International Journal of Automation and Computing  2018, Vol. 15 Issue (5): 603-615   PDF    
Complex Modified Projective Synchronization for Fractional-order Chaotic Complex Systems
Cui-Mei Jiang1, Shu-Tang Liu1, Fang-Fang Zhang2     
1 College of Control Science and Engineering, Shandong University, Jinan 250061, China;
2 School of Electrical Engineering and Automation, Qilu University of Technology, Jinan 250353, China
Abstract: The aim of this paper is to study complex modified projective synchronization (CMPS) between fractional-order chaotic nonlinear systems with incommensurate orders. Based on the stability theory of incommensurate fractional-order systems and active control method, control laws are derived to achieve CMPS in three situations including fractional-order complex Lorenz system driving fractional-order complex Chen system, fractional-order real Rössler system driving fractional-order complex Chen system, and fractionalorder complex Lorenz system driving fractional-order real Lü system. Numerical simulations confirm the validity and feasibility of the analytical method.
Key words: Fractional-order system     chaotic complex system     incommensurate order     complex modified projective synchronization (CMPS)     active control    
1 Introduction

In 1982, Flower et al.[1] paved the way for extending nonlinear systems into complex space. After that, complex chaos, as an important topic in nonlinear science, has received a great deal of attention among scientists due to its important applications arising in physical systems, image processing, and in particular in secure communications[2-6]. For example, a variety of physical phenomena could be described by the chaotic or hyperchaotic complex systems, such as, the detuned laser systems, rotating fluids, disk dynamos, electronic circuits, and the amplitudes of electromagnetic fields. In secure communication, the complex variables (doubling the number of variables) can be used to increase the content of transmitting information signals and enhance their security. More recently, the chaotic behavior of complex systems has been extensively studied. Some examples of the systems include the complex Lorenz system[7], the complex Chen system[8], the complex Lü system[8], the hyperchaotic complex Lü system[9], the hyperchaotic complex Lorenz system[10], the modified hyperchaotic complex Lü system[11], just to name a few examples.

Nowadays, study on chaos synchronization of complex nonlinear systems, including chaos and hyperchaos, has become a very hot research topic. And a lot of different approaches have been proposed for the synchronization of chaotic complex systems, which include complete synchronization (CS)[12], anti-synchronization (AS)[13], adaptive anti-synchronization (AAS)[14], lag synchronization (LS)[15], phase synchronization and anti-phase synchronization[16], projective synchronization (PS)[17], modified projective synchronization (MPS)[17], etc. PS is defined if the state variables of the drive and response systems synchronize up to a real scaling factor. MPS means the response of the synchronized dynamical states synchronize up to a real constant scaling matrix. On the basis of MPS, a new type of complex synchronization which is called complex modified projective synchronization (CMPS) is put forward[18, 19]. CMPS is a situation in which the state variables of the drive and response systems synchronize up to a complex constant scaling matrix. Additionally, the complex scaling matrices establish a link between the integer-order chaotic real systems and complex systems[20].

However, to the best of our knowledge, most of the studies about CMPS mainly focus on the integer-order chaotic complex systems. There are few papers on CMPS of the incommensurate fractional-order chaotic complex systems. During the past few years, fractional-order chaotic complex systems have been widely investigated because of their high nonlinearity and wide power spectrum. Luo and Wang[21] studied the fractional-order complex Lorenz system and its complete synchronization. The fractional-order complex Chen system was proposed and its hybrid synchronization was discussed in digital secure communication[22]. Liu et al.[23] introduced the fractional-order complex T system and achieved its function projective synchronization. Recently, Liu[24] studied complex modified hybrid projective synchronization (CMHPS) of fractional-order complex chaos and real hyper-chaos. Nevertheless, in [21-24], the fractional orders of the complex systems are commensurate. There exist many incommensurate fractional-order systems in practical applications. The synchronization will not be approached under the effects of incommensurate orders. Therefore, it is meaningful and valuable to study CMPS of fractional-order chaotic complex systems with incommensurate orders.

Motivated by the above discussions, in this paper, we have made an endeavor to study and analyze CMPS between fractional-order chaotic nonlinear systems with incommensurate orders. And corresponding nonlinear controllers are designed. By virtue of the complex scaling matrices in CMPS, we establish a link between the incommensurate fractional-order chaotic real systems and complex systems. CMPS of fractional-order chaotic systems will contain CS, AS, PS, MPS and extend previous works.

The remainder of this paper is organized as follows. Section 2 reviews some preliminaries. CMPS of the incommensurate fractional-order chaotic systems is discussed in detail in Sections 3 to 5. Finally, a concluding remark is given in Section 6.

2 Fractional derivative and preliminaries 2.1 Definition

As we know, there are several definitions of fractional derivative including the Grunwald-Letnikov definition, the Riemann-Liouville definition, and the Caputo definition[25-27].

Definition 1. The $\alpha$ order Riemann-Liouville fractional integration is given by

$ \begin{eqnarray*} _aI^{\alpha}_{t}f(t)=\frac{1}{\Gamma(\alpha)}\int^{\rm T}_{a}\frac{f(\tau)}{(t-\tau)^{1-\alpha}}{\rm d}{\tau}, \ \ t>a \end{eqnarray*} $

where $\Gamma(\cdot)$ is the Gamma function, determined by

$ \Gamma(\alpha)=\int^{\infty}_{0}t^{\alpha-1}{\rm e}^{-t}{\rm d}t. $

Definition 2. For $n-1 < \alpha < n$, the Riemann-Liouville (RL) fractional derivative of order $\alpha$ is defined as

$ \begin{eqnarray*} &&^{R}_aD^{\alpha}_{t}f(t)=\frac{1}{\Gamma(n-\alpha)}\frac{{\rm d}^{n}}{{\rm d}t^{n}}\int^ {\rm T}_{a}\frac{f(\tau)}{(t-\tau)^{\alpha-n+1}}{\rm d}{\tau}=\\ &&\qquad \qquad \quad \frac{{\rm d}^{n}}{{\rm d}t^{n}}I^{n-\alpha}f(t). \end{eqnarray*} $

Definition 3. The Grunwald-Letnikov (GL) derivative with fractional order $\alpha$ is given by

$ \begin{eqnarray*}~~~~~ ^{G}_aD^{\alpha}_{t}f(t)=\lim\limits_{h\rightarrow0}h^{-\alpha}\sum\limits_{i=0}^{\left[\frac{t-a}{h}\right]}(-1)^{i}\left( \begin{array}{ccccc} \alpha \\ i \\ \end{array} \right)f(t-ih) \end{eqnarray*} $

where $\left(\begin{array}{ccccc} \alpha \\ i \\ \end{array}\right)=\frac{\alpha(\alpha-1)(\alpha-2)\cdots(\alpha-i+1)}{i!}$ is the usual notation for the binomial coefficients and $[\cdot]$ means the integer part.

Definition 4. The Caputo fractional derivative of order $\alpha$ is defined as

$ \begin{eqnarray*} \begin{array}{lll} &^{C}_aD^{\alpha}_{t}f(t)\, =\\&\qquad\left\{ \begin{array}{lll} \dfrac{1}{\Gamma(n-\alpha)}\displaystyle\int^{\rm T}_{a}\dfrac{f^{(n)}(\tau)}{(t-\tau)^{\alpha-n+1}}{\rm d} {\tau}, \ n-1<\alpha<n \\[3mm] \dfrac{{\rm d}^{n}}{{\rm d}t^{n}}f(t), \ \ \ \alpha=n \end{array} \right. \end{array} \end{eqnarray*} $

where $n=[\alpha]$ and $^{C}_aD^{\alpha}_{t}$ is called the Caputo derivative operator.

