Chaos Synchronization between Two Different Fractional Systems of Lorenz Family

Abstract

This work investigates chaos synchronization between two different fractional order chaotic systems of Lorenz family. The fractional order Lü system is controlled to be the fractional order Chen system, and the fractional order Chen system is controlled to be the fractional order Lorenz-like system. The analytical conditions for the synchronization of these pairs of different fractional order chaotic systems are derived by utilizing Laplace transform. Numerical simulations are used to verify the theoretical analysis using different values of the fractional order parameter.

1. Introduction

Fractional calculus has been known since the early 17th century [1, 2]. It has useful applications in many fields of science like physics [3], engineering [4], mathematical biology [5, 6], and finance [7, 8].

The fractional order derivatives have many definitions; one of them is the Riemann-Liouville definition [9] which is given by 𝐷^𝛼𝑑𝑓(𝑡)=^𝑙𝑑𝑡^𝑙𝐽^𝑙−𝛼𝑓(𝑡),𝛼>0, (1.1) where 𝐽^𝜃 is the 𝜃-order Riemann-Liouville integral operator which is given as 𝐽^𝜃1𝑢(𝑡)=Γ(𝜃)^𝑡₀(𝑡−𝜏)^𝜃−1𝑢(𝜏)𝑑𝜏,𝜃>0. (1.2)

However, the most common definition is the Caputo definition [10], since it is widely used in real applications: 𝐷^𝛼_∗𝑓(𝑡)=𝐽^𝑙−𝛼𝑓^(𝑙)(𝑡), (1.3) where 𝑓^(𝑙) represents the 𝑙-order derivative of 𝑓(𝑡) and 𝑙=[𝛼]; this means that 𝑙 is the first integer which is not less than 𝛼. The operator 𝐷^𝛼_∗ is called “the Caputo differential operator of order 𝛼.” Hence, I choose the Caputo type throughout this paper.

On the other hand, chaos has been studied and developed with much interest by scientists since the birth of Lorenz chaotic attractor in 1963 [11]. Chen attractor is similar to Lorenz attractor but not topologically equivalent [12]. Recently, Lü et al. found a new chaotic system which connects the Lorenz and Chen attractors, according to the conditions formulated by Vaněček and Čelikovský, and it is called Lü system [13]. Afterwards, chaos in fractional order dynamical systems has become an interesting topic. In [14] chaotic behaviors of the fractional order Lorenz system are studied. Moreover, chaotic behaviors have also been found in the fractional order Chen system [15] and the fractional order Lü system [16]. Furthermore, Chaos synchronization in fractional order chaotic systems starts to attract increasing attention [16–20]. However, it has been studied very well in the case of integer order chaotic systems, due to its potential applications in physical, chemical, and biological systems [21–24] and secure communications [25].

The generalized synchronization between two different fractional order systems is investigated in [26]. However, in this paper, I investigate the conditions of chaos synchronization between two different fractional order chaotic systems of Lorenz family by designing suitable linear controllers. I give examples to achieve chaos synchronization of two pairs of different fractional order chaotic systems (fractional Chen & fractional Lü, fractional Lorenz-like, and fractional Chen) in drive-response structure. Conditions for achieving chaos synchronization using linear control method are further discussed using Laplace transform theory.

2. Systems Description

The fractional order Chen system isgiven as follows: 𝑑^𝛼𝑥𝑑𝑡^𝛼𝑑=𝑎(𝑦−𝑥),^𝛼𝑦𝑑𝑡^𝛼𝑑=(𝑐−𝑎)𝑥−𝑥𝑧+𝑐𝑦,^𝛼𝑧𝑑𝑡^𝛼=𝑥𝑦−𝑏𝑧. (2.1) Here and throughout, (𝑎,𝑏,𝑐)=(35,3,28) where 𝛼 is the fractional order. In the following I choose 𝛼=0.9 at which system (2.1) exhibits chaotic attractor (see Figure 1).

Figure 1 

Chaotic attractor of the fractional order Chen system (2.1) with 𝛼=0.9 and (𝑎,𝑏,𝑐)=(35,3,28).

