# Chaos Generated by Switching Fractional Systems

Authors: Wang Xingyuan; Qin Xue
Journal: Mathematical Problems in Engineering
 (2012)

---

## Abstract

We, for the first time, investigate the basic behaviours of a chaotic switching fractional system via both theoretical and numerical ways. To deeply understand the mechanism of the chaos generation, we also analyse the parameterization of the switching fractional system and the dynamics of the system's trajectory. Then we try to write down some detailed rules for designing chaotic or chaos-like systems by switching fractional systems, which can be used in the future application. At last, for the first time, we proposed a new switching fractional system, which can generate three attractors with the positive largest Lyapunov exponent.


---

## Body

## 1. Introduction

Over the last two decades, since chaos has been demonstrated that it can be useful and well controlled [1–3], increasing interests have been found in many areas such as secure communication [4], data encryption [5], nonlinear optimization [6], synchronization [7], and biology [8] (for more areas, see [3]). Therefore, generating chaos, especially generating chaos via simple physical devices, has been paid extensive and massive attention [9–14]. Just like the n-scroll Chua’s circuit [15], switching piecewise-linear function can easily generate various chaos dynamic behaviours. Especially the literatures [16–18] and give great details to generate chaos via switching systems.On the other hand, fractional calculus is a mathematical branch which has more than 300 years of history but just been interested recently in physics [19], chemistry, biology, and engineering [20]. Since fractional-order calculus can be treated as an expanding concept of integer-order calculus, there are many chaos systems that are expanded from integer-order ones: the fractional-order Chua system [21, 22], the fractional-order Duffing system [23], the fractional jerk model [24], the fractional-order Lorenz system [25], the fractional-order Chen system [26], the fractional-order Lü system [27], the fractional-order Rössler system [28], the fractional-order Arneodo system [29], the fractional-order Newton-Leipnik system [30], and the fractional-order Genesio-Tesi system [31]. And also, the fractional-order Ikeda delay system [32], non-integer-order cellular neural networks [33], and the fractional-order systems proposed in [34, 35] all can generate chaos.Generating a new chaotic or chaos-like system is always on the core of the chaos research with great theoretical and applied meanings. In addition a new chaotic system usually gets more complex dynamic behaviours being less recognized by people who does not catch up with the nonlinear dynamics. Thus, a new chaos system may be popular used in chaotic secure communication and encryption. Not like lots of the existing switching systems literatures [15–18], there are few literatures about switching fractional systems, even generating a fractional chaos easily is an increasing topic of many applied and theoretical fields. Also, as far as we know, there have not been basic dynamical behaviours given to any chaotic or chaos-like switching fractional systems ever. A significant work of switching fractional systems is proposed by S. Mohammad and H. Mohammad in [36]. The paper discussed the switching rules and how to choose the parameters to generate chaos and the rule of designing switching function have been discussed. However, the detailed relationship between the parameters and the behaviours of the chaos has not been discussed, not even being part of some analysis of the basic behaviours of the systems. In the present paper, we try to research the relationship between the parameters of the system and the behaviours of the chaos in both analytic and numerical ways. And we even epitomized some more detailed rules of generating chaotic or chaos-like dynamic behaviours via switching fractional systems then Mohammad did. Under these rules, we can generate chaotic or chaos-like dynamic behaviours easily via simple switching fractional systems. At last, we propose a new chaotic or chaos-like switching fractional system and count out its Lyapunov exponent.The paper is organized as follows. Section2 introduces basic definitions and some theories and lemmas, which are useful in following sections. In Section 3, we analyse an existing chaos generator and do numerical simulations to research quantificationally the relationship between the parameter and the dynamic behaviours. And we propose a new switching fractional systems, which can generate chaotic or chaos-like dynamic behaviours. Finally Section 4 concludes the paper.


## 2. Background of Fractional Calculus

### 2.1. Basic Definitions

There exits three main definitions of fractional-order derivatives. They are Grünwald-Letnikov fractional derivatives (G-L):(2.1)Dtαaf(t)=lim⁡h→0nh=t-ah-α∑r=0n(-1)r(αr)f(t-rh)(α>0),
Riemann-Liouville fractional derivatives (R-L):
(2.2)Dtααf(t)=1Γ(m-α)dmdtm∫at(t-τ)m-α-1f(τ)dτ(m-1≤α<m,α>0),
and Caputo’s fractional derivative:
(2.3)DaCtαf(t)=1Γ(α-n)∫atf(n)(τ)dτ(t-τ)α+1-n(n-1≤α<n,α>0).And since they can transform to each other, our use of fractional derivatives can be free for all of these three definitions.