Since the initial conditions for the fractional differential equations with Caputo derivatives take on the same form as those for integer-order ones, we employ the Caputo definition to describe the fractional-order chaotic systems. In the rest of this paper, the notation $\frac {d^{\alpha}}{{\rm d}t^{\alpha}}$ is chosen as the Caputo derivative operator $^{C}_aD^{\alpha}_{t}$ and we mainly consider the order $0 < \alpha < 1$.

2.2 Numerical algorithm

Unlike the numerical algorithm for solving an ordinary differential equation, the numerical simulation of a fractional differential equation is not so easy. In the literatures of fractional chaos field, there are two approximation methods which can be used to solve numerically fractional-order differential equations. The first one is based on the approximation of the fractional-order system behavior in the frequency domain[28]. The other algorithm to find an approximation for fractional-order systems is based on the predictor-correctors scheme[29, 30]. This method is an improved version of Adams-Bashforth-Moulton algorithm. Here, we choose the Adams-Bashforth-Moulton predictor-correctors for numerical computation. Next, we interpret the approximate solution of nonlinear fractional-order differential equations using this algorithm in the following way.

The differential equation can be described by

$ \begin{eqnarray*} \begin{array}{lll} \left\{ \begin{array}{lll} \dfrac{d^{\alpha}}{{\rm d}t^{\alpha}}x=f(t, x), \ \ \ 0\leq t\leq T \cr x^{k}(0)=x^{(k)}_{0}, \ \ \ k=0, 1, \cdots, [\alpha]-1 \cr \end{array} \right. \end{array} \end{eqnarray*} $

which is equivalent to the Volterra integral equation

$ \begin{align*} x(t)=\, &\sum^{[\alpha]-1}_{k=0}x^{(k)}_{0}\frac{t^{k}}{k!}+\\ &\frac{1}{\Gamma(\alpha)}\int^{\rm T}_{0}(t-\tau)^{(\alpha-1)}f(\tau, x){\rm d}\tau. \end{align*} $

Now, set $h=\frac{T}{N}$, $t_{j}=jh$ ($j=0, 1, \cdots, N$). Then, the corrector formula for the latter equation can be discretized as

$ \begin{align*} x_{h}(t_{n+1})=\, &\sum^{[\alpha]-1}_{k=0}x^{(k)}_{0}\frac{t^{k}_{n+1}}{k!}+\\ & \frac{h^{\alpha}}{\Gamma(\alpha+2)}f(t_{n+1}, x^{p}_{h}(t_{n+1}))+\\ &\frac{h^{\alpha}}{\Gamma(\alpha+2)}\sum^{n}_{j=0}\beta_{j, n+1}f(t_{j}, x_{h}(t_{j})) \end{align*} $

where predicted value $x^{p}_{h}(t_{n+1})$ is determined by the following formula:

$ \begin{align*} x_{h}^{p}(t_{n+1})=\, &\sum^{[\alpha]-1}_{k=0}x^{(k)}_{0}\frac{t^{k}_{ n+1}}{k!}+\\ &\frac{1}{\Gamma(\alpha)}\sum^{n}_{j=0}\gamma_{j, n+1}f(t_{j}, x_{h} (t_{j})) \end{align*} $
$ \begin{align*} \begin{array}{lll} \beta_{j, n+1}=\left\{ \begin{array}{lll} n^{\alpha+1}-(n-\alpha)(n+1)^{\alpha}, \ \ \ j=0 \cr (n-j+2)^{\alpha+1}+(n-j)^{\alpha+1}-\cr ~~~~~2(n-j+1)^{\alpha+1}, \ \ \ 1\leq j \leq n \cr 1, \ \ \ \ \ \ j=n+1 \end{array} \right. \end{array} \end{align*} $

and

$ \begin{align*} ~~~~\gamma_{j, n+1}=\frac{h^{\alpha}}{\alpha}\left((n-j+1)^{\alpha}-(n-j)^{\alpha}\right), \ \ 1\leq j\leq n. \end{align*} $

The error estimate of this approach is

$ \max\limits_{j=0, 1, \cdots, N}|x(t_{j})-x_{h}(t_{j})|=O(h^{p}) $

where $p=\min(2, 1+\alpha)$.

2.3 The stability of fractional-order systems

Next, we recall the main stability properties of the linear fractional-order systems. For a given linear fractional-order system:

$ \begin{eqnarray} \frac{{\rm d}^{\alpha}}{{\rm d}t^{\alpha}}x={\it\Phi}{x}, \ \ \ \ x(0)=x_{0} \end{eqnarray} $ (1)

where $x\in{\bf{R}}^{n}$, the matrix ${\it\Phi}\in{{\bf{R}}^{n\times{n}}}$, $\alpha=[\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}]$ denotes the fractional orders, $\frac{{\rm d}^{\alpha}}{{{\rm d}t^{\alpha}}}=[\frac{{\rm d}^{\alpha_{1}}}{{{\rm d}t^{\alpha_{1}}}}, \frac{{\rm d}^{\alpha_{2}}}{{{\rm d}t^{\alpha_{2}}}}, \cdots, \frac{{\rm d}^{\alpha_{n}}}{{{\rm d}t^{\alpha_{n}}}}]$ and $\frac{{\rm d}^{\alpha_{i}}}{{{\rm d}t^{\alpha_{i}}}}$ is the Caputo fractional derivative of order $\alpha_{i}$, where $0 < \alpha_{i} < {1}$ for $i=1, 2, \cdots, n$.

If $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{n}$, then the stability of the fractional-order system (1) is stated in the following.

Lemma 1.[31] When $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{n}$, the fractional-order system (1) is asymptotically stable if $|\arg(\lambda_{l}({\it\Phi}))| > \frac{\alpha\pi}{2}$, where $\arg(\lambda_{l}({\it\Phi}))$ denotes the argument of the eigenvalue $\lambda_{l}$ of ${\it\Phi}$. In this case, the components of the state decay towards $0$ like $t^{-\alpha}$.

When $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}$ are rational positive numbers, we have the following result.

Lemma 2.[32] Assume that $\alpha_{i}{'s}$ are rational numbers between $0$ and $1$, for $i=1, 2, \cdots, n$. Let $\gamma=\frac{1}{m}$, where $m$ is the least common multiple of the denominators $m_{i}$ of $\alpha_{i}{'s}$, $\alpha_{i}\frac{k_{i}}{m_{i}}$, $k_{i}, m_{i}\in\bf{N}$, $i=1, 2, \cdots, n$. Then, system (1) is asymptotically stable if all roots $\lambda$ of the equation $\det({\rm diag}(\lambda^{m{\alpha_{1}}}, \lambda^{m{\alpha_{2}}}, \cdots, \lambda^{m{\alpha_{n}}})-{\it\Phi})=0$ satisfy $|\arg(\lambda)| > \frac{\gamma\pi}{2}$.

