\Lambda=\Lambda^{r}+j\Lambda ^{i}\) and initial conditions \(x(0)\), \(w(0)\), if the controller is designed as
then CMGPS between the fractionalorder complex chaotic response system (15) and the real chaotic drive system (14) can be achieved with the desired complex transformation matrix Λ asymptotically if and only if all the eigenvalues of \(B  K\) satisfy \(\vert {\arg({\lambda_{\ell}}(B  K))} \vert > \frac{{\alpha\pi }}{2}\) (\(\ell = 1,2, \ldots,n\)), where \(K\in\mathbb{R}^{n\times n}\) is the control gain matrix.
Proof
Equation (17) can be written as
Substituting Eq. (14) and Eq. (15) into Eq. (20), one can get the derivative of the error system
Insertion of (19) into Eq. (20) and separation of the real and imaginary parts give
Due to Lemma 1, the error system (22) is asymptotically stable if and only if all the eigenvalues of \(B  K\) satisfy \(\vert {\arg ({\lambda_{\ell}}(B  K))} \vert > \frac{{\alpha\pi}}{2} \) (\(\ell = 1,2, \ldots,n\)), where \(K\in\mathbb{R}^{n\times n}\) is the control gain matrix. That is, \(\lim_{t\to+\infty}\e^{r}(t)\ =0\), and \(\lim_{t\to+\infty}\e^{i}(t)\=0\). Therefore, \(\lim_{t \to+\infty} \Vert e(t) \Vert =0\), CMGPS between the fractionalorder systems (14) and (15) is realized. This completes the proof. □
CMGPS scheme of fractionalorder chaotic complex drive system and real response system
Mathematical model and problem descriptions
Now, a class of ndimensional fractionalorder complex chaotic drive systems is considered as
where \(z=z^{r}+jz^{i} \in\mathbb{C}^{n}\) and \(z^{r}=(z^{r}_{1},z^{r}_{2},\ldots ,z^{r}_{n})^{\mathrm{T}}\in\mathbb{R}^{n}\), \(z^{i}=(z^{i}_{1},z^{i}_{2},\ldots ,z^{i}_{n})^{\mathrm{T}} \in\mathbb{R}^{n}\), \(A\in\mathbb{R}^{n\times n}\) are the coefficient matrix of z, while \(f=(f_{1},f_{2},\ldots, f_{n})^{\mathrm{T}}\) is a vector of complex nonlinear function.
The fractionalorder real chaotic response system with the controller is written as
where \(y=(y_{1},y_{2},\ldots, y_{n})^{\mathrm{T}}\in\mathbb{R}^{n}\) is a real state vector, \(H\in\mathbb{R}^{n\times n}\) is the coefficient matrix of y, while \(p=(p_{1},p_{2},\ldots, p_{n})^{\mathrm{T}}\) is a vector of complex nonlinear function, \(v=(v_{1},v_{2},\ldots, v_{n})^{\mathrm{T}}\in \mathbb{R}^{n}\) is the controller to be designed.
Next the definition of CMGPS with complex transformation matrix is introduced for the fractionalorder complex chaotic drive system and the real chaotic response system based on that of MGPS with real transformation matrix for two fractionalorder real chaotic systems [13].
Definition 2
For the fractionalorder complex chaotic drive system (23) and the real chaotic response system (24), it is said to be CMGPS with constant matrix \(\Lambda={\Lambda^{r}} +j{\Lambda^{i}}\) between \(z(t)\) and \(y(t)\) if there exists a controller v such that
while the matrix \(\Lambda\in\mathbb{C}^{n\times n}\) is defined as a complex transformation matrix of the fractionalorder complex chaotic drive system (23).