### 2.2. Some Theories and Lemmas

For the requirement in the next part, we list several theories and lemmas. All of them come from [37].We first defineγ(ε,φ)[36]. γ(ε,φ)(ε>0,0<φ<π) denotes the contour consisting of the following three parts:(i)argτ=φ,|τ|≥ε,(ii)-φ≤argτ≤φ,|τ|=ε,(iii)argτ=-φ,|τ|≥ε.More details ofγ(ε,φ) can be found in literature [36].Lemma 2.1.
Ifα<2, πα/2<μ<min{π,πα}and ε>0 is arbitrary, for arbitrary complex z the following expansion holds:
(2.4)1Γ(z)=12παi∫γ(ε,μ)exp⁡(ς1/α)ς(1-z-α)/αdς,
here Γ(z) is Euler’s gamma function.Lemma 2.2.
Ifα>0, β>0, one obtains
(2.5)∫0zEα,β(λtα)tβ-1dt=zβEα,β+1(λzα).Theorem 2.3.
If0<α<2, β is an arbitrary complex number, and μ is an arbitrary real number such that
(2.6)πα2<μ<min⁡{π,πα},
then for an arbitrary integer p≥1 the following expansion holds:
(2.7)Eα,β(z)=-∑k=1pz-kΓ(β-αk)+Ip(z),|z|⟶∞,μ≤|arg(z)|≤π,
where
(2.8)Ip(z)=12παizp∫γ(1,ϕ)exp⁡⁡(ς1/α)ς(1-β)/α+pdς.Using Lemma2.1 and Theorem 2.3, if 0<α<2, |z|→∞, πα/2<|arg(z)|≤π,we can quickly obtain the following: (2.9)Eα,α(z)=z-1Γ(-α),(2.10)Eα,α+1(z)=(1Γ(1-α)-1)z-1,
which will be used in the next section of our paper.Consider the following initial value problem for a nonhomogeneous fraction differential equation under nonzero initial conditions:(2.11)D0tαy(t)-λy(t)=h(t)(t>0),[D0tα-ky(t)]t=0=bk(k=1,2,…,n),n-1<α<n,
then we obtain the following solution:(2.12)y(t)=∑k=1nbktα-kEα,α-k+1(λtα)+∫0t(t-τ)α-1Eα,α(λ(t-τ)α)h(τ)dτ.


## 2.1. Basic Definitions

There exits three main definitions of fractional-order derivatives. They are Grünwald-Letnikov fractional derivatives (G-L):(2.1)Dtαaf(t)=lim⁡h→0nh=t-ah-α∑r=0n(-1)r(αr)f(t-rh)(α>0),
Riemann-Liouville fractional derivatives (R-L):
(2.2)Dtααf(t)=1Γ(m-α)dmdtm∫at(t-τ)m-α-1f(τ)dτ(m-1≤α<m,α>0),
and Caputo’s fractional derivative:
(2.3)DaCtαf(t)=1Γ(α-n)∫atf(n)(τ)dτ(t-τ)α+1-n(n-1≤α<n,α>0).And since they can transform to each other, our use of fractional derivatives can be free for all of these three definitions.


## 2.2. Some Theories and Lemmas

For the requirement in the next part, we list several theories and lemmas. All of them come from [37].We first defineγ(ε,φ)[36]. γ(ε,φ)(ε>0,0<φ<π) denotes the contour consisting of the following three parts:(i)argτ=φ,|τ|≥ε,(ii)-φ≤argτ≤φ,|τ|=ε,(iii)argτ=-φ,|τ|≥ε.More details ofγ(ε,φ) can be found in literature [36].Lemma 2.1.
Ifα<2, πα/2<μ<min{π,πα}and ε>0 is arbitrary, for arbitrary complex z the following expansion holds:
(2.4)1Γ(z)=12παi∫γ(ε,μ)exp⁡(ς1/α)ς(1-z-α)/αdς,
here Γ(z) is Euler’s gamma function.Lemma 2.2.
Ifα>0, β>0, one obtains
(2.5)∫0zEα,β(λtα)tβ-1dt=zβEα,β+1(λzα).Theorem 2.3.
If0<α<2, β is an arbitrary complex number, and μ is an arbitrary real number such that
(2.6)πα2<μ<min⁡{π,πα},
then for an arbitrary integer p≥1 the following expansion holds:
(2.7)Eα,β(z)=-∑k=1pz-kΓ(β-αk)+Ip(z),|z|⟶∞,μ≤|arg(z)|≤π,
where
(2.8)Ip(z)=12παizp∫γ(1,ϕ)exp⁡⁡(ς1/α)ς(1-β)/α+pdς.Using Lemma2.1 and Theorem 2.3, if 0<α<2, |z|→∞, πα/2<|arg(z)|≤π,we can quickly obtain the following: (2.9)Eα,α(z)=z-1Γ(-α),(2.10)Eα,α+1(z)=(1Γ(1-α)-1)z-1,
which will be used in the next section of our paper.Consider the following initial value problem for a nonhomogeneous fraction differential equation under nonzero initial conditions:(2.11)D0tαy(t)-λy(t)=h(t)(t>0),[D0tα-ky(t)]t=0=bk(k=1,2,…,n),n-1<α<n,
then we obtain the following solution:(2.12)y(t)=∑k=1nbktα-kEα,α-k+1(λtα)+∫0t(t-τ)α-1Eα,α(λ(t-τ)α)h(τ)dτ.