3 CMPS schemes of incommensurate fractional-order chaotic complex systems 3.1 Mathematical model and problem descriptions

Consider a fractional-order chaotic complex drive system of the following form:

$ \begin{eqnarray} \frac{{\rm d}^{\alpha}}{{\rm d}t^{\alpha}}z=Cz+p(z) \end{eqnarray} $ (2)

and the fractional-order chaotic complex response system is described as

$ \begin{eqnarray} \frac{{\rm d}^{\alpha}}{{\rm d}t^{\alpha}}y=By+g(y)+U \end{eqnarray} $ (3)

where $\alpha=[\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}]$ denotes the fractional orders. $z=z^{r}+{\rm j}z^{i}, \, y=y^{r}+{\rm j}y^{i}\in{\bf{C}}^{n}$ are the complex state vectors of systems (2) and (3), respectively. $p=(p_{1}, p_{2}, \cdots, p_{n})^{\rm T}$ and $g=(g_{1}, g_{2}, \cdots, g_{n})^{\rm T}$ are vectors of nonlinear complex functions. $C, B\in{{\bf{R}}^{n\times{n}}}$ are two real matrices of system parameters. $U$ is a nonlinear controller, $U=(U_{1}, U_{2}, \cdots, U_{n})^{\rm T}=U^{r}+{\rm j}U^{i}$, $U^{r}=(u_{1}, u_{3}, \cdots, u_{2n-1})^{\rm T}$, $U^{i}=(u_{2}, u_{4}, \cdots, u_{2n})^{\rm T}$. Superscripts $r$ and $i$ stand for the real and imaginary parts of a state complex vector.

Based on two fractional-order chaotic complex systems with incommensurate orders, we introduce the definition of CMPS as follows.

Definition 5. For the drive system (2) and the response system (3), it is said to be CMPS if there exists a complex constant matrix $M=M^{r}+{\rm j}M^{i}\in{\bf{C}}^{n\times{n}}$ such that

$ \begin{align*} \lim\limits_{t\rightarrow\infty}||e||^{2} =&\, \lim\limits_{t\rightarrow\infty}||y-Mz||^{2} =\\ &\lim\limits_{t\rightarrow\infty}(||y^{r}-M^{r}z^{r}+M^{i}z^{i}||^{2} + \\&||y^{i}-M^{r}z^{i}- M^{i}z^{r}||^{2})=0 \end{align*} $

where $||\cdot||$ is the Euclidean norm, $e=e^{r}+{\rm j}e^{i}$ is called the error vector between systems (2) and (3) for CMPS, $e^{r}=(e_{1}, e_{3}, \cdots, e_{2n-1})^{\rm T}= y^{r}-M^{r}z^{r}+M^{i}z^{i}$, $e^{i}=(e_{2}, e_{4}, \cdots, e_{2n})^{\rm T}= y^{i}-M^{r}z^{i}-M^{i}z^{r}$.

For the convenience of the following discussions, we assume $M=M^{r}+{\rm j}M^{i}$ =${\rm diag}(\varphi_{1}+{\rm j}\varphi_{2}, \varphi_{3}+{\rm j}\varphi_{4}, \cdots, \varphi_{2n-1}+{\rm j}\varphi_{2n})$ in our synchronization scheme and $\varphi_{1}+{\rm j}\varphi_{2}, \varphi_{3}+{\rm j}\varphi_{4}, \cdots, \varphi_{2n-1}+{\rm j}\varphi_{2n}$ are called complex scaling factors. If there exists $y_{l}\in{\bf R}$, then we select $\varphi_{2l}=0$ to avoid increasing a new imaginary part in the response system $(l=1, 2, \cdots, n)$.

Remark 1. It is important to note that many fractional-order chaotic complex systems can be described by (2), such as the fractional-order complex Lorenz system[21], the fractional-order complex Chen system[22], the fractional-order complex $T$ system[23], etc.

Remark 2. When $M=M^{r}={\rm diag}(\varphi_{1}, \varphi_{3}, \cdots, \varphi_{2n-1})$, CMPS will be reduced to MPS. In particular, if $\varphi_{1}=\varphi_{3}=\cdots=\varphi_{2n-1}$, then projective synchronization (PS) will appear, if $\varphi_{1}=\varphi_{3}=\cdots=\varphi_{2n-1}=1$ and $\varphi_{1}=\varphi_{3}=\cdots=\varphi_{2n-1}=-1$, then CS and AS will be achieved, respectively. Therefore, CMPS of fractional-order chaotic systems can be reduced to CS[33], AS[33], PS[34], and MPS.

Remark 3. When $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{n}$, CMPS is realized between two fractional-order chaotic complex systems with commensurate orders, and this is a special case of CMHPS[24].

Now, in this article, we obtain control laws to approach CMPS between two incommensurate fractional-order chaotic nonlinear systems. Next, we consider that the fractional-order complex Lorenz system drives the fractional-order complex Chen system.

3.2 CMPS between the fractional-order complex Lorenz system and complex Chen system

The fractional-order complex Lorenz system reads as follows:

$ \begin{eqnarray} \begin{array}{lll} \left\{ \begin{array}{lll} \dfrac{{\rm d}^{\alpha_{1}}{z_{1}}}{{\rm d}t^{\alpha_{1}}}=c_{1}(z_{2}-z_{1})\\[2mm] \dfrac{{\rm d}^{\alpha_{2}}{z_{2}}}{{\rm d}t^{\alpha_{2}}}=c_{2}z_{1}-z_{2}-z_{1}z_{3}\\[2mm]\dfrac{{\rm d}^{\alpha_{3}}{z_{3}}}{{\rm d}t^{\alpha_{3}}}=\frac{1}{2}(\bar{z}_{1}z_{2}+z_{1}\bar{z}_{2})-c_{3}z_{3} \end{array} \right. \end{array}\nonumber\\[-6mm] \end{eqnarray} $ (4)

where $z_{1}=m_{1}+{\rm j}m_{2}$, $z_{2}=m_{3}+{\rm j}m_{4}$ are complex variables, $z_{3}=m_{5}$ is a real variable, and $c_{i}$ are real parameters $(i=1, 2, 3)$. When $(c_{1}, c_{2}, c_{3})=(10, 180, 1)$, $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.98, 0.99, 0.97)$, system (4) generates chaotic attractors as displayed in Fig. 1.

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Fig. 1. 3D projections of attractors of fractional-order complex Lorenz system with $(c_{1}, c_{2}, c_{3})=(10, 180, 1)$ and $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.98, 0.99, 0.97)$

The fractional-order complex Chen system is defined as follows:

$ \begin{eqnarray} \begin{array}{lll} \left\{ \begin{array}{lll} \dfrac{{\rm d}^{\alpha_{1}}{y_{1}}}{{\rm d}t^{\alpha_{1}}}=b_{1}(y_{2}-y_{1})+U_{1}\\[2mm] \dfrac{{\rm d}^{\alpha_{2}}{y_{2}}}{{\rm d}t^{\alpha_{2}}}=(b_{2}-b_{1})y_{1}+b_{2}y_{2}-y_{1}y_{3}+U_{2}\\[2mm] \dfrac{{\rm d}^{\alpha_{3}}{y_{3}}}{{\rm d}t^{\alpha_{3}}}=\dfrac{1}{2}(\bar{y}_{1}y_{2}+y_{1}\bar{y}_{2})-b_{3}y_{3}+U_{3} \end{array} \right. \end{array}\nonumber\\[-6mm] \end{eqnarray} $ (5)

where $y_{1}=s_{1}+{\rm j}s_{2}$, $y_{2}=s_{3}+{\rm j}s_{4}$ and $y_{3}=s_{5}$, $b_{i}$ are real parameters $(i=1, 2, 3)$, $U_{1}=u_{1}+{\rm j}u_{2}$, $U_{2}=u_{3}+{\rm j}u_{4}$ are complex functions and $U_{3}=u_{5}$ is a real control function. System (5) is chaotic when $(b_{1}, b_{2}, b_{3})=(35, 28, 3)$, $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.98, 0.99, 0.97)$, see Fig. 2.

