Dynamic analysis of the fractional-order Liu system and its synchronization

The fractional differential calculus dates back to the 17th century. Although it has a long history, its applications to physics and engineering are just a recent focus of interest.^1–3 It was found that many systems in interdisciplinary fields can be described by the fractional differential equations, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, and electromagnetic waves.^4–7 There are essential differences between ordinary differential equation (ODE) systems and fractional-order differential systems, so properties and conclusions of ODE systems cannot be simply extended to that of the fractional-order differential systems. Nowadays, fractional-order systems have attracted more and more attention. It is known that many fractional-order differential systems behave chaotically, such as the fractional Lorenz system,⁸ fractional Duffing system,⁹ fractional Chua system,¹⁰ fractional Chen system,^11,12 and fractional Lü system,¹³ etc. Two problems should be answered. Firstly, when an ODE system is chaotic, under what conditions may the corresponding fractional-order system be also chaotic? More exactly, for what orders may the fractional-order system exhibit chaos? Secondly, how should one design a scheme to make chaotic fractional-order differential systems arrive synchronization? In the paper, the dynamic behaviors in different fractional-order Liu systems are analyzed based on the stability theory of fractional-order systems. A simple criterion for linear coupling synchronization of two identical fractional-order chaotic systems is also presented. It is believed that our methods are universally adaptable ways for other fractional-order systems as well. We can easily extend them to the dynamic analysis and coupling synchronization of other fractional-order chaotic systems.

At present, there are several definitions of the fractional-order differential system. Here we present the most common one of them:

where $m$ is the minimum integer which is not less than $α$⁠, $x(m)$ is the $m$-order derivative in usual sense, and $Jβ(β>0)$ is the $β$-order Reimann-Liouville integral operator which satisfies

where $Γ$ denotes the gamma function, and $Dα$ is generally called the “$α$-order Caputo differential operator.”¹⁴ 

The Liu system¹⁵ is described as

When $a=10$⁠, $b=40$⁠, $c=2.5$⁠, $h=4$⁠, and $k=1$⁠, system (1) exhibits a chaotic behavior. Its attractor is shown in Fig. 1.

The fractional-order Liu system is described as

where $dqi∕dtqi=Dqi(i=1,2,3)$ and $q1,q2,q3∊(0,1)$⁠. System (2) has three equilibriums: $S0(0,0,0)$⁠, $S1(bc∕hk,bc∕hk,b∕k)$⁠, and $S2(−bc∕hk,−bc∕hk,b∕k)$⁠. Choosing $a=10$⁠, $b=40$⁠, $c=2.5$⁠, $h=4$⁠, and $k=1$⁠, we obtain $S0(0,0,0)$⁠, $S1(5,5,40)$⁠, and $S2$(⁠$−5$⁠,$−5$⁠,40).

Consider a three-dimensional fractional-order system

where $q1,q2,q3∊(0,1)$⁠, the operator $Dθ$ is generally called the $θ$-order Caputo differential operator. The Jacobian matrix of system (3) is

Theorem 1. For $n$-dimensional fractional system, if all the eigenvalues $(λ1,λ2,…,λn)$ of the Jacobian matrix of some equilibrium satisfy

then the fractional-order system is asymptotically steady at the equilibrium.

According to the stability theory of fractional-order systems,¹⁶ it is easy to prove Theorem 1. Figure 2 illustrates Theorem 1. Obviously, if one equilibrium is stable, the fractional-order system will be steady at one point; and only when no equilibrium is stable, is it possible for the fractional-order system to exhibit chaos.

When $S0(0,0,0)$ is chosen to study, the eigenvalues of its corresponding Jacobian matrix are $λ1=−25.6155$⁠, $λ2=15.6155$⁠, and $λ3=−2.5$⁠. We can obtain $arg(λ1)=π$⁠, $arg(λ2)=0$⁠, and $arg(λ3)=π$⁠. According to Theorem 1, we can easily conclude that the equilibrium $S0$ of Eq. (2) is unstable when $0<α<1$⁠; i.e., $S0$ will never be stable.