## 3. Chaos Generation

### 3.1. Analysis of an Existing Chaos Generator

First, we discuss the chaos generated by the switching fractional systemS1 and S2 with the switching function (3.2):(3.1)S1:{Dα11x1=a1x1+b1y1,Dα12y1=-b1x1+a1y1,Dα13z1=-c1z1,S2:{Dα21x2=a2x2+b2y2,Dα22y2=-b2x2+a2y2,Dα23z2=-c2z2+p,
where 0<α11,α12,α13,α21,α22,α23<1,a1,a2,b1,b2,c1,c2,p≠0,(3.2)g(x,y,z)=x2+y2+z2-1.This kind of chaos was proposed by the literature [36]. That paper has discussed how to design the switching rule to generate chaos, how to choose the parameters to generate chaos, and how to design the switching function to generate chaos. Here we want to discuss the dynamic behaviours of the chaos more carefully and research the relationship between the parameters of the system and the dynamic behaviours of the chaos more carefully and quantificationally.Here we design the switching rule as follows: whenS1 is active, the system will switch to S2 at the time g(x1(t),y1(t),z1(t))≥0 with the initial condition of S2 being (x1(t),y1(t),z1(t)). Similarly, when S2 is active, the system will switch to S1 at the time g(x2(t),y2(t),z2(t))<0with the initial condition of S1 being (x2(t),y2(t),z2(t)).And we takeS2 to be asymptotically stable and take S1 to be unstable. Then we get the restricted conditions of the parameters: c2>0, |b2|>a2max{tan(α21π/2),tan(α22π/2)}, |p/c2|<1, and c1<0 or |b1|<a1tan(α11π/2) or |b1|<a1tan(α12π/2), only one of the establishments of the last three conditions is enough. Here we choose |b1|<a1min{tan(α11π/2),tan(α12π/2)} and make c1>0, so that, under the discussion in paper [36], the switching fractional system will perform chaos.Then we want to discuss the dynamic behaviours of the chaos more carefully. The switching function (3.2) and the switching rules divide the whole space into two regions, which we denote Σ={(x,y,z)∣x2+y2+z2-1≤0} andΣ̅={(x,y,z)∣x2+y2+z2-1>0}, respectively. When the system is in Σ̅, S2 is active. Since S2 is asymptotically stable, S2 will converge to its fixed point, which will be discussed in the following. Either the system orbits reach or does not reach the fixed point, because g(0,0,p/c2)<0, as we take above, the system orbits will go through the plane x2+y2+z2=1 and then switch to Σ. When the system is in Σ, S1 is active. Since S1 is unstable, the system orbits will diverge and go through the plane x2+y2+z2=1 and then switch toΣ̅. We will discuss S1 and S2 in the following analytically, respectively.We first rewriteS2 as follows:(3.3)S2:Dαν=Aν+U,
where
(3.4)ν=(x2,y2,z2),A=[a2b20-b2a2000-c2],U=[00000000p].And we can obtain the eigenvalues:λ21,22=a2±ib2,λ23=-c2. Under the constricted conditions of the parameters, we can obtain(3.5)αijπ2<|arg⁡(λij)|(i=2,j=1,2,3).Since the eigenvalues are different, we obtain a transformation matrix:(3.6)T=[-ii0110001],T-1=[12i120-12i120001],
so that we obtain(3.7)Dαν′=Λν′+U,
where
(3.8)ν′=Tν,(3.9)Λ=TAT-1=diag⁡(λ21,λ22,λ23).Because of (3.5), we can solve the transformed fractional differential equations (3.7) in the solution (2.12) with (2.9), so that we obtain(3.10)x2′=b21λ21t-1Γ(-α21),(3.11)b21=[Dtα21-10x2(t)]t=0,(3.12)y2′=b22λ22t-1Γ(-α22),(3.13)b22=[Dtα22-10y2(t)]t=0,(3.14)z2′=b23λ23-1t-1Γ(-α23)+p∫0t(t-τ)α23-1Eα23,α23(λ23(t-τ)α23)dτ,(3.15)b23=[Dtα23-10z2(t)]t=0,By using Lemma2.2, we obtain(3.16)z2′=b23λ23-1t-1Γ(-α23)+ptα23Eα23,α23+1(λ23tα23).Taking (2.10) into account, we obtain(3.17)z2′=b23λ23-1t-1Γ(-α23)+pλ23(1Γ(1-α23)-1).Using (3.8), we can obtain(3.18)x22+y22=12(-x2′2+y2′2),z2=z2′,taking (3.10) and (3.12) into account, we can obtain (3.19)x22+y22=12t-2(-1Γ2(-α21)b212(a2+ib2)2+1Γ2(-α22)b222(a2-ib2)2).
when
(3.20)t⟶+∞,x22+y22⟶0,z2⟶pλ23(1Γ(1-α23)-1).So we see that the trajectory ofS2 is a spiral line. And (0,0,(p/λ23)(1/Γ(1-α23)-1)) is the fixed point of S2. And we can slightly change the restricted condition of the parameters |p/c2|<1 to(3.21)|p-c2(1Γ(1-α23)-1)|<1.ForS1, taking c1>0, we can obtain(3.22)α13π2<|arg(λ13)|.Then we can quickly write(3.23)z1=b13λ13-1t-1Γ(-α13),
when(3.24)t⟶+∞,z1⟶0So we see that the trajectory ofS1 will fall into the plane z=0. Forx1,y1, because of their complexity, we do not give their analytic solutions.