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Fig. 2. 3D projections of attractors of fractional-order complex Chen system with $(b_{1}, b_{2}, b_{3})=(35, 28, 3)$ and $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.98, 0.99, 0.97)$

In the following discussions, we assume $M={\rm diag}(\varphi_{1}+{\rm j}\varphi_{2}, \varphi_{3}+{\rm j}\varphi_{4}, \varphi_{5})$ in our synchronization scheme. Defining the synchronization error as

$ \begin{eqnarray*} \begin{array}{lll} \left\{ \begin{array}{lll} e_{1}=s_{1}-\varphi_{1}m_{1}+\varphi_{2}m_{2}\\[2mm] e_{2}=s_{2}-\varphi_{1}m_{2}-\varphi_{2}m_{1}\\[2mm] e_{3}=s_{3}-\varphi_{3}m_{3}+\varphi_{4}m_{4}\\[2mm] e_{4}=s_{4}-\varphi_{4}m_{3}-\varphi_{3}m_{4}\\[2mm] e_{5}=s_{5}-\varphi_{5}m_{5}. \end{array} \right. \end{array}\nonumber\\[-6mm] \end{eqnarray*} $

Combining with systems (4) and (5), we get the error system as

$ \begin{eqnarray*} \begin{array}{lll} ~~~\left\{ \begin{array}{lll} \dfrac{{\rm d}^{\alpha_{1}}e_{1}}{{\rm d}t^{\alpha_{1}}}=b_{1}e_{3}-b_{1} e_{1}+(b_{1}\varphi_{3}-c_{1}\varphi_{1})m_{3}+c_{1}\varphi_{2}m_{4}-\\ \ \ \ \ \ \ \ \ \ \ b_{1}\varphi_{4}m_{4}-(b_{1}-c_{1})(\varphi_{1} m_{1}-\varphi_{2}m_{2})+u_{1}\cr \dfrac{{\rm d}^{\alpha_{1}} e_{2}}{{\rm d}t^{\alpha_{1}}}=b_{1} e_{4}-b_{1}e_{2}+(b_{1}\varphi_{4}-c_{1}\varphi_{2})m_{3}-c_{1} \varphi_{1}m_{4}+\\ \ \ \ \ \ \ \ \ \ \ b_{1}\varphi_{3}m_{4}-(b_{1}-c_{1}) (\varphi_{1}m_{2}+\varphi_{2}m_{1})+u_{2}\cr \dfrac{{\rm d}^{\alpha_{2}}e_{3}}{{\rm d}t^{\alpha_{2}}}=(b_{2}-b_{1}) e_{1}+b_{2}e_{3}+[(b_{2}-b_{1})\varphi_{1}-\\ \ \ \ \ \ \ \ \ \ \ c_{2}\varphi_{3}]m_{1}-[(b_{2}-b_{1}) \varphi_{2}-c_{2}\varphi_{4}]m_{2}- s_{1}s_{5}+\\ \ \ \ \ \ \ \ \ \ \ \varphi_{3}m_{1}m_{5}+ (b_{2}+1)(\varphi_{3}m_{3}-\\ \ \ \ \ \ \ \ \ \ \ \varphi_{4}m_{4})- \varphi_{4}m_{2}m_{5}+u_{3}\cr \dfrac{{\rm d}^{\alpha_{2}}e_{4}}{{\rm d}t^{\alpha_{2}}}=(b_{2}-b_{1}) e_{2}+b_{2}e_{4}+[(b_{2}-b_{1})\varphi_{1}-\\ \ \ \ \ \ \ \ \ \ \ c_{2}\varphi_{3}]m_{2}+ [(b_{2}-b_{1})\varphi_{2}-c_{2}\varphi_{4}]m_{1}- s_{2}s_{5}+\\ \ \ \ \ \ \ \ \ \ \ \varphi_{4}m_{1}m_{5}+(b_{2}+1)(\varphi_{4}m_{3}+ \varphi_{3}m_{4})+\\ \ \ \ \ \ \ \ \ \ \ \varphi_{3}m_{2}m_{5}+u_{4}\cr \dfrac{{\rm d}^{\alpha_{3}}e_{5}}{{\rm d}t^{\alpha_{3}}}=-b_{3}e_{5}+ s_{1}s_{3}+s_{2}s_{4}-(b_{3}-c_{3})\varphi_{5}m_{5}-\\ \ \ \ \ \ \ \ \ \ \ \varphi_{5}m_{1}m_{3}-\varphi_{5}m_{2}m_{4}+u_{5}.\cr \end{array} \right. \end{array} \end{eqnarray*} $

Thus, we define the active control inputs $u_{i}(t)$ $(i=1, 2, \cdots, 5)$ as

$ \begin{eqnarray*} \begin{array}{lll} \left\{ \begin{array}{lll} u_{1}(t)=-(b_{1}\varphi_{3}-c_{1}\varphi_{1})m_{3}-(c_{1}\varphi_{2}-b_{1}\varphi_{4})m_{4}+\\ \ \ \ \ \ \ \ \ \ \ (b_{1}-c_{1})(\varphi_{1}m_{1}-\varphi_{2}m_{2})+v_{1}(t)\cr u_{2}(t)=-(b_{1}\varphi_{4}-c_{1}\varphi_{2})m_{3}-(b_{1}\varphi_{3}-c_{1}\varphi_{1})m_{4}+\\ \ \ \ \ \ \ \ \ \ \ (b_{1}-c_{1})(\varphi_{1}m_{2}+\varphi_{2}m_{1})+v_{2}(t)\cr u_{3}(t)=-[(b_{2}-b_{1})\varphi_{1}-c_{2}\varphi_{3}]m_{1}+s_{1}s_{5}-\varphi_{3}m_{1}m_{5}-\\ \ \ \ \ \ \ \ \ \ \ (b_{2}+1)(\varphi_{3}m_{3}-\varphi_{4}m_{4})+\varphi_{4}m_{2}m_{5}+\\ \ \ \ \ \ \ \ \ \ \ [(b_{2}-b_{1})\varphi_{2}-c_{2}\varphi_{4}]m_{2}+v_{3}(t)\cr u_{4}(t)=-[(b_{2}-b_{1})\varphi_{1}-c_{2}\varphi_{3}]m_{2}+s_{2}s_{5}-\varphi_{4}m_{1}m_{5}-\\ \ \ \ \ \ \ \ \ \ \ (b_{2}+1)(\varphi_{4}m_{3}+\varphi_{3}m_{4}) -\varphi_{3}m_{2}m_{5}-\\ \ \ \ \ \ \ \ \ \ \ [(b_{2}-b_{1})\varphi_{2}-c_{2}\varphi_{4}]m_{1}+v_{4}(t)\cr u_{5}(t)=-s_{1}s_{3}-s_{2}s_{4}+(b_{3}-c_{3})\varphi_{5}m_{5}+\varphi_{5}m_{1}m_{3}+\\ \ \ \ \ \ \ \ \ \ \ \varphi_{5}m_{2}m_{4}+v_{5}(t)\cr \end{array} \right. \end{array} \end{eqnarray*} $

which leads to

$ \begin{eqnarray} \begin{array}{lll} \left\{ \begin{array}{lll} \dfrac{{\rm d}^{\alpha_{1}}e_{1}}{{\rm d}t^{\alpha_{1}}}=b_{1}e_{3}-b_{1}e_{1}+v_{1}(t)\\[2mm] \dfrac{{\rm d}^{\alpha_{1}} e_{2}}{{\rm d}t^{\alpha_{1}}}=b_{1}e_{4}-b_{1}e_{2}+v_{2}(t)\\[2mm] \dfrac{{\rm d}^{\alpha_{2}}e_{3}}{{\rm d}t^{\alpha_{2}}}=(b_{2}-b_{1})e_{1}+b_{2}e_{3}+v_{3}(t)\\[2mm] \dfrac{{\rm d}^{\alpha_{2}}e_{4}}{{\rm d}t^{\alpha_{2}}}=(b_{2}-b_{1})e_{2}+b_{2}e_{4}+v_{4}(t)\\[2mm] \dfrac{{\rm d}^{\alpha_{3}}e_{5}}{{\rm d}t^{\alpha_{3}}}=-b_{3}e_{5}+v_{5}(t). \end{array} \right. \end{array}\nonumber\\[-6mm] \end{eqnarray} $ (6)