The fractional order Lü system is given as follows 𝑑^𝛼𝑥𝑑𝑡^𝛼𝑑=𝑟(𝑦−𝑥),^𝛼𝑦𝑑𝑡^𝛼𝑑=−𝑥𝑧+𝑝𝑦,^𝛼𝑧𝑑𝑡^𝛼=𝑥𝑦−𝑞𝑧. (2.2) Here and throughout, (𝑟,𝑝,𝑞)=(35,28,3). By choosing 𝛼=0.9, system (2.2) has chaotic attractor (see Figure 2).

Figure 2 

Chaotic attractor of the fractional order Lü system (2.2) with 𝛼=0.9 and (𝑟,𝑝,𝑞)=(35,28,3).

The fractional order Lorenz-like system [27] is described by 𝑑^𝛼𝑥𝑑𝑡^𝛼𝑑=𝜎(𝑦−𝑥),^𝛼𝑦𝑑𝑡^𝛼𝑑=𝜌𝑥−𝑥𝑧+𝛾𝑦,^𝛼𝑧𝑑𝑡^𝛼=𝑥𝑦−𝛽𝑧, (2.3) which has a chaotic attractor as shown in Figure 3 when 𝛽=2.8,𝛾=10.6,𝜌=14,𝜎=20, and 𝛼=0.9.

Figure 3 

Chaotic attractor of the fractional order Lorenz-like system (2.3) with 𝛼=0.9 and (𝛽,𝛾,𝜌,𝜎)=(2.8,10.6,14,20).

It should be also noted that, the systems (2.1), (2.2), and (2.3) are still chaotic at the fractional order values 𝛼=0.95 and 𝛼=0.99.

3. Synchronization between Two Different Fractional Order Systems

Consider the master-slave (or drive-response) synchronization scheme of two autonomous different fractional order chaotic systems:

𝑑^𝛼𝑋𝑑𝑡^𝛼𝑑=𝑓(𝑋),^𝛼𝑌𝑑𝑡^𝛼=𝑔(𝑌)+𝑈(𝑡), (3.1) where 𝛼 is the fractional order, 𝑋∈𝑅^𝑛,𝑌∈𝑅^𝑛 represent the states of the drive and response systems, respectively, 𝑓∶𝑅^𝑛→𝑅^𝑛,𝑔∶𝑅^𝑛→𝑅^𝑛 are the vector fields of the drive and response systems, respectively. The aim is to choose a suitable linear control function 𝑈(𝑡)=(𝑢₁,…,𝑢_𝑛)^𝑇 such that the states of the drive and response systems are synchronized (i.e., lim_𝑡→∞‖𝑋−𝑌‖=0, where ‖⋅‖ is the Euclidean norm).

3.1. Synchronization between Chen and Lü Fractional Order Systems

In this subsection, the goal is to achieve chaos synchronization between the fractional order Chen system and the fractional order Lü system by using the fractional order Chen system to drive the fractional order Lü system. The drive and response systems are given as follows:

𝑑^𝛼𝑥_𝑚𝑑𝑡^𝛼=𝑎(𝑦_𝑚−𝑥_𝑚𝑑),^𝛼𝑦_𝑚𝑑𝑡^𝛼=(𝑐−𝑎)𝑥_𝑚−𝑥_𝑚𝑧_𝑚+𝑐𝑦_𝑚,𝑑^𝛼𝑧_𝑚𝑑𝑡^𝛼=𝑥_𝑚𝑦_𝑚−𝑏𝑧_𝑚𝑑_𝛼,𝑥_𝑠𝑑𝑡^𝛼=𝑟(𝑦_𝑠−𝑥_𝑠)+𝑢₁,𝑑^𝛼𝑦_𝑠𝑑𝑡^𝛼=−𝑥_𝑠𝑧_𝑠+𝑝𝑦_𝑠+𝑢₂,𝑑^𝛼𝑧_𝑠𝑑𝑡^𝛼=𝑥_𝑠𝑦_𝑠−𝑞𝑧_𝑠+𝑢₃, (3.2)(3.3) where 𝑢₁,𝑢₂, and 𝑢₃ are the linear control functions. Define the error variables as follows:

𝑒₁=𝑥_𝑠−𝑥_𝑚,𝑒₂=𝑦_𝑠−𝑦_𝑚,𝑒₃=𝑧_𝑠−𝑧_𝑚. (3.4)

By subtracting (3.2) from (3.3) and using (3.4), we obtain 𝑑^𝛼𝑒₁𝑑𝑡^𝛼=𝑟(𝑒₂−𝑒₁)+(𝑟−𝑎)(𝑦_𝑚−𝑥_𝑚)+𝑢₁,𝑑^𝛼𝑒₂𝑑𝑡^𝛼=𝑝𝑒₂−𝑧_𝑚𝑒₁−𝑥_𝑚𝑒₃−𝑒₁𝑒₃−(𝑐−𝑎)𝑥_𝑚+(𝑝−𝑐)𝑦_𝑚+𝑢₂,𝑑^𝛼𝑒₃𝑑𝑡^𝛼=−𝑞𝑒₃+𝑦_𝑚𝑒₁+𝑥_𝑚𝑒₂+𝑒₁𝑒₂−(𝑞−𝑏)𝑧_𝑚+𝑢₃. (3.5)

Now, by letting

𝑢₁=(𝑎−𝑟)(𝑦_𝑚−𝑥_𝑚𝑢),₂=(𝑐−𝑎)𝑥_𝑚+(𝑐−𝑝)𝑦_𝑚−𝑘₁(𝑦_𝑠−𝑦_𝑚𝑢),₃=(𝑞−𝑏)𝑧_𝑚−𝑘₂(𝑧_𝑠−𝑧_𝑚), (3.6) where 𝑘₁,𝑘₂≥0, then the error system (3.5) is reduced to

𝑑^𝛼𝑒₁𝑑𝑡^𝛼=𝑟(𝑒₂−𝑒₁𝑑),_𝛼𝑒₂𝑑𝑡^𝛼=(𝑝−𝑘₁)𝑒₂−𝑧_𝑚𝑒₁−𝑥_𝑚𝑒₃−𝑒₁𝑒₃,𝑑^𝛼𝑒₃𝑑𝑡^𝛼=−(𝑞+𝑘₂)𝑒₃+𝑦_𝑚𝑒₁+𝑥_𝑚𝑒₂+𝑒₁𝑒₂. (3.7)

By taking the Laplace transform in both sides of (3.7), letting 𝐸_𝑖(𝑠)=𝐿{𝑒_𝑖(𝑡)} where (𝑖=1,2,3), and applying 𝐿{𝑑^𝛼𝑒_𝑖/𝑑𝑡^𝛼}=𝑠^𝛼𝐸_𝑖(𝑠)−𝑠^𝛼−1𝑒_𝑖(0), we obtain

𝑠^𝛼𝐸₁(𝑠)−𝑠^𝛼−1𝑒₁(0)=𝑟(𝐸₂(𝑠)−𝐸₁𝑠(𝑠)),_𝛼𝐸₂(𝑠)−𝑠^𝛼−1𝑒₂(0)=(𝑝−𝑘₁)𝐸₂(𝑠)−𝐿{𝑥_𝑚𝑒₃}−𝐿{𝑧_𝑚𝑒₁}−𝐸₁(𝑠)𝐸₃𝑠(𝑠),_𝛼𝐸₃(𝑠)−𝑠^𝛼−1𝑒₃(0)=−(𝑞+𝑘₂)𝐸₃(𝑠)+𝐿{𝑦_𝑚𝑒₁}+𝐿{𝑥_𝑚𝑒₂}+𝐸₁(𝑠)𝐸₂(𝑠). (3.8)

Proposition 3.1. If 𝐸₁(𝑠),𝐸₂(𝑠) are bounded and 𝑝−𝑘₁≠0, then the drive and response systems (3.2) and (3.3) will be synchronized under a suitable choice of 𝑘₁ and 𝑘₂.