### 3.2. Numerical Simulations

Now we can do some numerical simulations and research in quantities of relationship between the parameters and the dynamic behaviours of the chaos. We take all the parameters under the restricted conditions we proposed above, so that the switching fractional system (3.1) will show chaos behaviours. Below in this part, we take α11=α12=α13=α21=α22=α23=0.9. Taking the parameters: a1=1, b1=2, c1=5, a2=0.7, b2=6, c2=1, p=1, the switching fractional system (3.1) has a chaotic attractor, which is shown in Figure 1. The maximum Lyapunov exponent of this attractor is LE=0.0155. And Figure 2 shows the directions of the trajectory of the switching fractional system (3.1) under the parameters: a1=1, b1=2, c1=5, a2=0.6, b2=5, c2=1, p=1. The trajectory is denoted by the arrows.Figure 1
The chaotic attractor generated by the switching fractional system (3.1).Figure 2
The trajectory of the switching fractional system (3.1).We first focus on the parametersa2 and b2. In this segment, we fix the parameters: a1=1, b1=2, c1=5, c2=1, p=1. We first take the parameter a2=0.7 and increase b2 from 6 to 7 with step 1. Figure 3 shows the phase portraits of the switching fractional system (3.1) under these parameters. From Figure 3, we can conclude that the increase of b2 makes S2 converge faster. This conclusion is established in all conditions, which satisfy a2∈(-∞,+∞) and |b2|>a2max{tan(α21π/2),tan(α22π/2)}. Then we take the parameter b2=6 and increase a2 from 0.5 to 0.6 with step 0.1. Figure 4 shows the phase portraits of the switching fractional system (3.1) under these parameters. From Figure 4, we can conclude that the increase of a2 makes S2 converge slower. This conclusion is established in all conditions, which satisfy a2∈(-∞,+∞) and |b2|>a2max{tan(α21π/2),tan(α22π/2)}.Phase portraits of the switching fractional system (3.1).
(a)
b2=6(b)
b2=7Phase portraits of the switching fractional system (3.1).
(a)
a2=0.5(b)
a2=0.6Then we focus on the parametersc2 and p. In this segment, we fix the parameters: a2=0.7, b2=6, a1=1, b1=2, c1=5. We increase c2 and p simultaneously from 1 to 2 with step 1. Figure 5 shows the phase portraits of the switching fractional system (3.1) under these parameters. From Figure 5, we can conclude that the increase of c2 makes S2 reach the plane (3.25)XYZ′=(0,0,p-c2(1Γ(1-α23)-1))
faster. This conclusion is established in all conditions, which satisfy c2>0.Phase portraits of the switching fractional system (3.1).
(a)
c2=1(b)
c2=2ForS1, we first focus on the parameters a1 and b1. In this segment, we fix the parameters: c1=4, a2=0.7, b2=6, c2=1, p=1. We first take the parameter b1=6 and increase a1 from 1 to 3 with step 2. Figure 6 shows the phase portraits of the switching fractional system (3.1) under these parameters. From Figure 6, we can conclude that the increase of a1 makes S1 diverge faster. This conclusion is established in all conditions, which satisfy a1∈(-∞,+∞) and |b1|<a1min{tan(α11π/2),tan(α12π/2)}.Then we take the parameter a1=1 and increase b1 from 3 to 6 with step 3. Figure 7 shows the phase portraits of the switching fractional system (3.1) under these parameters. From Figure 7, we can conclude that the increase of b1 makes S1 diverge slower. This conclusion is established in all conditions, which satisfy a1∈(-∞,+∞) and |b1|<a1min{tan(α11π/2),tan(α12π/2)}.Phase portraits of the switching fractional system (3.1).
(a)
a1=1(b)
a1=3Phase portraits of the switching fractional system (3.1).
(a)
b1=3(b)
b1=6Then we focus on the parametersc1. In this segment, we fix the parameters: a1=1, b1=2, a2=0.7, b2=6, c2=1, p=1. We increase c1 from 1 to 2 with step 1. Figure 8 shows the phase portraits of the switching fractional system (3.1) under these parameters. From Figure 8, we can conclude that the increase of c1 makes S1 reach the plane(3.26)XYZ=(0,0,0)Phase portraits of the switching fractional system (3.1).
(a)
c1=2(b)
c1=1faster. This conclusion is established in all conditions, which satisfyc1>0.