If the error of CMGPS is defined as
the objective of this section is to design a controller v to ensure that synchronization error (26) tends to zero asymptotically, i.e.,
General method of CMGPS
Theorem 2
For the given complex transformation matrix \(\Lambda=\Lambda^{r}+j\Lambda ^{i}\) and initial conditions \(z(0)\), \(y(0)\), if the designed controller is
then CMGPS between the fractionalorder real chaotic response system (24) and the complex drive system (23) can be achieved with the desired complex transformation matrix Λ asymptotically if and only if all the eigenvalues of \(HK\) satisfy \(\vert {\arg({\lambda_{\ell}}(HK))} \vert > \frac{{\alpha\pi}}{2}\) (\(\ell = 1,2, \ldots,n\)), where \(K\in\mathbb{R}^{n\times n}\) is the control gain matrix.
Proof
Substituting Eq. (23) and Eq. (24) into Eq. (26), one can get the derivative of the error system
Insertion of Eq. (27) into Eq. (28) gives
Due to Lemma 1, the error system (29) is asymptotically stable if and only if all the eigenvalues of \(H  K\) satisfy \(\vert {\arg ({\lambda_{\ell}}(H  K))} \vert > \frac{{\alpha\pi}}{2} \) (\(\ell = 1,2, \ldots,n\)), where \(K\in\mathbb{R}^{n\times n}\) is the control gain matrix. That is, \(\lim_{t \to+\infty} \Vert e(t) \Vert = 0\), the fractionalorder real chaotic response system (24) and the complex chaotic drive system (23) realize CMGPS . This completes the proof. □
CMGPS scheme of two fractionalorder chaotic complex systems
Mathematical model and problem descriptions
Now, the definition of CMGPS with complex transformation matrix is introduced for two fractionalorder complex chaotic systems based on that of MGPS with real transformation matrix for two fractionalorder real chaotic systems [13].
Definition 3
For the fractionalorder complex chaotic drive system (23) and the response system (15), it is said to be CMGPS with complex matrix \(\Lambda={\Lambda^{r}} +j{\Lambda^{i}}\) between \(z(t)\) and \(w(t)\) if there exists a controller \(v=v^{r}+jv^{i}\in\mathbb{C}^{n}\) such that
i.e.,
and
while the matrix \(\Lambda\in\mathbb{C}^{n\times n}\) is defined as a complex transformation matrix of the fractionalorder complex chaotic drive system (23).
If the error of CMGPS is defined as
then
the objective of this section is to design a controller v to ensure that synchronization error tends to zero asymptotically, i.e.,
and
General method of CMGPS
Theorem 3
For the given complex transformation matrix \(\Lambda={\Lambda^{r}} +j{\Lambda^{i}}\) and initial conditions \(z(0)\), \(w(0)\), if the controller is designed as
then CMGPS between the fractionalorder complex chaotic response system (15) and the drive system (23) can be achieved with the desired complex transformation matrix Λ asymptotically if and only if all the eigenvalues of \(B  K\) satisfy \(\vert {\arg({\lambda_{\ell}}(B  K))} \vert > \frac{{\alpha\pi}}{2}\) (\(\ell = 1,2, \ldots,n\)), where \(K\in\mathbb{R}^{n\times n}\) is the control gain matrix.
Proof
It is similar to the proof in Theorem 1 and thus is omitted. □
Corollary 1
If the structure of systems (15) and (23) is identical, i.e., \(A=B\), and \(f=g\), and the controller is designed as
then CMGPS between the fractionalorder complex chaotic response system (15) and the drive system (23) can be achieved with the desired complex transformation matrix Λ asymptotically if and only if all the eigenvalues of \(B  K\) satisfy \(\vert {\arg({\lambda_{\ell}}(B  K))} \vert > \frac{{\alpha\pi}}{2}\) (\(\ell = 1,2, \ldots,n\)), where \(K\in\mathbb{R}^{n\times n}\) is the control gain matrix.
Remark 4
In particular, if the transformation matrix \(\Lambda=\operatorname{diag}\{1,1,\ldots ,1\}\), then CMGPS in Corollary 1 is reduced to complete synchronization (CS) of identical fractionalorder complex chaotic systems in [14].
Numerical examples
Now, three examples are worked out to illustrate the theoretical results in this paper.
