Chaos Generated by Switching Fractional Systems

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.

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. Section 2 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): _𝑎𝐷^𝛼_𝑡𝑓(𝑡)=lim_{ℎ→0𝑛ℎ=𝑡−𝑎}ℎ^−𝛼𝑛_𝑟=0(−1)^𝑟𝛼𝑟𝑓(𝑡−𝑟ℎ)(𝛼>0), (2.1) Riemann-Liouville fractional derivatives (R-L): _𝛼𝐷^𝛼_𝑡𝑓(𝑡)=1Γ()𝑑𝑚−𝛼𝑑𝑡^𝑚𝑚𝑡_𝑎(𝑡−𝜏)^{𝑚−𝛼−1}𝑓(𝜏)𝑑𝜏(𝑚−1≤𝛼<𝑚,𝛼>0), (2.2) and Caputo’s fractional derivative: ^𝐶_𝑎𝐷^𝛼_𝑡𝑓(𝑡)=1Γ(𝛼−𝑛)^𝑡_𝑎𝑓^(𝑛)(𝜏)𝑑𝜏(𝑡−𝜏)^{𝛼+1−𝑛}(𝑛−1≤𝛼<𝑛,𝛼>0). (2.3)

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 define 𝛾(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 of 𝛾(𝜀,𝜑) can be found in literature [36].

Lemma 2.1. If 𝛼<2, 𝜋𝛼/2<𝜇<min{𝜋,𝜋𝛼}and 𝜀>0 is arbitrary, for arbitrary complex 𝑧 the following expansion holds: 1Γ(𝑧)=12𝜋𝛼𝑖_{𝛾𝜀,𝜇}exp𝜍^1/𝛼𝜍^{(1−𝑧−𝛼)/𝛼}𝑑𝜍, (2.4) here Γ(𝑧) is Euler’s gamma function.

Lemma 2.2. If 𝛼>0, 𝛽>0, one obtains ^𝑧₀𝐸_𝛼,𝛽𝜆𝑡^𝛼𝑡^𝛽−1𝑑𝑡=𝑧^𝛽𝐸_𝛼,𝛽+1𝜆𝑧^𝛼. (2.5)

Theorem 2.3. If 0<𝛼<2, 𝛽 is an arbitrary complex number, and 𝜇 is an arbitrary real number such that 𝜋𝛼2<𝜇<min{𝜋,𝜋𝛼}, (2.6) then for an arbitrary integer 𝑝≥1 the following expansion holds: 𝐸_𝛼,𝛽(𝑧)=−^𝑝_𝑘=1𝑧^−𝑘Γ𝛽−𝛼𝑘+𝐼_𝑝(𝑧),|𝑧|⟶∞,𝜇≤arg(𝑧)≤𝜋, (2.7) where 𝐼_𝑝(𝑧)=12𝜋𝛼𝑖𝑧^𝑝_𝛾1,𝜙exp𝜍^1/𝛼𝜍^{(1−𝛽)/𝛼+𝑝}𝑑𝜍. (2.8)

Using Lemma 2.1 and Theorem 2.3, if 0<𝛼<2, |𝑧|→∞, 𝜋𝛼/2<|arg(𝑧)|≤𝜋,we can quickly obtain the following: 𝐸_𝛼,𝛼(𝑧)=𝑧⁻¹Γ(),𝐸−𝛼_𝛼,𝛼+1(𝑧)=1Γ()𝑧1−𝛼−1⁻¹, (2.9)(2.10) 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:₀𝐷^𝛼_𝑡𝑦(𝑡)−𝜆𝑦(𝑡)=ℎ(𝑡)(𝑡>0₀),𝐷_𝑡^𝛼−𝑘𝑦(𝑡)_𝑡=0=𝑏_𝑘(𝑘=1,2,…,𝑛),𝑛−1<𝛼<𝑛, (2.11) then we obtain the following solution:𝑦(𝑡)=^𝑛_𝑘=1𝑏_𝑘𝑡^𝛼−𝑘𝐸_{𝛼,𝛼−𝑘+1}𝜆𝑡^𝛼+^𝑡₀(𝑡−𝜏)^𝛼−1𝐸_𝛼,𝛼𝜆(𝑡−𝜏)^𝛼ℎ(𝜏)𝑑𝜏. (2.12)