### 3.3. More Detailed Rules to Design Chaotic or Chaos-Like Fractional Switching Systems

Here we epitomize some more detailed rules of relationship of the dynamic behaviours of the switching fractional systems and the parameters of the switching fractional systems. Take the switching fractional system (3.1) with the parameters under the restricted conditions of the parameters, which we proposed previously, for example. There are two fixed points of the switching fractional system (3.1). The parameters c1, c2, and p control the speed of the switching fractional system to reach the fixed points, respectively. The parameters a1 and b1 control the speed of S1 to diverge. The parameters a2 and b2 control the speed of S2 to converge. Taking the experiments, we did above, into account, we conclude that if we want the switching fractional system to arrive at the fixed points faster, we can increase the parameters c1, c2, and p. However, c1, c2, and p should also not be so large that the orbits of the switching fractional system will cross through the plane x2+y2+z2=1, because the sum x12+y12+z12-1 will be bigger or smaller than 0, respectively; if we want S1to diverge faster, we can increase the parameter a1 or decrease the parameter b1; if we want S2to converge faster, we can increase the parameter b2 or decrease the parameter a2.So the rules are that we should balance all the parameters when we apply the rules obtained in the literature [36]. If we find that S2 converges too fast, we may decrease the value of either b2 or c2 or increase a2; it is also very similar to S1. But remember if either S1 or S2 converges or diverges so fast that it cannot reach its own fixed point plane, the pattern we get may not show two attractors. Even if we obey these rules, we can design a chaotic or chaos-like switching fractional system with a wide range of the parameter values.


### 3.4. A New Chaotic or Chaos-Like Switching Fractional System

Under the rules above, we here propose a new chaotic or chaos-like switching fractional systemS1, S2, and S3 with the switching function (3.2). Actually, under the rules above, lots of chaotic or chaos-like switching fractional systems can also be proposed: (3.27)S1:{Dα11x1=a1x1+b1y1,Dα12y1=-b1x1+a1y1,Dα13z1=-c1z1+p1,S2:{Dα21x2=a2x2+b2y2,Dα22y2=-b2x2+a2y2,Dα23z2=-c2z2,S3:{Dα31x3=a3x3+b3y3,Dα32y3=-b3x3+a3y3,Dα33z3=-c3z3+p2,
where 0<α11,α12,α13,α21,α22,α23,α31,α32,α33<1,a1,a2,a3,b1,b2,b3,c1,c2,c3,p1,p2≠0,(3.28)g1(x,y,z)=x2+y2+z2-1,g2(x,y)=xy.We take the switching rule as follows: whenS2 is active, the system will switch to S1 at the time g1(x2(t),y2(t),z2(t))≥0 and g2(x2(t),y2(t))≥0 with the initial condition of S1being (x2(t),y2(t),z2(t)) and the system will switch to S3 at the time g1(x2(t),y2(t),z2(t))≥0 and g2(x2(t),y2(t))<0 with the initial condition of S1 being (x2(t),y2(t),z2(t)). When S1 is active, the system will switch to S2 at the time g1(x1(t),y1(t),z1(t))<0with the initial condition of S2 being (x1(t),y1(t),z1(t)). When S3 is active, the system will switch to S2 at the time g1(x3(t),y3(t),z3(t))<0 with the initial condition of S2 being (x3(t),y3(t),z3(t)).We takeS1, S3 to be asymptotically stable, while we take S2 to be unstable. Then we determine the parameters:(3.29)|b1|>a1max⁡{tan(α11π2),tan(α12π2)},c1>0,|p1-c1(1Γ(1-α13)-1)|<1,|b2|<a2min⁡{tan(α21π2),tan(α22π2)},c2>0,|b3|>a3max⁡{tan(α31π2),tan(α32π2)},c3>0,|p3-c3(1Γ(1-α33)-1)|<1.We take the parameters:a1=0.6, b1=9, c1=5, p1=5, a2=1, b2=2, c2=5, a3=0.6, b3=9, c3=5, p2=-5, α11=α12=α13=α21=α22=α23=α31=α32=α33=0.9. The switching fractional system (3.27), has a chaotic attractor, which is shown in Figure 9. The maximum Lyapunov exponent of this attractor is LE=0.0434.Figure 9
The chaotic attractor generated by the switching fractional system (3.27).We take the parameters:a1=0.6, b1=9, c1=5,p1=5, a2=1, b2=2, c2=5, a3=0.6, b3=9, c3=5, p2=-5, α11=α12=α21=α22=α31=α32=0.9,α13=α23=α33=0.5.The switching fractional system (3.27), has a chaotic attractor, which is shown in Figure 10. The maximum Lyapunov exponent of this attractor isLE=0.0255. From Figure 10, we can obtain that the switching fractional system (3.27), under variant parameters αij(i=1,2,3,j=1,2,3) can also perform chaotic or chaos-like dynamic behaviours.Figure 10
The chaotic attractor generated by the switching fractional system (3.27).