Proof. Rewrite (3.8) as follows: 𝐸₁(𝑠)=𝑟𝐸₂(𝑠)𝑠^𝛼+𝑠+𝑟^𝛼−1𝑒₁(0)𝑠^𝛼,𝐸+𝑟₂(𝑠)=−𝐿{𝑧_𝑚𝑒₁}𝑠^𝛼−𝑝+𝑘₁−𝐿{𝑥_𝑚𝑒₃}𝑠^𝛼−𝑝+𝑘₁−𝐸₁(𝑠)𝐸₃(𝑠)𝑠^𝛼−𝑝+𝑘₁+𝑠^𝛼−1𝑒₂(0)𝑠^𝛼−𝑝+𝑘₁,𝐸₃(𝑠)=𝐿{𝑦_𝑚𝑒₁}𝑠^𝛼+𝑞+𝑘₂+𝐿{𝑥_𝑚𝑒₂}𝑠^𝛼+𝑞+𝑘₂+𝐸₁(𝑠)𝐸₂(𝑠)𝑠^𝛼+𝑞+𝑘₂+𝑠^𝛼−1𝑒₃(0)𝑠^𝛼+𝑞+𝑘₂. (3.9)
Using the final value theorem of the Laplace transform, it follows that lim_𝑡→∞𝑒₁(𝑡)=lim_𝑠→0+𝑠𝐸₁(𝑠)=lim_𝑠→0+𝑠𝐸₂(𝑠)=lim_𝑡→∞𝑒₂(𝑡),lim_𝑡→∞𝑒₂(𝑡)=lim_𝑠→0+𝑠𝐸₂1(𝑠)=𝑝−𝑘₁lim_𝑠→0+𝑠𝐿{𝑥_𝑚𝑒₃1}+𝑝−𝑘₁lim_𝑠→0+𝑠𝐿{𝑧_𝑚𝑒₁1}+𝑝−𝑘₁lim_𝑡→∞𝑒₁(𝑡)⋅lim_𝑡→∞𝑒₃(𝑡),lim_𝑡→∞𝑒₃(𝑡)=lim_𝑠→0+𝑠𝐸₃1(𝑠)=𝑞+𝑘₂lim_𝑠→0+𝑠𝐿{𝑦_𝑚𝑒₁1}+𝑞+𝑘₂lim_𝑠→0+𝑠𝐿{𝑥_𝑚𝑒₂1}+𝑞+𝑘₂lim_𝑡→∞𝑒₁(𝑡)⋅lim_𝑡→∞𝑒₂(𝑡). (3.10)
Since 𝐸₁(𝑠),𝐸₂(𝑠) are bounded and 𝑝−𝑘₁≠0 then lim_𝑡→∞𝑒₁(𝑡)=lim_𝑡→∞𝑒₂(𝑡)=0. Now, owing to the attractiveness of the attractors of systems (2.1) and (2.2), there exists 𝜂>0 such that |𝑥_𝑖(𝑡)|≤𝜂<∞,|𝑦_𝑖(𝑡)|≤𝜂<∞, and |𝑧_𝑖(𝑡)|≤𝜂<∞ where 𝑖 refers to the index of the drive or response variables. Therefore, lim_𝑡→∞𝑒₃(𝑡)=0. This implies that lim_𝑡→∞𝑒_𝑖(𝑡)=0,𝑖=1,2,3. (3.11)
Consequently, the synchronization between the drive and response systems (3.2) and (3.3) is achieved.

3.1.1. Numerical Results

An efficient method for solving fractional order differential equations is the predictor-correctors scheme or more precisely, PECE (Predict, Evaluate, Correct, Evaluate) technique which has been investigated in [28, 29], and represents a generalization of the Adams-Bashforth-Moulton algorithm. It is used throughout this paper.

Based on the above mentioned discretization scheme, the drive and response systems (3.2) and (3.3) are integrated numerically with the fractional orders 𝛼=0.9,0.95,0.99 and using the initial values 𝑥_𝑚(0)=15,𝑦_𝑚(0)=20,𝑧_𝑚(0)=29 and 𝑥_𝑠(0)=10,𝑦_𝑠(0)=15,𝑧_𝑠(0)=25. From Figure 4, it is clear that the synchronization is achieved for all these values of fractional order when 𝑘₁=20 and 𝑘₂=10.