CMGPS of fractionalorder chaotic real ChuaHartley’s drive system and complex Lorenz response system
In order to illustrate CMGPS behaviors of a fractionalorder real chaotic drive system and a complex chaotic response system, it is assumed that the fractionalorder chaotic real ChuaHartley’s system [5] drives the fractionalorder chaotic complex Lorenz system [14]. Therefore, the drive system is given in the form
where \(x= (x_{1},x_{2},x_{3} )^{T} \in\mathbb{R}^{3}\) is a real state vector. When \(m_{1} = 12.75\), \(m_{2}=\frac{100}{7}\), the fractional order ChuaHartley’s system (35) is chaotic as in Figure 1(a) and Figure 1(b) at \(\alpha=0.94\) and \(\alpha =0.96\), respectively.
The response system with the controller is written in the form
where \(w_{1}=w^{r}_{1}+jw^{i}_{1}\), \(w_{2}=w^{r}_{2}+jw^{i}_{2}\) are complex state variables, and \(w_{3}\) is a real state variable. System (36) is chaotic when \({b_{1}} = 10\), \({b_{2}} = 160\), \({b_{3}} = 1\), \(\alpha=0.96\) and in the absence of the controller \(v=v^{r}+jv^{i}\) as in Figure 1(c), see [14] for more details.
The complex transformation matrix can be taken as
and the error system \(e(t)=w(t)\Lambda x(t)\) is obtained as
The control gain matrix is chosen as
and the controller is designed according to (19) in Theorem 1 as follows:
The parameters of drive system (35) and response system (36) are selected as \(\alpha = 0.96\), \({m_{1}} = 12.75\), \({m_{2}} =\frac {100}{7}\), and \({b_{1}} = 10\), \({b_{2}} = 160\), \({b_{3}} = 1\), respectively. The initial values are randomly chosen as \(x_{0}=(0.01,0.01,0.1)^{T}\) and \(w_{0}=w^{r}_{0}+jw^{i}_{0}=(2+3j,5+6j,4)^{T}\), respectively. Therefore, all of the eigenvalues of \(BK\) are \(\lambda_{1}=10+2j\), \(\lambda _{2}=102j\), \(\lambda_{3}=1\), which satisfies \(\vert {\arg({\lambda_{\ell}}(B  K))} \vert > \frac{{\alpha\pi}}{2}\) (\(\ell= 1,2,3\)). The errors of CMGPS converge asymptotically to zero as in Figure 1(d). Hence, CMGPS has been achieved between the fractionalorder real chaotic ChuaHartley’s drive system (35) and the complex chaotic Lorenz response system (36).
CMGPS of fractionalorder chaotic complex Chen drive system and real ChuaHartley’s response system
In order to illustrate CMGPS behaviors of a fractionalorder complex chaotic drive system and a real chaotic response system, it is assumed that the fractionalorder complex chaotic Chen system [15] drives the fractionalorder real chaotic ChuaHartley’s system [6]. Therefore, the drive system is written in the form
where \(z_{1}=z^{r}_{1}+jz^{i}_{1}\), \(z_{2}=z^{r}_{2}+jz^{i}_{2}\) are complex state variables and \(z_{3}\) is a real state variable. System (39) is chaotic when \({a_{1}} = 35\), \({a_{2}} = 3\), \({a_{3}} = 28\), \(\alpha=0.96\) in Figure 2(a), see [15] for more details.
The response system with the controller is given in the form
where \(y= (y_{1},y_{2},y_{3} )^{T} \in\mathbb{R}^{3}\) is a real state vector, \(v= (v_{1},v_{2},v_{3} )^{T} \in\mathbb{R}^{3}\) is the controller.
