Synchronization of Butterfly Fractional Order Chaotic System

Abstract

In this paper, we study the synchronization of a nonlinear fractional system, and analyze its time response and chaotic behaviors. We represent a solution for considered system by employing the Mittag-Leffler matrix function and Jacobian matrix. Thereafter, we prove synchronization of error system between drive-response systems using stability theory and linear feedback control methods. Finally, numerical simulations are presented to show the effectiveness of the theoretical results.

1. Introduction

Fractional calculus has over 300 years of history. Recently, it has attracted increasing interest due to its potential applications physics and engineering such as viscoelastic systems [1], dielectric polarization [2], electrode-electrolyte polarization [3], electromagnetic waves [4], quantitative finance [5], and quantum evolution of complex systems [6]. Synchronization of a chaotic system occurs when two or more chaotic systems are connected and it has attracted wide research because of its applications in science and engineering such as biology, economics, secure communication and signal generator design (see [7,8]). Sufficient conditions for synchronization of fractional-order chaotic systems via linear control were investigated in [9], synchronization of fractional chaotic systems with different orders was studied in [10] using stability concepts and synchronization of the Lotka-Volterra chaotic system was studied using the active control method in [11]. Robust synchronization for chaotic and hyperchaotic fractional order systems with model uncertainties and disturbances was investigated in [12].

Fractional chaotic system, global asymptotic synchronization and adaptive sliding mode synchronization have been studied in [13,14,15] and sufficient conditions were presented for exponential synchronization of fractional order chaotic systems in [16]. Finite time stability and synchronization of fractional order chaotic system with uncertainties and disturbance was studied in [17].

In this paper, we consider the following drive system of fractional order

$$\left\{\begin{array}{c}{}^{C}D_0^{q}u\left(t\right)=Au\left(t\right)+f(u,u),\hfill \\ u\left(0\right)=u_0\hfill \end{array}\right.$$

(1)

and a corresponding response system of (1) is

$$\left\{\begin{array}{c}{}^C{D}_0^{q}v\left(t\right)=Av\left(t\right)+f(u,v)+h\left(t\right),\hfill \\ v\left(0\right)=v_0,\hfill \end{array}\right.$$

(2)

where $t\in {\mathbb{R}}_{+}:=[0,\infty ),q\in (0,1),$${}^{C}D_0^q$ denotes the Caputo fractional derivative [18] with lower limit at $0,$ $u\left(t\right)\in {\mathbb{R}}^n$ is a state, A is a matrix of dimension $n\times n,$ $f:{\mathbb{R}}^n\times {\mathbb{R}}^n⟶{\mathbb{R}}^n$ is a $C^1$-smooth and $h:{\mathbb{R}}_{+}\to {\mathbb{R}}^n$ is a controller.

Set the error system by $e\left(t\right)=v\left(t\right)-u\left(t\right).$ Define

$$h\left(t\right)=Be\left(t\right)$$

(3)

for an $n\times n$ matrix $B.$ Then we get

$$\left\{\begin{array}{c}{}^{C}D_0^{q}e\left(t\right)=(A+B+F(v,u))e\left(t\right),\hfill \\ e\left(0\right)=e_0=v_0-u_0\hfill \end{array}\right.$$

(4)

where

$$F(u,v)={\int }_0^1\frac{\partial f}{\partial v}(u,\theta v+(1-\theta )u)d\theta$$

(5)

is a Jacobian matrix.

From [18,19] the solution $e(·)\in C({\mathbb{R}}_{+},{\mathbb{R}}^n)$ of the system (4) is given by

$$\begin{array}{cc}\hfill e\left(t\right)& =E_q\left(t^{q}M\right)e_0+{\int }_0^t{(t-s)}^{q-1}{E}_{q,q}\left({(t-s)}^{q}M\right)F(v\left(s\right),u\left(s\right))e\left(s\right)ds\hfill \end{array}$$

(6)

where $M=A+B$, $E_q\left(M\right)$ and $E_{q,q}\left(M\right)$ are the well-known Mittag-Leffler functions in [18,20,21].

