# The Proposed Modified Liu System with Fractional Order

Authors: Alireza K. Golmankhaneh; Roohiyeh Arefi; Dumitru Baleanu
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

The chaos in a new system with order 3 is studied. We have shown that this chaotic system again will be chaotic when the order of system is less than 3. Generalized Adams-Bashforth algorithm has been used for investigating in stability of fixed points and existence of chaos.


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## Body

## 1. Introduction

It is well known that the nonlinear equations of dynamical systems with special condition have chaotic behavior [1]. Subsequently, additional studies were performed on the chaos and chaotic systems. Therefore solutions of different systems display their chaotic behavior such as Chen's system, Chua's dynamical system, the motion of double pendulum, and Rossler system amongst others. At first, it was thought that the chaos exists only when the order of system of differential equation is exactly 3. When the system of differential equation is composed of three first order differential equations, the order of the system is the sum of orders. But later on, a very interesting thing was realized; that is, it is also possible to observe chaotic behavior in a fractional order system. The system is composed of differential equations with fractional order derivatives [1–28]. For example, Sheu et al. reviewed the chaotic behavior of the Newton-Leipnik system with fractional order [10]. The important thing in the study of fractional-order systems is the minimum effective dimension of the system for which the system remains chaotic. This quantity has been numerically calculated for different systems including fractional order Lorenz system [11], fractional order Chua's system [12], and fractional order Rössler system [13]. Recently, the chaos has been studied in fractional ordered Liu system, where the numerical investigations on the dynamics of this system have been carried out, and properties of the system have been analyzed by means of Lyapunov exponent [14]. In this paper we study the chaotic behavior of a generalization of the Liu system with fractional order.The framework of the paper is as follows.In Section2, we study the behavior of a new fractional order system (modification of Liu system), and we study commensurate and incommensurate ordered systems and find lowest order at which chaos exists by numerical experiments. We have investigated the instability of fixed points and used Lyapunov exponent for the existence of chaos. In Section 3, we state the main conclusions.


## 2. The Proposed Modified Liu System

In this section, we review the condition for asymptotic stability of the commensurate and incommensurate fractional ordered systems. We suggest the readers to see [15–21] for the following section.


### 2.1. Asymptotic Stability of the Commensurate Fractional Ordered System

Let us consider the commensurate fractional ordered dynamical system equation(1)Dαxi=fi(x1,x2,x3),1≤i≤3.
An equilibrium point of (1) is p≡(x1*,x2*,x3*) which is fi(p)=0, 1≤i≤3, and a small disturbance from a fixed point is given by ξi=xi-xi*. Thus, we have
(2)Dαξi=Dαxi=fi(x1,x2,x3)=fi(ξ1+x1*,ξ2+x2*,ξ3+x3*)=fi(x1*,x2*,x3*)+ξ1∂fi(p)∂x1+ξ2∂fi(p)∂x2=+ξ3∂fi(p)∂x3+higher ordered terms≈ξ1∂fi(p)∂x1+ξ2∂fi(p)∂x2+ξ3∂fi(p)∂x3.
Write the system in the matrix form
(3)Dαξ=Jξ,
where
(4)J=(∂1f1(p)∂2f1(p)∂3f1(p)∂1f2(p)∂2f2(p)∂3f2(p)∂1f3(p)∂2f3(p)∂3f3(p)),
where J is Jacobian matrix of the system, and if J does not have purely imaginary eigenvalues, therefore the trajectories of the nonlinear system in the neighborhood of the equilibrium point have the same form as the trajectories of the linear system [18]. So we arrive at the following linear autonomous system:
(5)Dαξ=Jξ,ξ(0)=ξ0,
where J is n×n matrix and 0<α<1. The system (5) is asymptotically stable if and only if |arg(λ)|>απ/2 for all eigenvalues λ of J. So for this condition the solutions ξi(t) of (5) tend to 0 as t→∞. Therefore, the equilibrium point p of the system is asymptotically stable if |arg(λ)|>απ/2, for all eigenvalues λ of J. For example,(6)mini|arg(λi)|>απ2.


