# Synchronization of fractional chaotic systems based on a simple Lyapunov function

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## Abstract

In this paper the synchronization of fractional-order chaotic systems and a new property of fractional derivatives are studied. Then we propose a new fractional-order extension of Lyapunov direct method to control the fractional-order chaotic systems. A new synchronization method and a linear feedback controller are given to achieve the synchronization of fractional-order chaotic systems based on a simple Lyapunov candidate function. The proposed synchronization method can be applied to the synchronization of an arbitrary fractional-order chaotic system. This method is universal, simple, and theoretically rigorous. Numerical simulations of three fractional-order chaotic systems to verify the effectiveness and the universality of the proposed method.

## Keywords

Lyapunov function fractional order synchronization chaos## 1 Introduction

Fractional calculus is a topic of more than 300 years old. The idea of fractional calculus has been known since the regular calculus, with the first reference probably being associated with Leibniz and L’Hospital in 1695 where a half-order derivative was mentioned. As the generalization of integer-order dynamic systems, the fractional-order dynamic systems provide better mathematical models for some actual physical and engineering systems [1, 2]. The fractional-order nonlinear dynamic systems have many dynamic behaviors which are similar to the integer-order systems, such as chaos, bifurcation, and attractor [3, 4, 5, 6]. The fractional-order chaotic systems are extensively studied due to their potential applications in biology, information, chemistry, physics, and other fields [7, 8, 9, 10]. We can find numerous applications in electrochemistry, viscoelasticity, porous media, control, and electromagnetics [11, 12, 13, 14].

As is well known, synchronization control of chaos is very important but also very difficult for the chaotic systems. There are some methods which are proposed to control fractional-order chaotic systems [5, 15, 16, 17], such as backstepping controller [15], linear feedback controller [5], the Lyapunov equation-based method [16, 18], sliding mode method [17], the iterative learning control method [19, 20, 21] and the monotone iterative method [22, 23]. In most of these cases, the stability of the whole controlled system has to be analyzed using the fractional-order techniques as well. For the fractional-order linear time invariant systems, the stability can easily be proved using the method proposed by Matignon [24]. The stability of a fractional-order nonlinear time varying system was proposed by Diethelm [25], but the result is valid only for scalar fractional-order systems. Hence, in order to prove the stability of fractional-order nonlinear and time varying systems, some other methods must be applied. The important one of these methods is the fractional-order extension of the Lyapunov direct method, which was proposed by Li *et al.* [26]. And in [27] the authors proposed the Lyapunov direct method to prove the stability of fractional-order nonlinear system with time delay. As is well known, it is difficult to find a suitable Lyapunov candidate function to prove the stability of fractional systems. Some authors have presented Lyapunov functions to prove the stability of fractional-order systems [28, 29]. And some Lyapunov functions have been proposed related to fractional-order sliding mode control [30]. However, we find that the Lyapunov functions which were proposed in these papers are not simple, and they are valid for fractional-order system with specific characteristics.

In this paper, we first propose a new property of fractional-order derivatives which can help us find a simple Lyapunov function. Then we propose a new fractional-order extension of Lyapunov direct method and a control method for synchronizing an arbitrary fractional-order chaotic system. Some sufficient conditions of synchronization for the fractional-order chaotic systems are proposed based on a simple Lyapunov function. The proposed approach is simple, universal and theoretically rigorous. Numerical simulation results of the synchronization of the fractional-order unified system, the fractional-order Liu’s system, and the fractional-order Lorenz system demonstrate the effectiveness and the universality of the proposed method.

This paper is organized in the following manner: In Section 2 the preliminaries and some definitions are presented. Some synchronization criteria of fractional-order chaotic systems are proposed in Section 3. In Section 4, numerical simulation of three fractional chaotic systems shows the effectiveness and the universality of the control method. Finally, conclusions are in Section 5.

## 2 Preliminaries and definitions

### 2.1 Fractional derivative and numerical solution of differential equation

In the last decades, fractional calculus has been the subject of worldwide attention due to its broad range of applications in many fields, which indicates that it is an important role in modern science [1, 2]. Some definitions for fractional derivatives were studied in recent years. In this paper, two commonly used definitions of fractional derivatives, the Riemann-Liouville (RL) and the Caputo definition (C), are given as follows.