We present sufficient conditions for synchronization of nonlinear fractional order chaotic system by using asymptotic stability theory for Mittag-Leffler matrix function, Jacobian matrix and linear feedback controller. We introduce a Jacobian matrix for the nonlinear term and represent the solution for error system between drive-response systems. Note that the time response of drive system (1) makes the impact in response system (2) (see the nonlinear term of (2)). Moreover, we numerically check the time response for drive-response systems for different fractional orders and employ the linear feedback controller to synchronize the drive-response systems based on the chaotic behavior and state trajectories.

2. Main Results

We give the following explicit results for the norm of Mittag-Leffler matrix functions.

Lemma 1.

Assume

$$M=\mathrm{diag}\left(-{\lambda }_1,-{\lambda }_2,\cdots ,-{\lambda }_n\right),$$

(7)

for $0<{\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_n$. Then

$$\parallel E_q\left(t^{q}M\right)\parallel =E_q(-t^q{\lambda }_1),\parallel E_{q,q}\left({(t-s)}^{q}M\right)\parallel =E_{q,q}(-{(t-s)}^q{\lambda }_1).$$

Proof. 

From (7),

$$E_q\left(t^{q}M\right)=\mathrm{diag}(E_q(-t^q{\lambda }_1),E_q(-t^q{\lambda }_2),\cdots ,E_q(-t^q{\lambda }_n))$$

and

$$E_{q,q}\left({(t-s)}^{q}M\right)=\mathrm{diag}(E_{q,q}(-{(t-s)}^q{\lambda }_1),E_{q,q}(-{(t-s)}^q{\lambda }_2),\cdots ,E_{q,q}(-{(t-s)}^q{\lambda }_n)).$$

Since $E_q(-z)$ and $E_{q,q}(-z)$ are completely monotonous [21], we have

$$\begin{array}{c}0<E_q(-t^q{\lambda }_i)\le E_q(-t^q{\lambda }_1),\\ 0<E_{q,q}(-{(t-s)}^q{\lambda }_i)\le E_{q,q}(-{(t-s)}^q{\lambda }_1)\end{array}$$

for any $i=1,2,\cdots ,n$ and $0\le s\le t$. So considering any of the following standard norms on ${\mathbb{R}}^n$

$$\parallel x\parallel =\sqrt{\sum _{i=1}^n{x}_i^2},\parallel x\parallel =\underset{i=1,2,\cdots ,n}{max}|x_i|,\parallel x\parallel =\sum _{i=1}^n|x_i|,$$

one can derive the results. □

Definition 1.

System (4) is called stable if for any $ϵ>0$ there exists a $\delta >0$ such that $\parallel e_0\parallel < \delta$ guarantees that $\underset{t\ge 0}{sup}\parallel e\left(t\right)\parallel <ϵ.$

Definition 2.

The drive system (1) is said to be synchronized with the response system (2), if there exists an error system (4) which is stable.

We introduce the following conditions:

$\left[H1\right]$: $\underset{(u,v)\in {\mathbb{R}}^{2n}}{sup}\parallel F(u,v)\parallel =l<\infty$.

$\left[H2\right]$: $M=A+B$ satisfies (7).

Theorem 1.

Suppose that $\left[H1\right]$ and $\left[H2\right]$ hold. If $l<{\lambda }_1$ then the drive system (1) is synchronized with the response system (2) under the feedback controller (3).

Proof. 