(a)

(b)

(c)

Figure 4 

Synchronization errors of the drive system (3.2) and response system (3.3) using 𝑘1=20,𝑘2=10 and fractional orders: (a) 𝛼=0.9, (b) 𝛼=0.95, and (c) 𝛼=0.99.

3.2. Synchronization between Lorenz-Like and Chen Fractional Order Systems

In this case it is assumed that, the fractional order Lorenz-like system drives the fractional order Chen system. The drive and response systems are defined as follows:

𝑑^𝛼𝑥_𝑚𝑑𝑡^𝛼=𝜎(𝑦_𝑚−𝑥_𝑚𝑑),^𝛼𝑦_𝑚𝑑𝑡^𝛼=𝜌𝑥_𝑚−𝑥_𝑚𝑧_𝑚+𝛾𝑦_𝑚,𝑑^𝛼𝑧_𝑚𝑑𝑡^𝛼=𝑥_𝑚𝑦_𝑚−𝛽𝑧_𝑚𝑑_𝛼,𝑥_𝑠𝑑𝑡^𝛼=𝑎(𝑦_𝑠−𝑥_𝑠)+𝑣₁,𝑑^𝛼𝑦_𝑠𝑑𝑡^𝛼=(𝑐−𝑎)𝑥_𝑠−𝑥_𝑠𝑧_𝑠+𝑐𝑦_𝑠+𝑣₂,𝑑^𝛼𝑧_𝑠𝑑𝑡^𝛼=𝑥_𝑠𝑦_𝑠−𝑏𝑧_𝑠+𝑣₃, (3.12)(3.13) where 𝑣₁,𝑣₂, and 𝑣₃ are the linear control functions. The error variables are given by

𝑒₁=𝑥_𝑠−𝑥_𝑚,𝑒₂=𝑦_𝑠−𝑦_𝑚,𝑒₃=𝑧_𝑠−𝑧_𝑚. (3.14)

By subtracting (3.12) from (3.13) and using (3.14), we get

𝑑^𝛼𝑒₁𝑑𝑡^𝛼=𝑎(𝑒₂−𝑒₁)+(𝑎−𝜎)(𝑦_𝑚−𝑥_𝑚)+𝑣₁,𝑑^𝛼𝑒₂𝑑𝑡^𝛼=𝑐𝑒₂−𝑧_𝑚𝑒₁−𝑥_𝑚𝑒₃−𝑒₁𝑒₃+(𝑐−𝑎)𝑥_𝑠−𝜌𝑥_𝑚+(𝑐−𝛾)𝑦_𝑚+𝑣₂,𝑑^𝛼𝑒₃𝑑𝑡^𝛼=−𝑏𝑒₃+𝑦_𝑚𝑒₁+𝑥_𝑚𝑒₂+𝑒₁𝑒₂+(𝛽−𝑏)𝑧_𝑚+𝑣₃. (3.15)

Now, by choosing 𝑣₁=(𝜎−𝑎)(𝑦_𝑚−𝑥_𝑚),𝑣₂=𝜌𝑥_𝑚+(𝑎−𝑐)𝑥_𝑠+(𝛾−𝑐)𝑦_𝑚−𝑘₁𝑒₂,𝑣₃=(𝑏−𝛽)𝑧_𝑚−𝑘₂𝑒₃, (3.16) where 𝑘₁,𝑘₂≥0, then the error system (3.15) is rewritten as

𝑑^𝛼𝑒₁𝑑𝑡^𝛼=𝑎(𝑒₂−𝑒₁𝑑),_𝛼𝑒₂𝑑𝑡^𝛼=(𝑐−𝑘₁)𝑒₂−𝑧_𝑚𝑒₁−𝑥_𝑚𝑒₃−𝑒₁𝑒₃,𝑑^𝛼𝑒₃𝑑𝑡^𝛼=−(𝑏+𝑘₂)𝑒₃+𝑦_𝑚𝑒₁+𝑥_𝑚𝑒₂+𝑒₁𝑒₂. (3.17)