When $S1(5,5,40)$ is chosen to study, the eigenvalues of its corresponding Jacobian matrix are $λ1=−17.5614$⁠, $λ2=2.5307+10.3673j$⁠, and $λ3=2.5307−10.3673j$⁠. We can obtain $arg(λ1)=π$⁠, $arg(λ2)=1.3314$⁠, and $arg(λ3)=−1.3314$⁠. According to Theorem 1, we can conclude that when $q1,q2,andq3$ are all less than $0.8476[1.3314(2∕π)]$⁠, the equilibrium $S1$ of Eq. (2) is stable. On the contrary, when $q1,q2,andq3$ are all greater than $0.8476$⁠, the equilibrium $S1$ of Eq. (2) is unstable.

In the same way, when $q1,q2,andq3$ are all less than $0.8476$⁠, the equilibrium $S2(−5,−5,40)$ of Eq. (2) is stable; when $q1,q2,andq3$ are all greater than $0.8476$⁠, the equilibrium $S2$ of Eq. (2) is unstable.

To sum up, when $q1,q2,andq3$ are all less than $0.8476$⁠, there exists at least one stable equilibrium; i.e., system (2) will stabilize at one point (⁠$S1$ or $S2$⁠) finally. When $q1=q2=q3=0.8476$⁠, system (2) will exhibit a limit cycle around some equilibrium point; when $q1,q2,andq3$ are all greater than $0.8476$⁠, all the equilibriums of Eq. (2) are unstable, and it is possible for system (2) to exhibit chaos. (In fact, simulation results show that system (2) is chaotic under this condition, though other dynamic behaviors are still possible in theory.) When $qi<0.8476<qj(i≠j)$⁠, the problem will be complex, system (2) may be convergent, periodic, or chaotic. It seems hard to analyze its behavior theoretically now, so we will not consider such a case here.

Figures 3–9 show the dynamic behaviors of system (2) with different fractional orders. From these simulation results, we can see that when $q1,q2,andq3$ are all greater than $0.8476$⁠, system (2) will exhibit a chaotic behavior (Figs. 3–5). When $q1=q2=q3=0.8476$⁠, system (2) will exhibit a limit cycle around some equilibrium (Fig. 6). When $q1,q2,andq3$ are all less than $0.8476$⁠, system (2) will stabilize at one fixed point (Figs. 7–9). Because $S0$ is always unstable, depending upon the initial states, system (2) will stabilize at $S1$ or $S2$⁠.

A chaotic system is described as

and the unidirectional coupled drive-response system constructed from it can be described as

$B$ is a matrix that stands for a linear combination of $X2−X1$⁠, $α$ is the feedback coefficient. We often suppose $αB=diag(k1,k2,…,kn)$ (⁠$n$ is the number of variables). The error system is $E=X2−X1$⁠.

System (6) can be extended to fractional-order chaotic system. Choosing system (2) as the drive system, the response system can be described as

In Ref. 17, if a fractional function is defined as

then its numerical solution can be described as follows:

where $h$ is the time step length, $[t∕h]$ denotes the integer part of $t∕h$⁠, $ω0(βi)=1$⁠, $ωj(βi)=[1−(βi+1)∕j]ωj−1(βi)(j=1,2,…)$⁠.

Suppose the largest Lyapunov exponent of system (2) is $λ$⁠, the distance between system (2) and system (7) is $∥Ei−h∥$ at time $ti−h$⁠, then the distance will not go beyond $∥Ei−h∥eλh$ at time $ti$⁠. Because $λh$ is very small, we have $∥Ei−h∥eλh≈∥Ei−h∥(1+λh)$⁠; i.e., the distance will at most increase to $(1+λh)$ times comparing with the previous time. Suppose the feedback coefficients in the response system satisfy $k1=k2=,…,=kn$⁠. In Eq. (10), the feedback term is in $ut$⁠, so we can easily know that the feedback term will cause the distance between system (2) and system (7) to decrease by $hqjkj∥Ei−h∥$ at time $ti$⁠. Because $h$ is very small and $0<qj<1$⁠, we have $hqjkj∥Ei−h∥>hkj∥Ei−h∥$⁠; i.e., the distance will at least decrease to $(1−kjh)$ times comparing with the previous time. It is obvious that when $kj≥λ$⁠, we have $limt→∞∥E(t)∥=0$⁠; i.e., system (2) and system (7) will be synchronized.