From (6), $\left[H1\right]$, $\left[H2\right]$ and Lemma 1, we derive

$$\parallel e\left(t\right)\parallel \le E_q(-t^q{\lambda }_1)\parallel e_0\parallel +l{\int }_0^t{(t-s)}^{q-1}{E}_{q,q}(-{(t-s)}^q{\lambda }_1)\parallel e\left(s\right)\parallel ds.$$

(8)

Set $\theta (·)=\parallel e(·)\parallel$. Introducing a linear bounded operator $\Theta :C({\mathbb{R}}_{+},{\mathbb{R}}_{+})\to ({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ as

$$(\Theta \theta )\left(t\right)=E_q(-t^q{\lambda }_1)\theta \left(0\right)+l{\int }_0^t{(t-s)}^{q-1}{E}_{q,q}(-{(t-s)}^q{\lambda }_1)\theta \left(s\right)ds.$$

Then, (8) can be rewritten as

$$\theta \left(t\right)\le (\Theta \theta )\left(t\right),t\in {\mathbb{R}}_{+}.$$

Since $\Theta$ is non-decreasing, we see that the sequence ${\left\{{\Theta }^k\theta \right\}}_{k=0}^{\infty }$ is non-decreasing having a limit ${\theta }_{\infty }$ which satisfies ${\theta }_{\infty }=\Theta {\theta }_{\infty }$, i.e.,

$${\theta }_{\infty }\left(t\right)=E_q(-t^q{\lambda }_1){\theta }_{\infty }\left(0\right)+l{\int }_0^t{(t-s)}^{q-1}{E}_{q,q}(-{(t-s)}^q{\lambda }_1){\theta }_{\infty }\left(s\right)ds,$$

(9)

and $\theta \left(t\right)\le {\theta }_{\infty }\left(t\right)$ for any $t\in {\mathbb{R}}_{+}$. But (9) means

$${}^{C}D_0^q{\theta }_{\infty }\left(t\right)=-{\lambda }_1{\theta }_{\infty }\left(t\right)+l{\theta }_{\infty }\left(t\right)=(-{\lambda }_1+l){\theta }_{\infty }\left(t\right),$$

which has the solution

$${\theta }_{\infty }\left(t\right)=E_q\left(t^q(l-{\lambda }_1)\right){\theta }_{\infty }\left(0\right).$$

Summarizing, we obtain

$$\parallel e\left(t\right)\parallel \le E_q\left(t^q(l-{\lambda }_1)\right)\parallel e\left(0\right)\parallel .$$

(10)

Further, from [22] (Lemma 2.5), (10) reduces to

$$\parallel e\left(t\right)\parallel \le \frac{m(q,1,{\lambda }_1-l)}{t^q}\parallel e\left(0\right)\parallel ,$$

(11)

where

$$\begin{array}{c}\hfill m(q,1,{\lambda }_1-l)=\frac{sin\left(\pi (1-q)\right){\int }_0^{\infty }exp(-r^{\frac{1}{q}})dr}{{sin}^2\left(\pi q\right)\pi q({\lambda }_1-l)}.\end{array}$$

Finally, system (4) is stable with the convergence rate (11) and the drive system (1) is synchronized with the response system (2). □

Remark 1.

In [9,10,11,12,14,15,16,17], robust and exponential synchronization of fractional order deterministic chaotic systems was investigated using an adaptive scheme, an active control method, linear control, Lyapunov stability theory, linear feedback controller, sliding mode control and tracking control respectively. In this paper, we investigated the asymptotic synchronization for fractional order system using asymptotic stability theory, a feedback controller and Jacobian matrix.

Remark 2.

(i) $\left[H1\right]$ can be weakened in Theorem 1 to

$$\parallel f(u,v_1)-f(u,v_2)\parallel \le l\parallel v_1-v_2\parallel$$

for any $u,v_1,v_2\in {\mathbb{R}}^n$.

(ii) The above synchronization is based on the stability of an equilibrium of a linear fractional equation ${}^{C}D_0^{q}x\left(t\right)=Mx\left(t\right)$. We know that this holds if and only if

$$|arg\left(\lambda \right)|>\frac{q\pi }{2}$$

(12)

for any eigenvalue λ of M. But then it is not so clear an estimate like (11). This is a reason, why we consider a diagonal M. For a general M satisfying (12), we must follow the way from section 3.1 of [23], so a cumbersome approach based on real-valued Jordan form of M. Of course, the order in (7) is not important, so we can consider

$$M=\mathrm{diag}\left(m_{11},m_{22},\cdots ,m_{nn}\right)$$

with $m_{ii}<0$ and take ${\lambda }_1=max{m}_{ii}$.