## 3.1. Analysis of an Existing Chaos Generator

First, we discuss the chaos generated by the switching fractional systemS1 and S2 with the switching function (3.2):(3.1)S1:{Dα11x1=a1x1+b1y1,Dα12y1=-b1x1+a1y1,Dα13z1=-c1z1,S2:{Dα21x2=a2x2+b2y2,Dα22y2=-b2x2+a2y2,Dα23z2=-c2z2+p,
where 0<α11,α12,α13,α21,α22,α23<1,a1,a2,b1,b2,c1,c2,p≠0,(3.2)g(x,y,z)=x2+y2+z2-1.This kind of chaos was proposed by the literature [36]. That paper has discussed how to design the switching rule to generate chaos, how to choose the parameters to generate chaos, and how to design the switching function to generate chaos. Here we want to discuss the dynamic behaviours of the chaos more carefully and research the relationship between the parameters of the system and the dynamic behaviours of the chaos more carefully and quantificationally.Here we design the switching rule as follows: whenS1 is active, the system will switch to S2 at the time g(x1(t),y1(t),z1(t))≥0 with the initial condition of S2 being (x1(t),y1(t),z1(t)). Similarly, when S2 is active, the system will switch to S1 at the time g(x2(t),y2(t),z2(t))<0with the initial condition of S1 being (x2(t),y2(t),z2(t)).And we takeS2 to be asymptotically stable and take S1 to be unstable. Then we get the restricted conditions of the parameters: c2>0, |b2|>a2max{tan(α21π/2),tan(α22π/2)}, |p/c2|<1, and c1<0 or |b1|<a1tan(α11π/2) or |b1|<a1tan(α12π/2), only one of the establishments of the last three conditions is enough. Here we choose |b1|<a1min{tan(α11π/2),tan(α12π/2)} and make c1>0, so that, under the discussion in paper [36], the switching fractional system will perform chaos.Then we want to discuss the dynamic behaviours of the chaos more carefully. The switching function (3.2) and the switching rules divide the whole space into two regions, which we denote Σ={(x,y,z)∣x2+y2+z2-1≤0} andΣ̅={(x,y,z)∣x2+y2+z2-1>0}, respectively. When the system is in Σ̅, S2 is active. Since S2 is asymptotically stable, S2 will converge to its fixed point, which will be discussed in the following. Either the system orbits reach or does not reach the fixed point, because g(0,0,p/c2)<0, as we take above, the system orbits will go through the plane x2+y2+z2=1 and then switch to Σ. When the system is in Σ, S1 is active. Since S1 is unstable, the system orbits will diverge and go through the plane x2+y2+z2=1 and then switch toΣ̅. We will discuss S1 and S2 in the following analytically, respectively.We first rewriteS2 as follows:(3.3)S2:Dαν=Aν+U,
where
(3.4)ν=(x2,y2,z2),A=[a2b20-b2a2000-c2],U=[00000000p].And we can obtain the eigenvalues:λ21,22=a2±ib2,λ23=-c2. Under the constricted conditions of the parameters, we can obtain(3.5)αijπ2<|arg⁡(λij)|(i=2,j=1,2,3).Since the eigenvalues are different, we obtain a transformation matrix:(3.6)T=[-ii0110001],T-1=[12i120-12i120001],
so that we obtain(3.7)Dαν′=Λν′+U,
where
(3.8)ν′=Tν,(3.9)Λ=TAT-1=diag⁡(λ21,λ22,λ23).Because of (3.5), we can solve the transformed fractional differential equations (3.7) in the solution (2.12) with (2.9), so that we obtain(3.10)x2′=b21λ21t-1Γ(-α21),(3.11)b21=[Dtα21-10x2(t)]t=0,(3.12)y2′=b22λ22t-1Γ(-α22),(3.13)b22=[Dtα22-10y2(t)]t=0,(3.14)z2′=b23λ23-1t-1Γ(-α23)+p∫0t(t-τ)α23-1Eα23,α23(λ23(t-τ)α23)dτ,(3.15)b23=[Dtα23-10z2(t)]t=0,By using Lemma2.2, we obtain(3.16)z2′=b23λ23-1t-1Γ(-α23)+ptα23Eα23,α23+1(λ23tα23).Taking (2.10) into account, we obtain(3.17)z2′=b23λ23-1t-1Γ(-α23)+pλ23(1Γ(1-α23)-1).Using (3.8), we can obtain(3.18)x22+y22=12(-x2′2+y2′2),z2=z2′,taking (3.10) and (3.12) into account, we can obtain (3.19)x22+y22=12t-2(-1Γ2(-α21)b212(a2+ib2)2+1Γ2(-α22)b222(a2-ib2)2).
when
(3.20)t⟶+∞,x22+y22⟶0,z2⟶pλ23(1Γ(1-α23)-1).So we see that the trajectory ofS2 is a spiral line. And (0,0,(p/λ23)(1/Γ(1-α23)-1)) is the fixed point of S2. And we can slightly change the restricted condition of the parameters |p/c2|<1 to(3.21)|p-c2(1Γ(1-α23)-1)|<1.ForS1, taking c1>0, we can obtain(3.22)α13π2<|arg(λ13)|.Then we can quickly write(3.23)z1=b13λ13-1t-1Γ(-α13),
when(3.24)t⟶+∞,z1⟶0So we see that the trajectory ofS1 will fall into the plane z=0. Forx1,y1, because of their complexity, we do not give their analytic solutions.