### Definition 1

### Definition 2

*α*is defined as

### Definition 3

*α*is given as

### 2.2 Some properties of the fractional derivative

In this paper we mainly consider the fractional order of the chaotic system to be \(0<\alpha<1\). Then some general properties of the fractional-order derivative \({}_{a}D_{t}^{\alpha}\) (Riemann-Liouville and Caputo definition) are described as follows [1, 2].

### Property 1

Leibniz rule for fractional differentiation [1, 2]

In the following, two new properties of fractional-order derivatives is proposed, which can help us find a simple Lyapunov function.

### Property 2

Caputo definition \({}_{a}^{C}D_{t}^{\alpha}\)

*P*, then

### Proof

*P*is positive definite, then

*i.e.*the conclusion (12) is clearly true. □

### Property 3

Riemann-Liouville definition \({}_{a}^{R}D_{t}^{\alpha}\)

*P*, then

### Proof

*P*is positive definite, then

*i.e.*\(f(t)\leq0\), then the conclusion (20) is clearly true. □

### Remark 1

*i.e.*\(P=I\), the property (2) and (3) can be written as

### 2.3 Stability of fractional-order system

*α*is the fractional order of the derivative; \(\boldsymbol{x}(t)=(x_{1}(t), x_{2}(t),\ldots, x_{n}(t))^{T} \in\mathbb{R}^{n}\); \(\boldsymbol{x}(0)=(c_{1},c_{2},\ldots,c_{n})^{T}\) is the initial value; \({}_{0}D_{t}^{\alpha}\) denotes the Caputo (or Riemann-Liouville) fractional-order derivative operator. The equilibrium points of this system are calculated by solving \(\boldsymbol{f}( \boldsymbol{x}^{*})=0\). For this fractional-order nonlinear system, the fractional-order extension of Lyapunov direct method has been proposed, which is obtained as follows [26].

### Theorem 1

*Let*\(\boldsymbol{x}=0\)

*be an equilibrium point for the nonautonomous fractional*-

*order system*(31).

*Assume that there exists a Lyapunov function*\(V(t, \boldsymbol{x}(t))\)

*and class*-

*K*

*functions*\(\alpha_{i}\) (\(i=1,2,3\))

*satisfying*

*where*\(\beta\in(0, 1)\).

*Then the equilibrium point of system*(31)

*is asymptotically stable*.

In the following, based on the fractional-order extension of the Lyapunov direct method and the new property of fractional derivatives, we will find a suitable Lyapunov function and propose the stability condition of the fractional chaotic system.

### Theorem 2

*Take the fractional*-

*order system*

*where*\(\alpha\in(0, 1)\)

*and*\({}_{0}D_{t}^{\boldsymbol{\alpha}}\)

*denotes the Caputo*(

*or Riemann*-

*Liouville*)

*derivative*.

*Without loss of generality*,

*let*\(\boldsymbol{x}^{*}=0\)

*be the equilibrium point and*\(\boldsymbol{x}(t)\in\mathbb{R}^{n}\).

*If there exists a positive definite matrix*

*P*,

*which satisfies*

*then the origin of the system*(34)

*is asymptotically stable*.

### Proof

*P*is a positive definite matrix, we introduce a Lyapunov function

*K*functions in [33], it follows from Theorem 1 that the origin of the system (34) is asymptotically stable. □

## 3 Synchronization of fractional-order chaotic system

To realize the synchronization of two system (38) and (39), we add a linear feedback control input to the response system (39). It is well known that the linear controller has many advantages: (1) it is very simple; (2) it is easily realized experimentally; (3) it is more suitable for engineering applications than other controllers.

*and*

**x***.*

**y**Then we can easily see that systems (38) and (40) are synchronized if and only if the error system (41) is asymptotically stable at the origin. Therefore, our aim is to design a suitable feedback gain matrix *K* such that the error system (41) is asymptotically stable.