3. Chaos Generation

3.1. Analysis of an Existing Chaos Generator

First, we discuss the chaos generated by the switching fractional system 𝑆₁ and 𝑆₂ with the switching function (3.2):𝑆₁∶𝐷^𝛼11𝑥₁=𝑎₁𝑥₁+𝑏₁𝑦₁,𝐷^𝛼12𝑦₁=−𝑏₁𝑥₁+𝑎₁𝑦₁,𝐷^𝛼13𝑧₁=−𝑐₁𝑧₁,𝑆₂∶𝐷^𝛼21𝑥₂=𝑎₂𝑥₂+𝑏₂𝑦₂,𝐷^𝛼22𝑦₂=−𝑏₂𝑥₂+𝑎₂𝑦₂,𝐷^𝛼23𝑧₂=−𝑐₂𝑧₂+𝑝, (3.1) where 0<𝛼₁₁,𝛼₁₂,𝛼₁₃,𝛼₂₁,𝛼₂₂,𝛼₂₃<1,𝑎₁,𝑎₂,𝑏₁,𝑏₂,𝑐₁,𝑐₂,𝑝≠0,𝑔𝑥,𝑦,𝑧=𝑥²+𝑦²+𝑧²−1. (3.2)

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: when 𝑆₁ 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 take 𝑆₂ 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} andΣ={(𝑥,𝑦,𝑧)∣𝑥²+𝑦²+𝑧²−1>0}, respectively. When the system is in Σ, 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 toΣ. We will discuss 𝑆₁ and 𝑆₂ in the following analytically, respectively.

We first rewrite 𝑆₂ as follows:𝑆₂∶𝐷^𝛼𝝂=𝐀𝝂+𝐔, (3.3) where 𝝂=𝑥₂,𝑦₂,𝑧₂,𝑎𝐀=²𝑏₂0−𝑏₂𝑎₂000−𝑐₂,𝐔=00000000𝑝. (3.4)

And we can obtain the eigenvalues: 𝜆_21,22=𝑎₂±𝑖𝑏₂,𝜆₂₃=−𝑐₂. Under the constricted conditions of the parameters, we can obtain𝛼_𝑖𝑗𝜋2<arg𝜆_𝑖𝑗𝑖=2,𝑗=1,2,3. (3.5)

Since the eigenvalues are different, we obtain a transformation matrix: ,𝐓𝐓=−𝑖𝑖0110001₋₁=112𝑖20−112𝑖20,001 (3.6) so that we obtain𝐷^𝛼𝝂^′=𝚲𝝂^′+𝐔, (3.7) where 𝝂^′=𝐓𝝂,𝚲=𝐓𝐀𝐓⁻¹=diag𝜆₂₁,𝜆₂₂,𝜆₂₃. (3.8)(3.9)

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 obtain𝑥^′₂=𝑏₂₁𝜆₂₁𝑡⁻¹Γ−𝛼₂₁, (3.10)𝑏₂₁=₀𝐷^𝛼21_𝑡⁻¹𝑥₂(𝑡)_𝑡=0, (3.11)𝑦^′₂=𝑏₂₂𝜆₂₂𝑡⁻¹Γ−𝛼₂₂, (3.12)𝑏₂₂=₀𝐷^𝛼22_𝑡⁻¹𝑦₂(𝑡)_𝑡=0, (3.13)𝑧^′₂=𝑏₂₃𝜆⁻¹₂₃𝑡⁻¹Γ−𝛼₂₃+𝑝^𝑡₀(𝑡−𝜏)^𝛼23−1𝐸_{𝛼23,𝛼23}𝜆₂₃(𝑡−𝜏)^𝛼23𝑑𝜏, (3.14)𝑏₂₃=₀𝐷^𝛼23_𝑡⁻¹𝑧₂(𝑡)_𝑡=0, (3.15)

By using Lemma 2.2, we obtain𝑧^′₂=𝑏₂₃𝜆⁻¹₂₃𝑡⁻¹Γ−𝛼₂₃+𝑝𝑡^𝛼23𝐸_{𝛼23,𝛼23+1}𝜆₂₃𝑡^𝛼23. (3.16)

Taking (2.10) into account, we obtain𝑧^′₂=𝑏₂₃𝜆⁻¹₂₃𝑡⁻¹Γ−𝛼₂₃+𝑝𝜆₂₃1Γ1−𝛼₂₃−1. (3.17)

Using (3.8), we can obtain𝑥²₂+𝑦²₂=12−𝑥₂^′2+𝑦₂^′2,𝑧₂=𝑧^′₂, (3.18)

taking (3.10) and (3.12) into account, we can obtain 𝑥²₂+𝑦²₂=12𝑡⁻²−1Γ²−𝛼₂₁𝑏²₂₁𝑎₂+𝑖𝑏₂²+1Γ²−𝛼₂₂𝑏²₂₂𝑎₂−𝑖𝑏₂². (3.19) when 𝑡⟶+∞,𝑥²₂+𝑦²₂⟶0,𝑧₂⟶𝑝𝜆₂₃1Γ1−𝛼₂₃−1. (3.20)

So we see that the trajectory of 𝑆₂ 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 to𝑝−𝑐₂1Γ1−𝛼₂₃−1<1. (3.21)

For 𝑆₁, taking 𝑐₁>0, we can obtain𝛼₁₃𝜋2<arg𝜆₁₃. (3.22)

Then we can quickly write 𝑧₁=𝑏₁₃𝜆⁻¹₁₃𝑡⁻¹Γ−𝛼₁₃, (3.23) when𝑡⟶+∞,𝑧₁⟶0 (3.24)

So we see that the trajectory of 𝑆₁ 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 parameters 𝑎₂ 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)}.