Remark 3.

Here, we proposed a key problem to study the synchronization of a nonlinear fractional order system by employing the Mittag-Leffler matrix function, Jacobian matrix, asymptotic stability theory and linear feedback control methods. The main advantages of the consider model is the time response of drive system (1) makes the effect in response system (2) and combined the nonlinear terms by using Jacobian matrix to represent a solution for error system. This type of model is more applicable and reasonable to study the stability concepts via stable equilibrium, point, state, manifold and finite and infinite dimensional stochastic settings. Further, as an applications point of view we show the behaviors of the considered model for different fractional order through numerical simulations.

3. Examples

In this section, we present the numerical examples to verify the obtained theoretical results.

Example 1.

Consider the following fractional order drive system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{C}D_0^{q}u\left(t\right)=Au\left(t\right)+f(u,u),t\ge 0,q\in (0,1),\hfill \\ u\left(0\right)=u_0,\hfill \end{array}\right.\end{array}$$

(13)

where

$$\begin{array}{c}\hfill A=\left(\begin{array}{ccc}2& 1& 0\\ 0& 4& 1\\ 0& 0& -9\end{array}\right),\end{array}$$

and

$$\begin{array}{c}\hfill u\left(t\right)={(u_1\left(t\right),u_2\left(t\right),u_3\left(t\right))}^T,f(u,u)=\left(\begin{array}{c}1\\ cos{u}_2\\ 1\end{array}\right).\end{array}$$

The state trajectory of the drive system (13) for different fractional orders $q=0.828,0.9,0.95,0.99$ are given in Figure 1, Figure 2, Figure 3 and Figure 4. The trajectories of the system (13) is obviously stable for the fractional order $q<0.828.$ From Figure 1, Figure 2, Figure 3 and Figure 4, one can conclude that the behavior of the state trajectories $u_1\left(t\right),u_2\left(t\right),u_3\left(t\right)$ of the system (13) is unstable for the fractional order $q\ge 0.828.$

Thus, it is necessary to introduce the control parameter $h\left(t\right)$ to synchronize the error system (15), which is given in the following Example 2.

Example 2.

Consider the following fractional order response systems corresponding to the drive system (13)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{C}D_0^{q}v\left(t\right)=Av\left(t\right)+f(u,v)+h\left(t\right),t\ge 0,q\in (0,1),\hfill \\ v\left(0\right)=v_0,\hfill \end{array}\right.\end{array}$$

(14)

and the errors between (13) and (14) satisfying

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{C}D_0^{q}e\left(t\right)=(A+B+F(v,u))e\left(t\right),t\ge 0,q\in (0,1),\hfill \\ e\left(0\right)=e_0=v_0-u_0,\hfill \end{array}\right.\end{array}$$

(15)

where $M=A+B=\left(\begin{array}{ccc}-1& 0& 0\\ 0& -5& 0\\ 0& 0& -2\end{array}\right)$ and $({\lambda }_1,{\lambda }_2,{\lambda }_3)=(-1,-5,-2)$,

$$\begin{array}{c}\hfill f(u,v)=\left(\begin{array}{c}1+\frac{v-u}{2}\\ cos{u}_2\\ 1\end{array}\right),\end{array}$$

and A and $f(u,u)$ are already defined in Example 1. Further, as per Remark 2 (i), we can weakened in Theorem 1 to

$$\parallel f(u,v_1)-f(u,v_2)\parallel \le \frac{1}{2}\parallel v_1-v_2\parallel .$$

Here, $l=\frac{1}{2}$ and satisfies the statement of the Theorem 1 i.e., $l<{\lambda }_1=1.$