Take Laplace transform in both sides of (3.17), let 𝐸_𝑖(𝑠)=𝐿{𝑒_𝑖(𝑡)}, where (𝑖=1,2,3), and apply 𝐿{𝑑^𝛼𝑒_𝑖/𝑑𝑡^𝛼}=𝑠^𝛼𝐸_𝑖(𝑠)−𝑠^𝛼−1𝑒_𝑖(0). After that, by doing similar analysis like the previous subsection, we obtain

lim_𝑡→∞𝑒₁(𝑡)=lim_𝑠→0+𝑠𝐸₁(𝑠)=lim_𝑠→0+𝑠𝐸₂(𝑠)=lim_𝑡→∞𝑒₂(𝑡),lim_𝑡→∞𝑒₂(𝑡)=lim_𝑠→0+𝑠𝐸₂1(𝑠)=𝑐−𝑘₁lim_𝑠→0+𝑠𝐿{𝑥_𝑚𝑒₃1}+𝑐−𝑘₁lim_𝑠→0+𝑠𝐿{𝑧_𝑚𝑒₁1}+𝑐−𝑘₁lim_𝑡→∞𝑒₁(𝑡)⋅lim_𝑡→∞𝑒₃(𝑡),lim_𝑡→∞𝑒₃(𝑡)=lim_𝑠→0+𝑠𝐸₃1(𝑠)=𝑏+𝑘₂lim_𝑠→0+𝑠𝐿{𝑦_𝑚𝑒₁1}+𝑏+𝑘₂lim_𝑠→0+𝑠𝐿{𝑥_𝑚𝑒₂1}+𝑏+𝑘₂lim_𝑡→∞𝑒₁(𝑡)⋅lim_𝑡→∞𝑒₂(𝑡). (3.18)

If we assume that 𝑐−𝑘₁≠0 and 𝐸₁(𝑠),𝐸₂(𝑠) are bounded, then it follows that lim_𝑡→∞𝑒₁(𝑡)=lim_𝑡→∞𝑒₂(𝑡)=0. Now, owing to the attractiveness of the attractors of systems (2.1) and (2.3), there exists 𝜉>0 such that |𝑥_𝑖(𝑡)|≤𝜉<∞,|𝑦_𝑖(𝑡)|≤𝜉<∞, and |𝑧_𝑖(𝑡)|≤𝜉<∞ where 𝑖 refers to the index of the drive or response variables. Therefore, lim_𝑡→∞𝑒₃(𝑡)=0. Consequently,

lim_𝑡→∞𝑒_𝑖(𝑡)=0,𝑖=1,2,3. (3.19)

Thus, the states of the drive system (3.12) are synchronized with the states of the response system (3.13), as the controllers (3.16) are activated.

3.2.1. Numerical Results

Numerical simulations are carried out to integrate the drive and response systems (3.12) and (3.13) using the predictor-correctors scheme, with the fractional orders 𝛼=0.9,0.95,0.99 and the initial values 𝑥_𝑚(0)=10,𝑦_𝑚(0)=16,𝑧_𝑚(0)=25 and 𝑥_𝑠(0)=15,𝑦_𝑠(0)=20,𝑧_𝑠(0)=29. Thus, the drive and response systems (3.12) and (3.13) are synchronized in such a successful way for all at the above-mentioned fractional orders values, using the linear controllers (3.16) with 𝑘₁=20 and 𝑘₂=10 (see Figure 5).

(a)

(b)

(c)

Figure 5 

Synchronization errors of the drive system (3.12) and response system (3.13) using 𝑘1=20,𝑘2=10 and fractional orders: (a) 𝛼=0.9, (b) 𝛼=0.95, and (c) 𝛼=0.99.

4. Conclusion

Chaos synchronization between two different fractional order chaotic systems has been studied using linear control technique. Fractional order Chen system has been used to drive fractional order Lü system, and fractional order Lorenz-like system has been used to drive fractional order Chen system. Conditions for chaos synchronization have been investigated theoretically by using Laplace transform. Numerical simulations have been carried out using different fractional order values to show the effectiveness of the proposed synchronization techniques.

Acknowledgments

The author wishes to thank the reviewers and the associate editor for their careful reading and efforts and for providing some helpful suggestions. Also I wish to thank Professor E. Ahmed and Dr. Faycal Ben Adda for discussion and help.