Next, an appropriate feedback controller is designed to stabilize system (6). We choose

$ \begin{eqnarray*} \left( \begin{array}{cc} v_{1}(t) \\ v_{2}(t) \\ v_{3}(t) \\ v_{4}(t) \\ v_{5}(t) \\ \end{array} \right) =L\left( \begin{array}{ccccc} e_{1}(t) \\ e_{2}(t) \\ e_{3}(t) \\ e_{4}(t) \\ e_{5}(t) \\ \end{array} \right) \end{eqnarray*} $

where $L=(l_{ij})_{5\times{5}}$ is a $5\times{5}$ real matrix. Then system (6) is written as

$ \begin{eqnarray*} \frac{{\rm d}^{\alpha}}{{\rm d}t^{\alpha}}e={\it\Phi}{e} \end{eqnarray*} $

where ${\it\Phi}=$

$ \begin{eqnarray*} ~~~\left( \begin{array}{ccccc} l_{11}-b_{1}&l_{12}&l_{13}+b_{1}&l_{14}&l_{15}\\ l_{21}& l_{22}-b_{1}&l_{23}&l_{24}+b_{1}&l_{25}\\ l_{31}+\beta&l_{32}&l_{33}+b_{2}&l_{34}&l_{35}\\ l_{41}&l_{42}+\beta&l_{43}& l_{44}+b_{2} &l_{45}\\ l_{51}&l_{52}&l_{53}& l_{54} &l_{55}-b_{3}\\ \end{array} \right) \end{eqnarray*} $

with $\beta=b_{2}-b_{1}$.

In order to make the error system stable, matrix $L$ should be selected in such a way that the feedback system satisfies conditions of Lemma 2. There is not a unique choice for such matrix $L$, a good choice can be as

$ \begin{eqnarray*} L= \left( \begin{array}{ccccc} 15&l_{12}&l_{13}&l_{14}&l_{15}\\ 0& 10&l_{23}&l_{24}&l_{25}\\ 7&0&-30&l_{34}&l_{35}\\ 0&7&0&-29 &l_{45}\\ 0&0&0&0 &1\\ \end{array} \right). \end{eqnarray*} $

It is obvious that $l_{11} < b_{1}$, $l_{22} < b_{1}$, $l_{33} < -b_{2}$, $l_{44} < -b_{2}$, and $l_{55} < b_{3}$ when we choose $(b_{1}, b_{2}, b_{3})=(35, 28, 3)$. Since $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.98, 0.99, 0.97)$, $m=100$, and $\gamma=\frac{1}{100}$, then the characteristic equation

$ \begin{eqnarray*} \det({\rm diag}(\lambda^{m{\alpha_{1}}}, \lambda^{m{\alpha_{2}}}, \cdots, \lambda^{m{\alpha_{n}}})-{\it\Phi})=0 \end{eqnarray*} $

can be written as

$ \begin{align} \quad (\lambda^{98}+20)(\lambda^{98}+25)(\lambda^{99}+2)(\lambda^{99}+1)(\lambda^{97}+2)=0. \end{align} $ (7)

We can show that all roots of (7) satisfy $|\arg(\lambda)| > \frac{\gamma\pi}{2}$. According to Lemma 2, the error vector $e(t)$ asymptotically converges to zero as $t\rightarrow{\infty}$. Therefore, CMPS between the incommensurate fractional-order chaotic complex systems (4) and (5) is achieved.

In the numerical simulations, the initial values of systems (4) and (5) are chosen as $(z_{1}, z_{2}, z_{3})^{\rm T}=(2+3{\rm j}, 5+6{\rm j}, 9)^{\rm T}$, $(y_{1}, y_{2}, y_{3})^{\rm T}=(6+9{\rm j}, 5+7{\rm j}, 12)^{\rm T}$, and a scaling matrix is taken as $M={\rm diag}(4-2{\rm j}, 3+5{\rm j}, 2)$. Thus, the initial errors are $(-8+{\rm j}, 20-36{\rm j}, -6)^{\rm T}$. Further, taking $l_{13}=-38$, $l_{15}=-1$, $l_{24}=-33$, and $l_{12}=l_{14}=l_{23}=l_{25}=l_{34}=l_{35}=l_{45}=0$, we obtain simulation results as displayed in Figs. 3 and 4. The CMPS process of systems (4) and (5) is described in Fig. 3, where the solid line presents the states of drive system and the dashed line shows the states of response system. Fig. 4 shows all error states converge asymptotically to zero, i.e., CMPS between the incommensurate fractional-order complex Lorenz system and complex Chen system is achieved.

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Fig. 3. State variables of systems (4) and (5) for the case $M={\rm diag}(4-2{\rm j}, 3+5{\rm j}, 2)$ and $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.98, 0.99, 0.97)$

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Fig. 4. Errors of CMPS between systems (4) and (5) with the scaling matrix $M={\rm diag}(4-2{\rm j}, 3+5{\rm j}, 2)$ and $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.98, 0.99, 0.97)$

4 CMPS schemes between fractional-order chaotic real drive system and complex response system 4.1 Mathematical model and problem descriptions

The fractional-order chaotic complex response system is defined as in (3) and the fractional-order chaotic real drive system reads as

$ \begin{eqnarray} \frac{d^{\alpha}}{{\rm d}t^{\alpha}}x=Ax+f(x) \end{eqnarray} $ (8)

where $x=(x_{1}, x_{2}, \cdots, x_{n})^{\rm T}$ is a real state vector, $f$ is a vector of nonlinear function. $A\in{{\bf{R}}^{n\times{n}}}$ is a real matrix of system parameters.

The error of CMPS with a complex scaling matrix $M=M^{r}+{\rm j}M^{i}\in{\bf{C}}^{n\times{n}}$ between systems (8) and (3) is defined as

$ e(t)=e^{r}(t)+je^{i}(t)=y-Mx $

i.e.,

$ \begin{eqnarray*} \begin{array}{lll} \left\{ \begin{array}{lll} e^{r}=y^{r}-M^{r}x\cr e^{i}=y^{i}-M^{i}x. \cr \end{array} \right. \end{array} \end{eqnarray*} $

Then, in the following, we design a controller to achieve CMPS between the fractional-order real Rössler system[35] and the fractional-order complex Chen system with incommensurate orders.