### 2.2. Asymptotic Stability of the Incommensurate Fractional Ordered System

Consider the following incommensurate fractional ordered dynamical system [19, 20]. Now suppose the commensurate fractional ordered dynamical system equation
(7)Dαixi=fi(x1,x2,x3),1≤i≤3,
where 0<αi<1. One can write it as αi=vi/ui, (ui,vi)=1, so ui, vi are positive integers. The definition of common multiple of ui’s is M. Equilibrium point and small disturbance of the system p and ξi respectively as above. So we get
(8)Dαixi≈ξ1∂fi(p)∂x1+ξ2∂fi(p)∂x2+ξ3∂fi(p)∂x3,1≤i≤3,
where it can be written as
(9)(Dα1ξ1Dα2ξ2Dα3ξ3)=J(ξ1ξ2ξ3),
where J is Jacobian matrix evaluated at point p. The definition is(10)Δ(λ)=diag([λMα1λMα2λMα3])-J,
that is, if all the roots of equation Δ(λ)=0 satisfy the condition |arg(λ)|>π/2M [21], the solution of linear system is asymptotically stable as follows:
(11)π2M-mini|arg(λi)|<0.
The term in the left side of (11) is an instability measure for equilibrium point in fractional ordered system (IMFOS). Then, fractional order equation (7) exhibits chaotic attractor if the condition is [19, 20]
(12)IMFOS≥0.


### 2.3. Modified Liu System

In this section we introduce the following system and show that the system is chaotic:(13)x˙=-ax-by2,y˙=cy+dzx-ez2,z˙=fz+gxy,
where a=2, b=e=1,c=-3, d=-4, f=-7, and g=4 with the initial conditions (0.2,0,0.5) lead to the chaotic trajectories. Meanwhile, we want to show chaotic behavior of (13) involving fractional order. Also, we will calculate the minimum effective dimension by which the system remains chaotic. The corresponding fractional order system is
(14)Dα1=-ax-by2,Dα2=cy+dzx-ez2,Dα3=fz+gxy,
where αi∈(0,1). In (14) if we choose α1=α2=α3=α, the system is called commensurate, and otherwise it is incommensurate. Now, we have four real equilibrium points for (13) which are shown in Table 1. In Table 1, we see the equilibrium points and the eigenvalues of the corresponding Jacobian matrix
(15)J=(-a-2by0dzcdx-2ezgyxgxf).
A saddle point p is stable, if the Jacobian matrix has at least one eigenvalue with a negative real part. Otherwise, one eigenvalue with a nonnegative real part is called unstable. And saddle points have index one or two if there is exactly one or two unstable eigenvalues, respectively. It is established in the literature [22–26] that scrolls are generated only around the saddle points of an index one or two. Saddle points of index one are to connect scrolls. Table 1 shows that the equilibrium points E1 and E2 are saddle points of index two; therefor we have two-scroll attractor [22], in the fractional system given by (14).Table 1
Equilibrium points and corresponding eigenvalues.
Equilibrium point
Eigenvalue
Nature
E
0
(
0,0
,
0
)
(-7,3,-2)
Saddle point
E
1
(
-
24.392
,
-
6.98456
,
97.3529
)
(-64.4116,63.0432,-4.63163)
Saddle point
E
2
(
-
1.04307
,
1.44435
,
-
0.860895
)
(-6.694,0.347±5.1057i)
Saddle point
E
3
(
-
1.30636
,
-
1.61639
,
1.20662
)
(-6.82593,0.412964±4.75973i)
Saddle point


### 2.4. Commensurate Ordered System

Consider the system equation (14), and let α1=α2=α3=α, so it is called commensurate order for this case. In this case a system shows regular behavior if it satisfies mini|arg(λi)|>απ/2, then we have [15–21]
(16)α<2πmini|arg(λi)|≈0.95.
From Figure 1 we can see that the Lyapunov exponent for the case of commensurate ordered equation (14) is positive if α>0.95 [27, 28]. Thus we can realize that the system does not indicate chaotic behavior for the value α<0.95. This corollary has been figured out by numerical outcome. Moreover, Figure 2 illustrates the phase portrait in xy-plane for the α=0.95. Numerical experiments and Figures 3 and 4 demonstrate that the system has chaotic behavior for α=0.96. In addition, Figures 5 and 6 represent solutions x(t) and y(t) for α=0.96, respectively. Adams-Bashforth predictor-corrector algorithm is used for numerical result with step size 0.1.Figure 1
Lyapunov exponent.Figure 2
Phase portraitα=0.95.Figure 3
Phase portraitx-y  α=0.96.Figure 4
Phase portraity-zα=0.96.Figure 5
Solutionx(t) for α=0.96.Figure 6
Solutiony(t) for α=0.96.