In the simulations, Figs. 10–12 show the history of $e1(t)$⁠, $e2(t)$⁠, $e3(t)$ in the error system (8) when $k1$⁠, $k2$⁠, $k3$⁠, and $q1$⁠, $q2$⁠, $q3$ select different values. Choosing $q1=0.98,q2=0.98,andq3=0.98$⁠, according to the method presented by Bennetin et al.,¹⁸ we obtain system (2)’s largest Lyapunov exponent $λ1=1.632$⁠. Let $k1=1.632,k2=1.632,$ and $k3=1.632$⁠, from Fig. 10, we can see that system (2) and system (7) are synchronized. Choosing $q1=0.95,q2=0.95,andq3=0.95$⁠, in the same way, we obtain system (2)’s largest Lyapunov exponent $λ1=1.508$⁠. Letting $k1=1.508,k2=1.508,andk3=1.508$⁠, from Fig. 11, we can see that system (2) and system (7) achieve synchronization. Choosing $q1=0.9,q2=0.9,andq3=0.9$⁠, for another time, we obtain system (2)’s largest Lyapunov exponent $λ1=0.981$⁠. Letting $k1=0.981,k2=0.981,andk3=0.981$⁠, from Fig. 12, we can see that system (2) and system (7) arrive at synchronization as well. These simulations show that when each coupling coefficient is not less than the largest Lyapunov exponent of the drive system, the fractional-order chaotic synchronization will be achieved.

In this paper, the dynamic behaviors of the fractional order Liu system are studied theoretically based on the stability theory of fractional-order systems. Simulation results show the effectiveness of our method. With this method, a critical value $α̂$ can be obtained by calculation. When a $n$-dimensional ODE system is chaotic, as for the corresponding fractional-order system, we have some theoretical conclusions: If $q1,q2,…,qn<α̂$⁠, the system will stabilize at one of its equilibriums; if $q1,q2,…,qn=α̂$⁠, the system will exhibit a limit cycle around some equilibrium; if $q1,q2,…,qn>α̂$⁠, the system may exhibit a chaotic behavior. Applying the method to fractional-order Lorenz system and Chen system, we get the following results: for the fractional-order Lorenz system, the critical value $α̂=0.99$⁠; for the fractional-order Chen system, the critical value $α̂=0.69$⁠. These theoretical results are in accord with the simulation results in Refs. 8 and 11 within the experimental error range. Therefore, it is believed that our method is universal for other fractional-order systems. In the past, the dynamic behaviors of fractional-order systems can only be obtained by simulations, but now we can analyze them theoretically.

What deserves to be noticed is that if $q1,q2,…,qn>α̂$⁠, the system may exhibit a chaotic behavior. Now there is not enough theoretical proof to demonstrate that if $q1,q2,…,qn>α̂$⁠, the system will exhibit a chaotic behavior, while according to the simulation results of many fractional-order systems, the latter argument seems true in reality.

Linear coupling is the most common type of synchronization. We present a simple criterion for the synchronization of two identical fractional-order chaotic systems: When each coupling coefficient is not less than the largest Lyapunov exponent of the drive system, the coupling synchronization will be achieved. This criterion can be seemed as an expansion of the conclusion in Ref. 19, which is suitable for ODE systems. In numerical simulations, two identical fractional-order Liu systems are synchronized successfully through this strategy.

This research is supported by the Chinese National Natural Science Foundation (Contract No. 60573172), the Superior University Science Technology Research Project of Liao Ning province (Contract No. 20040081).