4.2 CMPS between fractional-order real Rössler system and complex Chen system

The fractional-order complex Chen system is described by (5), while the fractional-order real Rössler system is depicted as

$ \begin{eqnarray} \begin{array}{lll} \left\{ \begin{array}{lll} \dfrac{{\rm d}^{\alpha_{1}}x_{1}}{{\rm d}t^{\alpha_{1}}}=-(x_{2}+x_{3})\\[2mm] \dfrac{{\rm d}^{\alpha_{2}}x_{2}}{{\rm d}t^{\alpha_{2}}}=x_{1}+a_{1}x_{2}\\[2mm] \dfrac{{\rm d}^{\alpha_{3}}x_{3}}{{\rm d}t^{\alpha_{3}}}=a_{2}+x_{3}(x_{1}-a_{3}) \end{array} \right. \end{array}\nonumber\\[-6mm] \end{eqnarray} $ (9)

where $x_{i}$ are real state variables and $a_{i}$ are real parameters $(i=1, 2, 3)$. When $(a_{1}, a_{2}, a_{3})=(0.4, 0.2, 10)$, $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.9, 0.91, 0.92)$, chaotic behaviors can be displayed for system (9), see Fig. 5.

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Fig. 5. Chaotic attractor of fractional-order real Rössler system with $(a_{1}, a_{2}, a_{3})=(0.4, 0.2, 10)$, $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.9, 0.91, 0.92)$

For a given scaling matrix $M={\rm diag}(\varphi_{1}+{\rm j}\varphi_{2}, \varphi_{3}+{\rm j}\varphi_{4}, \varphi_{5})$, we define the synchronization error as

$ \begin{eqnarray*} \begin{array}{lll} \left\{ \begin{array}{lll} e_{1}=s_{1}-\varphi_{1}x_{1}\cr e_{2}=s_{2}-\varphi_{2}x_{1}\cr e_{3}=s_{3}-\varphi_{3}x_{2}\cr e_{4}=s_{4}-\varphi_{4}x_{2}\cr e_{5}=s_{5}-\varphi_{5}x_{3}.\cr \end{array} \right. \end{array} \end{eqnarray*} $

By virtue of systems (5) and (9), the error system becomes:

$ \begin{eqnarray*} \begin{array}{lll} \left\{ \begin{array}{lll} \dfrac{{\rm d}^{\alpha_{1}}e_{1}}{{\rm d}t^{\alpha_{1}}}=b_{1}e_{3}-b_{1}e_{1}+(b_{1}\varphi_{3}+\varphi_{1})x_{2}-b_{1}\varphi_{1}x_{1}+\\ \ \ \ \ \ \ \ \ \ \ \ \ \varphi_{1}x_{3}+u_{1}\cr \dfrac{{\rm d}^{\alpha_{1}} e_{2}}{{\rm d}t^{\alpha_{1}}}=b_{1}e_{4}-b_{1}e_{2}+(b_{1}\varphi_{4}+\varphi_{2})x_{2}-b_{1}\varphi_{2}x_{1}+\\ \ \ \ \ \ \ \ \ \ \ \ \ \varphi_{2}x_{3}+u_{2}\cr \dfrac{{\rm d}^{\alpha_{2}}e_{3}}{{\rm d}t^{\alpha_{2}}}=(b_{2}-b_{1})e_{1}+b_{2}e_{3}+[(b_{2}-b_{1})\varphi_{1}-\varphi_{3}]x_{1}-\\ \ \ \ \ \ \ \ \ \ \ \ \ s_{1}s_{5}+(b_{2}-a_{1})\varphi_{3}x_{2}+u_{3}\cr \dfrac{{\rm d}^{\alpha_{2}}e_{4}}{{\rm d}t^{\alpha_{2}}}=(b_{2}-b_{1})e_{2}+b_{2}e_{4}+[(b_{2}-b_{1})\varphi_{2}-\varphi_{4}]x_{1}-\\ \ \ \ \ \ \ \ \ \ \ \ \ s_{2}s_{5}+(b_{2}-a_{1})\varphi_{4}x_{2}+u_{4}\cr \dfrac{{\rm d}^{\alpha_{3}}e_{5}}{{\rm d}t^{\alpha_{3}}}=-b_{3}e_{5}+s_{1}s_{3}+s_{2}s_{4}+(a_{3}-b_{3})\varphi_{5}x_{3}-\\ \ \ \ \ \ \ \ \ \ \ \ \ \ a_{2}\varphi_{5}-\varphi_{5}x_{1}x_{3}+u_{5}. \end{array} \right. \end{array} \end{eqnarray*} $

Hence, we define active control functions $u_{i}(t)$ $(i=1, 2, \cdots, 5)$ as

$ \begin{eqnarray*} \begin{array}{lll} \left\{\!\! \begin{array}{lll} u_{1}(t)=-(b_{1}\varphi_{3}+\varphi_{1})x_{2}-\varphi_{1}x_{3}+b_{1}\varphi_{1}x_{1}+v_{1}(t)\cr u_{2}(t)=-(b_{1}\varphi_{4}+\varphi_{2})x_{2}-\varphi_{2}x_{3}+b_{1}\varphi_{2}x_{1}+v_{2}(t)\cr u_{3}(t)=-[(b_{2}-b_{1})\varphi_{1}-\varphi_{3}]x_{1}-(b_{2}-a_{1})\varphi_{3}x_{2}+\\ \ \ \ \ \ \ \ \ \ \ s_{1}s_{5}+v_{3}(t)\cr u_{4}(t)=-[(b_{2}-b_{1})\varphi_{2}-\varphi_{4}]x_{1}-(b_{2}-a_{1})\varphi_{4}x_{2}+\\ \ \ \ \ \ \ \ \ \ \ s_{2}s_{5}+v_{4}(t)\cr u_{5}(t)=-s_{1}s_{3}-s_{2}s_{4}-(a_{3}-b_{3})\varphi_{5}x_{3}+\varphi_{5}x_{1}x_{3}+\\ \ \ \ \ \ \ \ \ \ \ a_{2}\varphi_{5}+v_{5}(t)\cr \end{array} \right. \end{array} \end{eqnarray*} $

which implies that

$ \begin{eqnarray*} \begin{array}{lll} \left\{\!\! \begin{array}{lll} \dfrac{{\rm d}^{\alpha_{1}}e_{1}}{{\rm d}t^{\alpha_{1}}}=b_{1}e_{3}-b_{1}e_{1}+v_{1}(t)\\[2mm] \dfrac{{\rm d}^{\alpha_{1}} e_{2}}{{\rm d}t^{\alpha_{1}}}=b_{1}e_{4}-b_{1}e_{2}+v_{2}(t)\\[2mm]\dfrac{{\rm d}^{\alpha_{2}}e_{3}}{{\rm d}t^{\alpha_{2}}}=(b_{2}-b_{1})e_{1}+b_{2}e_{3}+v_{3}(t)\\[2mm] \dfrac{{\rm d}^{\alpha_{2}}e_{4}}{{\rm d}t^{\alpha_{2}}}=(b_{2}-b_{1})e_{2}+b_{2}e_{4}+v_{4}(t)\\[2mm] \dfrac{{\rm d}^{\alpha_{3}}e_{5}}{{\rm d}t^{\alpha_{3}}}=-b_{3}e_{5}+v_{5}(t). \end{array} \right. \end{array} \end{eqnarray*} $

In view of the fact that the term $v_{i}(t)$ are linear functions of the error terms $e_{i}(t)$ $(i=1, 2, \cdots, 5)$, we set