The complex transformation matrix can be taken as
and the error system \(e(t)=y(t) {\Lambda^{r}}{z^{r}}(t) + {\Lambda ^{i}}{z^{i}}(t)\) is obtained as
The control gain matrix is chosen as
and the real controller is designed according to (27) in Theorem 2 as follows:
The parameters of drive system (39) and response system (40) are selected as \(\alpha = 0.96\), \({a_{1}} = 35\), \({a_{2}} = 3\), \({a_{3}} = 28\), and \(h_{1} = 12.75\), \(h_{2}=\frac{100}{7} \), respectively. The initial values are randomly chosen as \(z_{0}=z^{r}_{0}+jz^{i}_{0}=(2 + 4j,6 + 3j,5)^{T}\) and \(y_{0}=(0.01,0.01,0.1)^{T}\), respectively. Therefore, all of the eigenvalues of \(HK\) are \(\lambda_{1}=1+j\), \(\lambda_{2}=1j\), \(\lambda_{3}=1\), which satisfies \(\vert {\arg({\lambda_{\ell}}(H  K))} \vert > \frac{{\alpha\pi }}{2}\) (\(\ell= 1,2,3\)). The errors of CMGPS converge asymptotically to zero as in Figure 2(b). Hence, CMGPS has been achieved between the fractionalorder complex chaotic Chen drive system (39) and the real chaotic ChuaHartley’s response system (40).
CMGPS of fractionalorder chaotic complex Chen drive system and complex Lorenz response system
In order to illustrate CMGPS behaviors of two fractionalorder chaotic complex systems, it is assumed that the fractionalorder complex chaotic Chen system (39) drives the fractionalorder complex chaotic Lorenz system (36). The parameters of drive system (39) and response system (36) are selected as \(\alpha = 0.96\), \({a_{1}} = 35\), \({a_{2}} = 3\), \({a_{3}} = 28\), and \({b_{1}} = 10\), \({b_{2}} = 160\), \({b_{3}} = 1\), respectively.
The complex transformation matrix can be taken as
and the error system \(e(t)=w(t)\Lambda z(t)\) is obtained as
The control gain matrix is chosen as
and the complex controller is designed according to (33) in Theorem 3 as follows:
The initial values are randomly chosen as \(z_{0}=(2 + 4j, 6 + 3j, 5)^{T}\) and \(w_{0}=w^{r}_{0}+jw^{i}_{0}=(2 + 3j,5 + 6j,4)^{T}\), respectively. Therefore, all of the eigenvalues of \(BK\) are \(\lambda_{1}=5+3j\), \(\lambda _{2}=53j\), \(\lambda_{3}=1\), which satisfies \(\vert {\arg({\lambda_{\ell}}(B  K))} \vert > \frac{{\alpha\pi}}{2}\) (\(\ell = 1,2,3\)). The errors of CMGPS converge asymptotically to zero as in Figure 3, where the blue line shows the real parts of the errors and the red line presents the imaginary parts of the errors. Hence, CMGPS has been achieved between the fractionalorder chaotic complex Chen drive system (39) and the complex Lorenz response system (36).
Conclusions
In this paper, the definitions of modified generalized projective synchronization with complex transformation matrix (CMGPS) are introduced, where the drive and response systems could be asymptotically synchronized up to a desired complex transformation matrix, not a diagonal matrix. Moreover, general methods of CMGPS are designed for the fractionalorder real chaotic drive system and the complex chaotic response system, and for the fractionalorder complex chaotic drive system and the real chaotic response system, and for two fractionalorder complex chaotic systems, respectively. It should be noticed that the Lyapunov function is not required to be calculated in this scheme; it is really simple and feasible in practical applications. Three numerical examples are worked out to illustrate the feasibility and effectiveness of the theoretical results.
The complex transformation matrix not only establishes a link between fractionalorder real chaotic systems and complex chaotic systems, but also sets up a bridge between two fractionalorder complex chaotic systems. It increases the range of choosing fractionalorder chaotic generators in the transmitters and receivers, thus an interceptor is harder to crack information sources.
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Acknowledgements
This research was partially supported by the National Nature Science Foundation of China (Grant Nos. 61273088, 61473133), the Nature Science Foundation of Shandong Province, China (No. ZR2014FL015), and the Foundation for University Young Key Teacher Program of Shandong Provincial Education Department, China.
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PACS Codes
 05.45.Gg
 05.45.Xt
 05.45.Pq
Keywords
 fractionalorder chaos
 chaos with complex variable
 modified generalized projective synchronization