(a) 𝑏2=6

(b) 𝑏2=7

Figure 3 

Phase portraits of the switching fractional system (3.1).

(a) 𝑎2=0.5

(b) 𝑎2=0.6

Figure 4 

Phase portraits of the switching fractional system (3.1).

Then we focus on the parameters 𝑐₂ 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 𝑋𝑌𝑍^′=𝑝0,0,−𝑐₂1Γ1−𝛼₂₃−1 (3.25) faster. This conclusion is established in all conditions, which satisfy 𝑐₂>0.

(a) 𝑐2=1

(b) 𝑐2=2

Figure 5 

Phase portraits of the switching fractional system (3.1).

For 𝑆₁, 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)}.

(a) 𝑎1=1

(b) 𝑎1=3

Figure 6 

Phase portraits of the switching fractional system (3.1).

(a) 𝑏1=3

(b) 𝑏1=6

Figure 7 

Phase portraits of the switching fractional system (3.1).

Then we focus on the parameters 𝑐₁. 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 plane𝑋𝑌𝑍=(0,0,0) (3.26)

(a) 𝑐1=2

(b) 𝑐1=1

Figure 8 

Phase portraits of the switching fractional system (3.1).

faster. This conclusion is established in all conditions, which satisfy 𝑐₁>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 𝑥²₁+𝑦²₁+𝑧²₁−1 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 system 𝑆₁, 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: 𝑆₁∶𝐷^𝛼11𝑥₁=𝑎₁𝑥₁+𝑏₁𝑦₁,𝐷^𝛼12𝑦₁=−𝑏₁𝑥₁+𝑎₁𝑦₁,𝐷^𝛼13𝑧₁=−𝑐₁𝑧₁+𝑝₁,𝑆₂∶𝐷^𝛼21𝑥₂=𝑎₂𝑥₂+𝑏₂𝑦₂,𝐷^𝛼22𝑦₂=−𝑏₂𝑥₂+𝑎₂𝑦₂,𝐷^𝛼23𝑧₂=−𝑐₂𝑧₂,𝑆₃∶𝐷^𝛼31𝑥₃=𝑎₃𝑥₃+𝑏₃𝑦₃,𝐷^𝛼32𝑦₃=−𝑏₃𝑥₃+𝑎₃𝑦₃,𝐷^𝛼33𝑧₃=−𝑐₃𝑧₃+𝑝₂, (3.27) where 0<𝛼₁₁,𝛼₁₂,𝛼₁₃,𝛼₂₁,𝛼₂₂,𝛼₂₃,𝛼₃₁,𝛼₃₂,𝛼₃₃<1,𝑎₁,𝑎₂,𝑎₃,𝑏₁,𝑏₂,𝑏₃,𝑐₁,𝑐₂,𝑐₃,𝑝₁,𝑝₂≠0,𝑔₁𝑥,𝑦,𝑧=𝑥²+𝑦²+𝑧²−1,𝑔₂𝑥,𝑦=𝑥𝑦. (3.28)

We take the switching rule as follows: when 𝑆₂ 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 take 𝑆₁, 𝑆₃ to be asymptotically stable, while we take 𝑆₂ to be unstable. Then we determine the parameters:𝑏₁>𝑎₁𝛼maxtan¹¹𝜋2𝛼,tan¹²𝜋2,𝑐₁𝑝>0,¹−𝑐₁1Γ1−𝛼₁₃𝑏−1<1,₂<𝑎₂𝛼mintan²¹𝜋2𝛼,tan²²𝜋2,𝑐₂𝑏>0,₃>𝑎₃𝛼maxtan³¹𝜋2𝛼,tan³²𝜋2,𝑐₃𝑝>0,³−𝑐₃1Γ1−𝛼₃₃−1<1. (3.29)

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 𝛼_𝑖𝑗(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.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (nos. 61173183, 60973152, and 60573172), the Superior University Doctor Subject Special Scientific Research Foundation of China (no. 20070141014), and the Natural Science Foundation of Liaoning Province (no. 20082165).