$ \begin{eqnarray*} \left( \begin{array}{cc} v_{1}(t) \\ v_{2}(t) \\ v_{3}(t) \\ v_{4}(t) \\ v_{5}(t) \\ \end{array} \right) =L\left( \begin{array}{ccccc} e_{1}(t) \\ e_{2}(t) \\ e_{3}(t) \\ e_{4}(t) \\ e_{5}(t) \\ \end{array} \right) \end{eqnarray*} $

where $L=(l_{ij})_{5\times{5}}$ is a $5\times{5}$ real matrix. $L$ can be chosen as

$ \begin{eqnarray*} L= \left( \begin{array}{ccccc} 0&l_{12}&l_{13}&l_{14}&l_{15}\\ 0&1&l_{23}&l_{24}&l_{25}\\ 7&0&-29&l_{34}&l_{35}\\ 0&7&0&-30 &l_{45}\\ 0&0&0&0 &0\\ \end{array} \right). \end{eqnarray*} $

It is clear that $l_{11} < b_{1}$, $l_{22} < b_{1}$, $l_{33} < -b_{2}$, $l_{44} < -b_{2}$, and $l_{55} < b_{3}$ for the choice of $(b_{1}, b_{2}, b_{3})=(35, 28, 3)$. Since $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.9, 0.91, 0.92)$, $m=100$, and $\gamma=\frac{1}{100}$, then the equation

$ \begin{eqnarray*} \det({\rm diag}(\lambda^{m{\alpha_{1}}}, \lambda^{m{\alpha_{2}}}, \cdots, \lambda^{m{\alpha_{n}}})-{\it\Phi})=0 \end{eqnarray*} $

can be depicted as

$ \begin{align} \quad(\lambda^{90}+35)(\lambda^{90}+34)(\lambda^{91}+1)(\lambda^{91}+2)(\lambda^{92}+3)=0. \end{align} $ (10)

By a simple calculation, we conclude that all roots of (10) lie in the region $|\arg(\lambda)| > \frac{\gamma\pi}{2}$. It follows from Lemma 2 that the error vector $e(t)$ asymptotically converges to zero as $t\rightarrow{\infty}$.

The initial values of systems (9) and (5) are chosen as $(x_{1}, x_{2}, x_{3})^{\rm T}=(1, 2, 3)^{\rm T}$ and $(y_{1}, y_{2}, y_{3})^{\rm T}=(6-9{\rm j}, -8+6{\rm j}, 1)^{\rm T}$, respectively. Taking a scaling matrix as $M={\rm diag}(1-{\rm j}, 1-{\rm j}, 1)$, we obtain the initial errors as $(5-8{\rm j}, -10+8{\rm j}, -2)^{\rm T}$. Further, selecting $l_{13}=70$, $l_{24}=70$, $l_{12}=l_{14}=l_{15}=l_{23}=l_{25}=l_{34}=l_{35}=l_{45}=0$, we obtain simulation results as displayed in Figs. 6-8. From Fig. 6, it can be observed that the errors of CMPS converge asymptotically to zero. The CMPS process of systems (9) and (5) is described in Fig. 7, where the solid line presents the states of drive system and the dotted line shows the states of response system. Fig. 8 illustrates that the real parts $(y_{1}^{r}, y_{2}^{r}, y_{3})$ of the response system (5) completely synchronize $(x_{1}, x_{2}, x_{3})$ of the drive system (9), while the imaginary parts $(y_{1}^{i}, y_{2}^{i})$ of the response system (5) anti-synchronize $(x_{1}, x_{2})$ of the drive system (9). Therefore, the fractional-order chaotic real Rössler system and complex Chen system achieve CMPS.

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Fig. 6. Errors of CMPS between systems (9) and (5) for the case $M={\rm diag}(1-{\rm j}, 1-{\rm j}, 1)$ and $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.9, 0.91, 0.92)$

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Fig. 7. State variables of systems (9) and (5) for the case $M={\rm diag}(1-{\rm j}, 1-{\rm j}, 1)$ and $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.9, 0.91, 0.92)$

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Fig. 8. Attractors of systems (5) and (9) in CMPS process

5 CMPS schemes between fractional-order chaotic complex drive system and real response system 5.1 Mathematical model and problem descriptions

The fractional-order chaotic complex drive system can be described by (2) and the fractional-order chaotic real response system is considered as

$ \begin{eqnarray} \frac{{\rm d}^{\alpha}}{{\rm d}t^{\alpha}}w=Dw+q(w)+U \end{eqnarray} $ (11)

where $w=(w_{1}, w_{2}, \cdots, w_{n})^{\rm T}$ is a real state vector, $q$ is a vector of nonlinear function, $D\in{{\bf{R}}^{n\times{n}}}$ is a coefficient matrix.

As $w(t)$ is real, a real $U$ is chosen to ensure CMPS of real parts and avoid increasing the imaginary parts of response system. And in this case, the error vector with a given scaling matrix $M=M^{r}+{\rm j}M^{i}\in{\bf{C}}^{n\times{n}}$ is defined as

$ e=w-M^{r}z^{r}+M^{i}z^{i}. $

Therefore, the complex drive system (2) and the real response system (11) are CMPS of real parts.

In what follows, our aim is to approach CMPS of the fractional-order complex Lorenz system and real Lü system[36] with incommensurate orders.

5.2 CMPS between fractional-order complex Lorenz system and real Lü system

The fractional-order complex Lorenz system is given by (4), while the fractional-order real Lü system is defined as

$ \begin{eqnarray} \begin{array}{lll} \left\{\!\!\! \begin{array}{lll} \dfrac{{\rm d}^{\alpha_{1}}{w_{1}}}{{\rm d}t^{\alpha_{1}}}=d_{1}(w_{2}-w_{1})+U_{1}\\[2mm] \dfrac{{\rm d}^{\alpha_{2}}{w_{2}}}{{\rm d}t^{\alpha_{2}}}=-w_{1}w_{3}+d_{2}w_{2}+U_{2}\\[2mm] \dfrac{{\rm d}^{\alpha_{3}}{w_{3}}}{{\rm d}t^{\alpha_{3}}}=w_{1}w_{2}-d_{3}w_{3}+U_{3} \end{array} \right. \end{array}\nonumber\\[-5mm] \end{eqnarray} $ (12)

where $w_{i}$ are real state variables, $d_{i}$ are real parameters, and $U_{i}$ are real control functions $(i=1, 2, 3)$. When $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.95, 0.96, 0.97)$, $(d_{1}, d_{2}, d_{3})=(35, 28, 3)$, system (12) behaves chaotically as shown in Fig. 9.