### 2.5. Incommensurate Ordered System

In this section we indicate that the condition for being chaotic system in the case of commensurate is not sufficient for the incommensurate case. So let us consider the fractional order system equation (14). Figures 7 and 8 display that the Lyapunov exponent is positive for α1≥0.89, α2=α3=1, and for the case α3≥0.88,α1=α3=1, respectively. Now, let us consider the following cases for (14).Figure 7
Lyapunov exponent-α1.Figure 8
Lyapunov exponent-α3.(1) In the first condition we chooseα1=22/25,α2=α3=1. Therefore, one can acquire M = LCM(25,1,1)=25. Since we have Δ(λ)=diag([λ22λ25λ25])-J(E1), then
(17)det(Δ(λ))=-1.9×10-4-9.5×106-3λ22+5.45×10-3λ25+4λ47+2λ50+λ72.
And the instability measure for equilibrium point in fractional ordered system (IMFOS) will be for the system
(18)π50-0.0=0.0628>0.
In Figure 9, there is no chaos, whereas IMFOS >0 [15–21]. This consequence leads to that the IMFOS >0 is not sufficient for existence of chaos.Figure 9
Phase portraitx-yα1=0.88,α2=α3=1.(2) As second case, supposeα1=89/100,α2=α3=1, so that in this case the M = LCM(100,1,1)=100. Also we have Δ(λ)=diag([λ89λ100λ100])-J(E1), as well as we obtain
(19)det(Δ(λ))=-1.9×10-4-9.5×10-3λ89+5.5×10-3λ100+4λ189+2λ200+λ289.
Therefore, the IMFOS of the system for this state is
(20)π200-0.0=0.0157>0.
If we look at Figure 10 we conclude the chaotic behavior for the system with the mentioned condition. Here we remark the lowest dimension of the system which has chaos (Figure 11).Figure 10
Phase portraitx-zα1=0.89,α2=α3=1.Figure 11
Phase portraity-zα3=0.87,α2=α1=1.(3) As third case, letα3=43/50,α1=α3=1. So with some manipulation we get M = LCM(1,1,50)=50,
(21)det(Δ(λ))=-18807.7+5430.19λ43-9484.56λ50-λ93+7λ100+λ143.
If we compute the IMFOS for the system, we will arrive at
(22)π100-0.0=0.0314>0.
Figure 11 and (22) point that the system does not have chaotic condition.(4) For the fourth case, we considerα3=22/25,α1=α3=1. Therefore, we will obtain M = LCM(1,1,50)=50,
(23)det(Δ(λ))=-1.9×10-4-9.5×106-3λ22+5.45×10-3λ25+4λ47+2λ50+λ72.
IMFOS for the fourth case is
(24)π100-0.0=0.0628>0.
Numerical results and Figure 12 indicate for the this case that the system is chaotic.Figure 12
Phase portraity-zα3=0.88,α2=α1=1.


## 2.1. Asymptotic Stability of the Commensurate Fractional Ordered System

Let us consider the commensurate fractional ordered dynamical system equation(1)Dαxi=fi(x1,x2,x3),1≤i≤3.
An equilibrium point of (1) is p≡(x1*,x2*,x3*) which is fi(p)=0, 1≤i≤3, and a small disturbance from a fixed point is given by ξi=xi-xi*. Thus, we have
(2)Dαξi=Dαxi=fi(x1,x2,x3)=fi(ξ1+x1*,ξ2+x2*,ξ3+x3*)=fi(x1*,x2*,x3*)+ξ1∂fi(p)∂x1+ξ2∂fi(p)∂x2=+ξ3∂fi(p)∂x3+higher ordered terms≈ξ1∂fi(p)∂x1+ξ2∂fi(p)∂x2+ξ3∂fi(p)∂x3.
Write the system in the matrix form
(3)Dαξ=Jξ,
where
(4)J=(∂1f1(p)∂2f1(p)∂3f1(p)∂1f2(p)∂2f2(p)∂3f2(p)∂1f3(p)∂2f3(p)∂3f3(p)),
where J is Jacobian matrix of the system, and if J does not have purely imaginary eigenvalues, therefore the trajectories of the nonlinear system in the neighborhood of the equilibrium point have the same form as the trajectories of the linear system [18]. So we arrive at the following linear autonomous system:
(5)Dαξ=Jξ,ξ(0)=ξ0,
where J is n×n matrix and 0<α<1. The system (5) is asymptotically stable if and only if |arg(λ)|>απ/2 for all eigenvalues λ of J. So for this condition the solutions ξi(t) of (5) tend to 0 as t→∞. Therefore, the equilibrium point p of the system is asymptotically stable if |arg(λ)|>απ/2, for all eigenvalues λ of J. For example,(6)mini|arg(λi)|>απ2.