### Theorem 3

*The controlled fractional*-

*order error system*(41)

*is asymptotically stable at the origin*,

*i*.

*e*.

*the systems*(38)

*and*(40)

*are asymptotically synchronized*,

*if the feedback gain matrix*

*K*

*makes the symmetric matrix*

*negative definite for all*\(\boldsymbol{x}(t)\)

*and*\(\boldsymbol{y}(t)\),

*where*

*P*

*is a positive definite matrix*.

### Proof

*P*is a positive definite matrix. It follows from Properties 2, 3 that

*S*is negative definite for all \(\boldsymbol{x}(t)\) and \(\boldsymbol{y}(t)\), we have

*i.e.*the systems (38) and (40) are asymptotically synchronized.

*C*, satisfying

### Corollary 1

*The controlled fractional*-

*order error system*(41)

*is asymptotically stable at the origin*,

*i*.

*e*.

*the systems*(38)

*and*(40)

*are asymptotically synchronized*,

*if the feedback gain matrix*

*K*

*makes the matrix*

*negative definite*,

*where*

*P*

*is a positive definite and*

*C*

*is defined as*(47).

This corollary can easily be proved by Theorem 3 and inequality (47).

If the positive definite matrix is \(P=I\), the constant matrix \(C=cI\) and the feedback gain matrix \(K=kI\), where *I* is identity matrix, the following corollary can be obtained.

### Corollary 2

*The controlled fractional*-

*order error system*(41)

*is asymptotically stable at the origin*,

*i*.

*e*.

*the systems*(38)

*and*(40)

*are asymptotically synchronized*,

*if the feedback gain matrix*\(K=kI\)

*makes the matrix*

*negative definite*.

*Particularly*,

*let*\(\lambda_{\max}\)

*be the maximal eigenvalue of the matrix*\(\frac{A^{T}+A}{2}\),

*if*\(K=kI\)

*satisfies*

*the controlled error system*(41)

*is asymptotically stable at the origin*.

### Remark 2

In these corollaries, these synchronization criteria are sufficient conditions for the synchronization of the fractional chaotic system. In the application, we only choose the feedback gain matrix \(K=kI\), which satisfies \(k>\lambda_{\max}+c\), then the controller can make the error system (41) asymptotically stable at the origin, *i.e.* it makes the systems (38) and (40) synchronize. This method is simple and universal.

### Remark 3

In Corollary 2, we suppose that the constant matrix is \(C=cI\) and the feedback gain matrix is \(K=kI\), then the conclusion is obtained. If we suppose that *C* and *K* are diagonal matrices, *i.e.* \(C=\operatorname{diag}(c_{1},c_{2},c_{3})\) and \(K=\operatorname{diag}(k_{1},k_{2},k _{3})\). According to Theorem 3 and Corollary 2, we must find a suitable \(k_{i}\) which satisfies the condition. In many cases, some \(k_{i}\) and \(c_{i}\) are equal to zero, which makes the linear controller simpler.

## 4 Simulation and analysis

In this section, three 3D fractional-order chaotic systems are used as examples to show the validity and effectiveness of the synchronization criterion. For the fractional-order error system in the form (41), by adding a linear controller and letting the feedback gain matrix satisfy the conditions of Corollary 2, the error system can be stabilized to the equilibrium point, *i.e.* the drive system and response system are asymptotically synchronized.

### 4.1 Synchronization of the fractional-order unified system

*et al.*introduced the unified system [34] in 2002, which can unify the Chen, Lü and Lorenz system. Its fractional expression is given as follows [35, 36]:

*α*is the fractional order of the derivative. If the parameter satisfies \(a\in[0, 0.8)\), system (51) is the generalized Lorenz system; if parameter satisfies \(a=0.8\), the system becomes the fractional-order Lü system; and if parameter satisfies \(a\in(0.8, 1]\), it becomes the fractional-order generalized Chen system. We select \(a=1\), there are three equilibrium points \((0, 0, 0)\), \((3\sqrt{7}, 3\sqrt{7},21)\) and \((-3\sqrt{7},-3 \sqrt{7},21)\) in this system (51). When we choose the fractional order \(\alpha=0.9\) and initial value \((2,2,1)\), the system (51) can generate chaos, which is shown in Figure 1.