## 3.2. Numerical Simulations

Now we can do some numerical simulations and research in quantities of relationship between the parameters and the dynamic behaviours of the chaos. We take all the parameters under the restricted conditions we proposed above, so that the switching fractional system (3.1) will show chaos behaviours. Below in this part, we take α11=α12=α13=α21=α22=α23=0.9. Taking the parameters: a1=1, b1=2, c1=5, a2=0.7, b2=6, c2=1, p=1, the switching fractional system (3.1) has a chaotic attractor, which is shown in Figure 1. The maximum Lyapunov exponent of this attractor is LE=0.0155. And Figure 2 shows the directions of the trajectory of the switching fractional system (3.1) under the parameters: a1=1, b1=2, c1=5, a2=0.6, b2=5, c2=1, p=1. The trajectory is denoted by the arrows.Figure 1
The chaotic attractor generated by the switching fractional system (3.1).Figure 2
The trajectory of the switching fractional system (3.1).We first focus on the parametersa2 and b2. In this segment, we fix the parameters: a1=1, b1=2, c1=5, c2=1, p=1. We first take the parameter a2=0.7 and increase b2 from 6 to 7 with step 1. Figure 3 shows the phase portraits of the switching fractional system (3.1) under these parameters. From Figure 3, we can conclude that the increase of b2 makes S2 converge faster. This conclusion is established in all conditions, which satisfy a2∈(-∞,+∞) and |b2|>a2max{tan(α21π/2),tan(α22π/2)}. Then we take the parameter b2=6 and increase a2 from 0.5 to 0.6 with step 0.1. Figure 4 shows the phase portraits of the switching fractional system (3.1) under these parameters. From Figure 4, we can conclude that the increase of a2 makes S2 converge slower. This conclusion is established in all conditions, which satisfy a2∈(-∞,+∞) and |b2|>a2max{tan(α21π/2),tan(α22π/2)}.Phase portraits of the switching fractional system (3.1).
(a)
b2=6(b)
b2=7Phase portraits of the switching fractional system (3.1).
(a)
a2=0.5(b)
a2=0.6Then we focus on the parametersc2 and p. In this segment, we fix the parameters: a2=0.7, b2=6, a1=1, b1=2, c1=5. We increase c2 and p simultaneously from 1 to 2 with step 1. Figure 5 shows the phase portraits of the switching fractional system (3.1) under these parameters. From Figure 5, we can conclude that the increase of c2 makes S2 reach the plane (3.25)XYZ′=(0,0,p-c2(1Γ(1-α23)-1))
faster. This conclusion is established in all conditions, which satisfy c2>0.Phase portraits of the switching fractional system (3.1).
(a)
c2=1(b)
c2=2ForS1, we first focus on the parameters a1 and b1. In this segment, we fix the parameters: c1=4, a2=0.7, b2=6, c2=1, p=1. We first take the parameter b1=6 and increase a1 from 1 to 3 with step 2. Figure 6 shows the phase portraits of the switching fractional system (3.1) under these parameters. From Figure 6, we can conclude that the increase of a1 makes S1 diverge faster. This conclusion is established in all conditions, which satisfy a1∈(-∞,+∞) and |b1|<a1min{tan(α11π/2),tan(α12π/2)}.Then we take the parameter a1=1 and increase b1 from 3 to 6 with step 3. Figure 7 shows the phase portraits of the switching fractional system (3.1) under these parameters. From Figure 7, we can conclude that the increase of b1 makes S1 diverge slower. This conclusion is established in all conditions, which satisfy a1∈(-∞,+∞) and |b1|<a1min{tan(α11π/2),tan(α12π/2)}.Phase portraits of the switching fractional system (3.1).
(a)
a1=1(b)
a1=3Phase portraits of the switching fractional system (3.1).
(a)
b1=3(b)
b1=6Then we focus on the parametersc1. In this segment, we fix the parameters: a1=1, b1=2, a2=0.7, b2=6, c2=1, p=1. We increase c1 from 1 to 2 with step 1. Figure 8 shows the phase portraits of the switching fractional system (3.1) under these parameters. From Figure 8, we can conclude that the increase of c1 makes S1 reach the plane(3.26)XYZ=(0,0,0)Phase portraits of the switching fractional system (3.1).
(a)
c1=2(b)
c1=1faster. This conclusion is established in all conditions, which satisfyc1>0.