## 2.2. Asymptotic Stability of the Incommensurate Fractional Ordered System

Consider the following incommensurate fractional ordered dynamical system [19, 20]. Now suppose the commensurate fractional ordered dynamical system equation
(7)Dαixi=fi(x1,x2,x3),1≤i≤3,
where 0<αi<1. One can write it as αi=vi/ui, (ui,vi)=1, so ui, vi are positive integers. The definition of common multiple of ui’s is M. Equilibrium point and small disturbance of the system p and ξi respectively as above. So we get
(8)Dαixi≈ξ1∂fi(p)∂x1+ξ2∂fi(p)∂x2+ξ3∂fi(p)∂x3,1≤i≤3,
where it can be written as
(9)(Dα1ξ1Dα2ξ2Dα3ξ3)=J(ξ1ξ2ξ3),
where J is Jacobian matrix evaluated at point p. The definition is(10)Δ(λ)=diag([λMα1λMα2λMα3])-J,
that is, if all the roots of equation Δ(λ)=0 satisfy the condition |arg(λ)|>π/2M [21], the solution of linear system is asymptotically stable as follows:
(11)π2M-mini|arg(λi)|<0.
The term in the left side of (11) is an instability measure for equilibrium point in fractional ordered system (IMFOS). Then, fractional order equation (7) exhibits chaotic attractor if the condition is [19, 20]
(12)IMFOS≥0.


## 2.3. Modified Liu System

In this section we introduce the following system and show that the system is chaotic:(13)x˙=-ax-by2,y˙=cy+dzx-ez2,z˙=fz+gxy,
where a=2, b=e=1,c=-3, d=-4, f=-7, and g=4 with the initial conditions (0.2,0,0.5) lead to the chaotic trajectories. Meanwhile, we want to show chaotic behavior of (13) involving fractional order. Also, we will calculate the minimum effective dimension by which the system remains chaotic. The corresponding fractional order system is
(14)Dα1=-ax-by2,Dα2=cy+dzx-ez2,Dα3=fz+gxy,
where αi∈(0,1). In (14) if we choose α1=α2=α3=α, the system is called commensurate, and otherwise it is incommensurate. Now, we have four real equilibrium points for (13) which are shown in Table 1. In Table 1, we see the equilibrium points and the eigenvalues of the corresponding Jacobian matrix
(15)J=(-a-2by0dzcdx-2ezgyxgxf).
A saddle point p is stable, if the Jacobian matrix has at least one eigenvalue with a negative real part. Otherwise, one eigenvalue with a nonnegative real part is called unstable. And saddle points have index one or two if there is exactly one or two unstable eigenvalues, respectively. It is established in the literature [22–26] that scrolls are generated only around the saddle points of an index one or two. Saddle points of index one are to connect scrolls. Table 1 shows that the equilibrium points E1 and E2 are saddle points of index two; therefor we have two-scroll attractor [22], in the fractional system given by (14).Table 1
Equilibrium points and corresponding eigenvalues.
Equilibrium point
Eigenvalue
Nature
E
0
(
0,0
,
0
)
(-7,3,-2)
Saddle point
E
1
(
-
24.392
,
-
6.98456
,
97.3529
)
(-64.4116,63.0432,-4.63163)
Saddle point
E
2
(
-
1.04307
,
1.44435
,
-
0.860895
)
(-6.694,0.347±5.1057i)
Saddle point
E
3
(
-
1.30636
,
-
1.61639
,
1.20662
)
(-6.82593,0.412964±4.75973i)
Saddle point