Set the suitable control $h\left(t\right)=Be\left(t\right)$, step size $\widehat{h}=0.001$ and $q=0.9,$

$$\begin{array}{cc}\hfill h\left(t\right)& ={(h_1\left(t\right),h_2\left(t\right),h_3\left(t\right))}^T,B=\left(\begin{array}{ccc}-3& -1& 0\\ 0& -9& -1\\ 0& 0& 7\end{array}\right),e_i\left(t\right)=v_i\left(t\right)-u_i\left(t\right),i=1,2,3.\hfill \end{array}$$

So

$$\begin{array}{cc}\hfill h_1\left(t\right)& =b_{11}{e}_1\left(t\right)+b_{12}{e}_2\left(t\right)+b_{13}{e}_3\left(t\right);b_{11}=-3,b_{12}=-1,b_{13}=0,\hfill \\ \hfill h_2\left(t\right)& =b_{21}{e}_1\left(t\right)+b_{22}{e}_2\left(t\right)+b_{23}{e}_3\left(t\right);b_{21}=0,b_{22}=-9,b_{23}=-1,\hfill \\ \hfill h_3\left(t\right)& =b_{31}{e}_1\left(t\right)+b_{32}{e}_2\left(t\right)+b_{33}{e}_3\left(t\right);b_{31}=0,b_{32}=0,b_{33}=7.\hfill \end{array}$$

Next, from the matrices $f(u,u)$ and $f(u,v)$ we get $\parallel F(u,v)\parallel =l=\frac{1}{2}.$ Hence, $\left[H1\right]$ is verified by using the Jacobian matrix (5). The chaotic behavior of the states $(u_1\left(t\right),u_2\left(t\right))$ and $(v_1\left(t\right),v_2\left(t\right)),(u_1\left(t\right),u_3\left(t\right))$ and $(v_1\left(t\right),v_3\left(t\right))$ and $(u_2\left(t\right),u_3\left(t\right))$ and $(v_2\left(t\right),v_3\left(t\right))$ of the drive-response systems (13) and (14) with fractional order $q=0.9$ are given in Figure 5, Figure 6 and Figure 7.

Moreover, three-dimensional phase diagram of the states $u_1\left(t\right),u_2\left(t\right),u_3\left(t\right)$ of (13) with fractional order $q=0.9$ and the time response of the states $(t,u_1\left(t\right))$ and $(t,v_1\left(t\right)),(t,u_2\left(t\right))$ and $(t,v_2\left(t\right))$ and $(t,u_3\left(t\right))$ and $(t,v_3\left(t\right))$ of the drive-response systems (13) and (14) with fractional order $q=0.9$ are showed in Figure 8, Figure 9, Figure 10 and Figure 11. In Figure 12, Figure 13 and Figure 14, we have showed synchronized time response for the same states of the drive-response systems (13) and (14) with fractional order $q=0.9.$ Finally, time response of the states $e_1\left(t\right),e_2\left(t\right),e_3\left(t\right)$ for the error system (15) with fractional order $q=0.9$ is showed in Figure 15.

From Theorem 1, system (13) is synchronized with (14) under the control $h\left(t\right)$ as shown in the Figure 12, Figure 13 and Figure 14. Time responses of the synchronization errors between (13) and (14) are shown in Figure 15, which provides the convergence of the synchronization errors to zero properly. Finally, all the hypothesis of Theorem 1 is verified numerically.

4. Conclusions

We presented synchronization criteria for nonlinear fractional order systems using Jacobian matrix and asymptotic stability estimation of the Mittag-Leffler matrix function, and a suitable linear feedback controller. The above arguments can be extended to fractional evolution equations of (1) and (2) with (3) on a Banach space X when A: $D\left(A\right)\subseteq X\to X$ is a generator of a $C_0$-semigroup $\{S_A\left(t\right),t\ge 0\}$ on X [24], $f:X\times X\to X$ is globally Lipschitz, $h:{\mathbb{R}}_{+}\to X$ is continuous and $B:X\to X$ is linear and continuous. We intend to study this case in our next paper.