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Fig. 9. Chaotic attractors of fractional-order real Lü system when $(d_{1}, d_{2}, d_{3})=(35, 28, 3)$ and $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.95, 0.96, 0.97)$

Assume that $M={\rm diag}(\varphi_{1}+{\rm j}\varphi_{2}, \varphi_{3}+{\rm j}\varphi_{4}, \varphi_{5})$ in our synchronization scheme. We define the synchronization error as

$ \begin{eqnarray*} \begin{array}{lll} \left\{ \begin{array}{lll} e_{1}=w_{1}-\varphi_{1}m_{1}+\varphi_{2}m_{2}\cr e_{2}=w_{2}-\varphi_{3}m_{3}+\varphi_{4}m_{4}\cr e_{3}=w_{3}-\varphi_{5}m_{5}.\cr \end{array} \right. \end{array} \end{eqnarray*} $

Then, we obtain the error system as

$ \begin{eqnarray*} \begin{array}{lll} \left\{\!\!\! \begin{array}{lll} \dfrac{{\rm d}^{\alpha_{1}}e_{1}}{{\rm d}t^{\alpha_{1}}}=d_{1}e_{2}-d_{1}e_{1}+(d_{1}\varphi_{3}-c_{1}\varphi_{1})m_{3}-d_{1}\varphi_{4}m_{4}+\\ \ \ \ \ \ \ \ \ \ \ \ \ c_{1}\varphi_{2}m_{4}-(d_{1}-c_{1})(\varphi_{1}m_{1}-\varphi_{2}m_{2})+U_{1}\cr \dfrac{{\rm d}^{\alpha_{2}} e_{2}}{{\rm d}t^{\alpha_{2}}}=d_{2}e_{2}+(d_{2}+1)(\varphi_{3}m_{3}-\varphi_{4}m_{4})-w_{1}w_{3}+\\ \ \ \ \ \ \ \ \ \ \ \ \ (\varphi_{3}m_{1}-\varphi_{4}m_{2})(m_{5}-c_{2})+U_{2}\cr \dfrac{{\rm d}^{\alpha_{3}}e_{3}}{{\rm d}t^{\alpha_{3}}}=-d_{3}e_{3}+w_{1}w_{2}-(d_{3}-c_{3})\varphi_{5}m_{5}-\\ \ \ \ \ \ \ \ \ \ \ \ \ \varphi_{5}(m_{1}m_{3}+m_{2}m_{4})+U_{3}.\cr \end{array} \right. \end{array} \end{eqnarray*} $

Thus, the active control inputs $U_{i}(t)$ $(i=1, 2, 3)$ are designed as

$ \begin{eqnarray*} \begin{array}{lll} \left\{\!\!\! \begin{array}{lll} U_{1}(t)=-(d_{1}\varphi_{3}-c_{1}\varphi_{1})m_{3}+(d_{1}\varphi_{4}-c_{1}\varphi_{2})m_{4}+\\ \ \ \ \ \ \ \ \ \ \ (d_{1}-c_{1})(\varphi_{1}m_{1}-\varphi_{2}m_{2})+v_{1}(t)\cr U_{2}(t)=-(d_{2}+1)(\varphi_{3}m_{3}-\varphi_{4}m_{4})+w_{1}w_{3}-\\ \ \ \ \ \ \ \ \ \ \ (\varphi_{3}m_{1}-\varphi_{4}m_{2})(m_{5}-c_{2})+v_{2}(t)\cr U_{3}(t)=(d_{3}-c_{3})\varphi_{5}m_{5}+\varphi_{5}(m_{1}m_{3}+m_{2}m_{4})-\\ \ \ \ \ \ \ \ \ \ \ \ w_{1}w_{2}+v_{3}(t).\cr \end{array} \right. \end{array} \end{eqnarray*} $

And the error system becomes

$ \begin{eqnarray*} \begin{array}{lll} \left\{ \begin{array}{lll} \dfrac{{\rm d}^{\alpha_{1}}e_{1}}{{\rm d}t^{\alpha_{1}}}=d_{1}e_{2}-d_{1}e_{1}+v_{1}(t)\\[2mm] \dfrac{{\rm d}^{\alpha_{2}} e_{2}}{{\rm d}t^{\alpha_{2}}}=d_{2}e_{2}+v_{2}(t)\\[2mm] \dfrac{{\rm d}^{\alpha_{3}}e_{3}}{{\rm d}t^{\alpha_{3}}}=-d_{3}e_{3}+v_{3}(t). \end{array} \right. \end{array} \end{eqnarray*} $

In order to satisfy conditions of Lemma 2, we choose

$ \begin{eqnarray*} \left( \begin{array}{cc} v_{1}(t) \\ v_{2}(t) \\ v_{3}(t) \\ \end{array} \right) =L\left( \begin{array}{ccccc} e_{1}(t) \\ e_{2}(t) \\ e_{3}(t) \\ \end{array} \right) \end{eqnarray*} $

where $L=(l_{ij})_{3\times{3}}$ is a $3\times{3}$ real matrix. Thus, we make a good choice as

$ \begin{eqnarray*} L= \left( \begin{array}{ccc} 30&-35&0\\ l_{21}&-36&0 \\ l_{31} &l_{32}&2 \\ \end{array} \right) \end{eqnarray*} $

We obtain that $l_{11} < d_{1}$, $l_{22} < -d_{2}$, and $l_{33} < d_{3}$ in the case where $(d_{1}, d_{2}, d_{3})=(35, 28, 3)$. It follows from $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.95, 0.96, 0.97)$, $m=100$, and $\gamma=\frac{1}{100}$ that the equation

$ \begin{eqnarray*} \det({\rm diag}(\lambda^{m{\alpha_{1}}}, \lambda^{m{\alpha_{2}}}, \cdots, \lambda^{m{\alpha_{n}}})-{\it\Phi})=0 \end{eqnarray*} $

can be described as

$ \begin{eqnarray} (\lambda^{95}+5)(\lambda^{96}+8)(\lambda^{97}+1)=0. \end{eqnarray} $ (13)

We can demonstrate that all roots of (13) lie in the region $|\arg(\lambda)| > \frac{\gamma\pi}{2}$. From Lemma 2, the error vector $e(t)$ asymptotically converges to zero as $t\rightarrow{\infty}$.

In the numerical simulations, the initial values of systems (4) and (12) are chosen as $(z_{1}, z_{2}, z_{3})^{\rm T}=(2+3{\rm j}, 5+6{\rm j}, 9)^{\rm T}$, $(w_{1}, w_{2}, w_{3})^{\rm T}=(1, 2, 3)^{\rm T}$, and a complex constant scaling matrix is taken as $M={\rm diag}(3+{\rm j}, 1-2{\rm j}, -2)$. Thus, the initial errors are $(-2, -15, 21)^{\rm T}$. Selecting $l_{21}=2$, $l_{31}=1$, $l_{32}=4$, we have simulation results as shown in Figs. 10 and 11. The phase portraits in drive system (4) and response system (12) with different initial conditions are presented in Fig. 10. From Fig. 11, it is obvious that the errors of CMPS converge asymptotically to zero, i.e., fractional-order complex Lorenz system and real Lü system are CMPS of real parts.

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Fig. 10. Phase portraits of the attractor for systems (4) and (12) in CMPS process

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Fig. 11. Errors of CMPS between systems (4) and (12) for the case $M={\rm diag}(3+{\rm j}, 1-2{\rm j}, -2)$ and $(\alpha_{1}, \alpha_{2}, \alpha_{3})=(0.95, 0.96, 0.97)$

6 Conclusions

We investigate complex modified projective synchronization (CMPS) between two incommensurate fractional-order chaotic nonlinear systems. On the basis of the stability result of incommensurate fractional-order systems, we design CMPS schemes in three situations including fractional-order complex Lorenz system driving fractional-order complex Chen system, fractional-order real Rössler system driving fractional-order complex Chen system, and fractional-order complex Lorenz system driving fractional-order real Lü system. Numerical simulations are performed to demonstrate the validity and feasibility of the schemes. Moreover, CMPS establishes a link between the incommensurate fractional-order real systems and complex systems. Thus in secure communications, CMPS can increase the diversity and the security of transmitted information. Therefore, CMPS of fractional-order chaotic complex systems will play an important role in practical applications. However, in practical chaotic synchronization, there are a lot of problems including the existence of unknown parameters and external disturbances, which is our future work.

Acknowledgements

This work was supported by Key Program of National Natural Science Foundation of China (No. 61533011) and National Natural Science Foundation of China (Nos. 61273088 and 61603203).

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