## 2.4. Commensurate Ordered System

Consider the system equation (14), and let α1=α2=α3=α, so it is called commensurate order for this case. In this case a system shows regular behavior if it satisfies mini|arg(λi)|>απ/2, then we have [15–21]
(16)α<2πmini|arg(λi)|≈0.95.
From Figure 1 we can see that the Lyapunov exponent for the case of commensurate ordered equation (14) is positive if α>0.95 [27, 28]. Thus we can realize that the system does not indicate chaotic behavior for the value α<0.95. This corollary has been figured out by numerical outcome. Moreover, Figure 2 illustrates the phase portrait in xy-plane for the α=0.95. Numerical experiments and Figures 3 and 4 demonstrate that the system has chaotic behavior for α=0.96. In addition, Figures 5 and 6 represent solutions x(t) and y(t) for α=0.96, respectively. Adams-Bashforth predictor-corrector algorithm is used for numerical result with step size 0.1.Figure 1
Lyapunov exponent.Figure 2
Phase portraitα=0.95.Figure 3
Phase portraitx-y  α=0.96.Figure 4
Phase portraity-zα=0.96.Figure 5
Solutionx(t) for α=0.96.Figure 6
Solutiony(t) for α=0.96.


## 2.5. Incommensurate Ordered System

In this section we indicate that the condition for being chaotic system in the case of commensurate is not sufficient for the incommensurate case. So let us consider the fractional order system equation (14). Figures 7 and 8 display that the Lyapunov exponent is positive for α1≥0.89, α2=α3=1, and for the case α3≥0.88,α1=α3=1, respectively. Now, let us consider the following cases for (14).Figure 7
Lyapunov exponent-α1.Figure 8
Lyapunov exponent-α3.(1) In the first condition we chooseα1=22/25,α2=α3=1. Therefore, one can acquire M = LCM(25,1,1)=25. Since we have Δ(λ)=diag([λ22λ25λ25])-J(E1), then
(17)det(Δ(λ))=-1.9×10-4-9.5×106-3λ22+5.45×10-3λ25+4λ47+2λ50+λ72.
And the instability measure for equilibrium point in fractional ordered system (IMFOS) will be for the system
(18)π50-0.0=0.0628>0.
In Figure 9, there is no chaos, whereas IMFOS >0 [15–21]. This consequence leads to that the IMFOS >0 is not sufficient for existence of chaos.Figure 9
Phase portraitx-yα1=0.88,α2=α3=1.(2) As second case, supposeα1=89/100,α2=α3=1, so that in this case the M = LCM(100,1,1)=100. Also we have Δ(λ)=diag([λ89λ100λ100])-J(E1), as well as we obtain
(19)det(Δ(λ))=-1.9×10-4-9.5×10-3λ89+5.5×10-3λ100+4λ189+2λ200+λ289.
Therefore, the IMFOS of the system for this state is
(20)π200-0.0=0.0157>0.
If we look at Figure 10 we conclude the chaotic behavior for the system with the mentioned condition. Here we remark the lowest dimension of the system which has chaos (Figure 11).Figure 10
Phase portraitx-zα1=0.89,α2=α3=1.Figure 11
Phase portraity-zα3=0.87,α2=α1=1.(3) As third case, letα3=43/50,α1=α3=1. So with some manipulation we get M = LCM(1,1,50)=50,
(21)det(Δ(λ))=-18807.7+5430.19λ43-9484.56λ50-λ93+7λ100+λ143.
If we compute the IMFOS for the system, we will arrive at
(22)π100-0.0=0.0314>0.
Figure 11 and (22) point that the system does not have chaotic condition.(4) For the fourth case, we considerα3=22/25,α1=α3=1. Therefore, we will obtain M = LCM(1,1,50)=50,
(23)det(Δ(λ))=-1.9×10-4-9.5×106-3λ22+5.45×10-3λ25+4λ47+2λ50+λ72.
IMFOS for the fourth case is
(24)π100-0.0=0.0628>0.
Numerical results and Figure 12 indicate for the this case that the system is chaotic.Figure 12
Phase portraity-zα3=0.88,α2=α1=1.


## 3. Conclusions

A fractional order of a new system is investigated. Numerical calculations are performed for different values of fractional order. Lyapunov exponents and analytical conditions given in the literature have been used to check the existence of chaos. A minimum effective dimension is calculated for commensurate fractional order. Mathematica 7 has been used for computations in this paper.



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*Source: 186037-2013-04-10.xml*