*and*

**x***.*

**y***i.e.*the systems (52) and (54) are asymptotically synchronized. And we can easily see that the maximal eigenvalue of matrix \(\frac{A^{T}+A}{2}\) is \(\lambda_{\max}\approx31\). So if the feedback gain matrix \(K=kI\) satisfies \(k>56\), the error system is asymptotically stable at the origin. When selecting \(k=57\), the numerical results, illustrated in Figure 2, show that the fractional-order error system (55) is driven to its equilibrium point \((0,0,0)\) asymptotically as \(t\rightarrow\infty\), which implies that the drive system (52) and response system (54) are synchronized.

### 4.2 Synchronization of the fractional-order Liu system

*et al.*introduced a three dimensional chaotic system in 2009, which is called Liu’s system [37]. Gejji and Bhalekar [38] studied the fractional-order chaotic Liu system, which is described as

*and*

**x***.*

**y***i.e.*the systems (57) and (59) are asymptotically synchronized. And we can easily see that the maximal eigenvalue of matrix \(\frac{A^{T}+A}{2}\) is \(\lambda_{\max}=2.5\). Hence, if the feedback gain matrix \(K=kI\) satisfies \(k>6\), the system is asymptotically stable at the origin. When selecting \(k=7\), the numerical results, illustrated in Figure 4, show that the fractional-order error system (60) is driven to its equilibrium point \((0,0,0)\) asymptotically as \(t\rightarrow\infty\), which implies that the drive system (57) and response system (59) are synchronized.

### 4.3 Synchronization of the fractional-order Lorenz system

*α*is the order of fractional derivative. With the order \(\alpha=0.995\) and initial value \((0.1, 0.2, 0.3)\), the system can generate chaotic behavior, as shown in Figure 5.

*and*

**x***.*

**y***i.e.*\(c\approx25\). Hence, according to Corollary 2, if the matrix \(S=\frac{A^{T}+A}{2}+(c-k)I\) is negative positive, the error system (66) is asymptotically stable,

*i.e.*the systems (63) and (65) are asymptotically synchronized. And we can easily see that the maximal eigenvalue of matrix \(\frac{A^{T}+A}{2}\) is \(\lambda_{\max}\approx14\). So if the feedback gain matrix \(K=kI\) satisfies \(k>39\), the system is asymptotically stable at the origin. When selecting \(k=40\), the numerical results, illustrated in Figure 6, show that the fractional-order unified system is driven to its equilibrium point \((0,0,0)\) asymptotically as \(t\rightarrow\infty\), which implies that the drive system (63) and response system (65) are synchronized.

## 5 Conclusion

In this letter we proposed a new synchronization method for fractional-order chaotic systems based on a simple Lyapunov function. We also proposed some sufficient conditions of synchronization for the fractional-order chaotic systems. The proposed method is simple, universal and theoretically rigorous. Furthermore, we have implemented and verified our method for other fractional-order chaotic systems [2, 41, 42, 43, 44], namely the Lü chaotic system, the fractional-order Newton-Leipnik system, the Rössler system, financial systems, etc. The numerical simulation results also indicate that the proposed controller can effectively make the fractional-order chaotic systems synchronized, and the proposed method provides a theoretical basis for the applications of synchronization method in fractional-order dynamic systems. In future work as regards this topic, we will consider whether the proposed synchronization method can be extended to control other complex chaotic systems, such as the networked fractional chaotic systems [45, 46] and fractional multi-scroll chaotic systems [47].

## Notes

### Acknowledgements

This work is partly supported by Natural Science Foundation of China (No. 11501390, 61573010), Sichuan Youth Science & Technology Foundation (No. 2016JQ0046), Artificial Intelligence Key Laboratory of Sichuan Province (No. 2016RYJ06), Found of Sichuan University of Science and Engineering (No. 2016RCL33, 2015RC10), the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (No. 2015QYY01), Opening Project of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (No. 2016WYJ04).

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