## 3.3. More Detailed Rules to Design Chaotic or Chaos-Like Fractional Switching Systems

Here we epitomize some more detailed rules of relationship of the dynamic behaviours of the switching fractional systems and the parameters of the switching fractional systems. Take the switching fractional system (3.1) with the parameters under the restricted conditions of the parameters, which we proposed previously, for example. There are two fixed points of the switching fractional system (3.1). The parameters c1, c2, and p control the speed of the switching fractional system to reach the fixed points, respectively. The parameters a1 and b1 control the speed of S1 to diverge. The parameters a2 and b2 control the speed of S2 to converge. Taking the experiments, we did above, into account, we conclude that if we want the switching fractional system to arrive at the fixed points faster, we can increase the parameters c1, c2, and p. However, c1, c2, and p should also not be so large that the orbits of the switching fractional system will cross through the plane x2+y2+z2=1, because the sum x12+y12+z12-1 will be bigger or smaller than 0, respectively; if we want S1to diverge faster, we can increase the parameter a1 or decrease the parameter b1; if we want S2to converge faster, we can increase the parameter b2 or decrease the parameter a2.So the rules are that we should balance all the parameters when we apply the rules obtained in the literature [36]. If we find that S2 converges too fast, we may decrease the value of either b2 or c2 or increase a2; it is also very similar to S1. But remember if either S1 or S2 converges or diverges so fast that it cannot reach its own fixed point plane, the pattern we get may not show two attractors. Even if we obey these rules, we can design a chaotic or chaos-like switching fractional system with a wide range of the parameter values.


## 3.4. A New Chaotic or Chaos-Like Switching Fractional System

Under the rules above, we here propose a new chaotic or chaos-like switching fractional systemS1, S2, and S3 with the switching function (3.2). Actually, under the rules above, lots of chaotic or chaos-like switching fractional systems can also be proposed: (3.27)S1:{Dα11x1=a1x1+b1y1,Dα12y1=-b1x1+a1y1,Dα13z1=-c1z1+p1,S2:{Dα21x2=a2x2+b2y2,Dα22y2=-b2x2+a2y2,Dα23z2=-c2z2,S3:{Dα31x3=a3x3+b3y3,Dα32y3=-b3x3+a3y3,Dα33z3=-c3z3+p2,
where 0<α11,α12,α13,α21,α22,α23,α31,α32,α33<1,a1,a2,a3,b1,b2,b3,c1,c2,c3,p1,p2≠0,(3.28)g1(x,y,z)=x2+y2+z2-1,g2(x,y)=xy.We take the switching rule as follows: whenS2 is active, the system will switch to S1 at the time g1(x2(t),y2(t),z2(t))≥0 and g2(x2(t),y2(t))≥0 with the initial condition of S1being (x2(t),y2(t),z2(t)) and the system will switch to S3 at the time g1(x2(t),y2(t),z2(t))≥0 and g2(x2(t),y2(t))<0 with the initial condition of S1 being (x2(t),y2(t),z2(t)). When S1 is active, the system will switch to S2 at the time g1(x1(t),y1(t),z1(t))<0with the initial condition of S2 being (x1(t),y1(t),z1(t)). When S3 is active, the system will switch to S2 at the time g1(x3(t),y3(t),z3(t))<0 with the initial condition of S2 being (x3(t),y3(t),z3(t)).We takeS1, S3 to be asymptotically stable, while we take S2 to be unstable. Then we determine the parameters:(3.29)|b1|>a1max⁡{tan(α11π2),tan(α12π2)},c1>0,|p1-c1(1Γ(1-α13)-1)|<1,|b2|<a2min⁡{tan(α21π2),tan(α22π2)},c2>0,|b3|>a3max⁡{tan(α31π2),tan(α32π2)},c3>0,|p3-c3(1Γ(1-α33)-1)|<1.We take the parameters:a1=0.6, b1=9, c1=5, p1=5, a2=1, b2=2, c2=5, a3=0.6, b3=9, c3=5, p2=-5, α11=α12=α13=α21=α22=α23=α31=α32=α33=0.9. The switching fractional system (3.27), has a chaotic attractor, which is shown in Figure 9. The maximum Lyapunov exponent of this attractor is LE=0.0434.Figure 9
The chaotic attractor generated by the switching fractional system (3.27).We take the parameters:a1=0.6, b1=9, c1=5,p1=5, a2=1, b2=2, c2=5, a3=0.6, b3=9, c3=5, p2=-5, α11=α12=α21=α22=α31=α32=0.9,α13=α23=α33=0.5.The switching fractional system (3.27), has a chaotic attractor, which is shown in Figure 10. The maximum Lyapunov exponent of this attractor isLE=0.0255. From Figure 10, we can obtain that the switching fractional system (3.27), under variant parameters αij(i=1,2,3,j=1,2,3) can also perform chaotic or chaos-like dynamic behaviours.Figure 10
The chaotic attractor generated by the switching fractional system (3.27).


## 4. Conclusions

In this paper, we for the first time study the basic behaviours of a chaotic switching fractional system both theoretically and numerically. We also analyse the parameterization of the switching fractional system and the dynamics of the system’s trajectory. Then we try to write down some more detailed rules for designing chaotic or chaos-like systems by switching fractional systems based on the basic rules given in the literature [36]. At last, for the first time, we proposed a new switching fractional system, which can generate three attractors with the positive largest Lyapunov exponent.



---

*Source: 601309-2012-01-05.xml*