# A New Series of Three-Dimensional Chaotic Systems with Cross-Product Nonlinearities and Their Switching

Authors: Xinquan Zhao; Feng Jiang; Zhigang Zhang; Junhao Hu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper introduces a new series of three-dimensional chaotic
systems with cross-product nonlinearities. Based on some conditions,
we analyze the globally exponentially or globally conditional
exponentially attractive set and positive invariant set of these
chaotic systems. Moreover, we give some known examples to show our
results, and the exponential estimation is explicitly derived.
Finally, we construct some three-dimensional chaotic systems with
cross-product nonlinearities and study the switching system between
them.


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## Body

## 1. Introduction

Since Lorenz discovered the well-known Lorenz chaotic system, many other chaotic systems have been found, including the well-known Rössler system and Chua’s circuit, which  serve as models of the study of chaos [1–12].The Lorenz system plays an important role in the study of nonlinear science and chaotic dynamics [13–18]. We know that it is extremely difficult to obtain the information of chaotic attractor directly from system. Most of the results in the literature are based on computer simulations. When calculating the Lyapunov exponents of the system, one needs to assume that the system is bounded in order to conclude chaos. Therefore, the study of the globally attractive set of the Lorenz system is not only theoretically significant but also practically important. Moreover, Liao et al. [19, 20] gave globally exponentially attractive set and positive invariant set for the classical Lorenz system and the generalized system by constructive proofs. In addition, Yu et al. [21] studied the problem of invariant set of systems, which was considered as a more generalized Lorenz system.In this paper, we consider the following three-dimensional autonomous systems with cross-product nonlinearities:(1)x˙=Ax+f(x)+C,
where xT=(x1,x2,x3) and
(2)A=[a11a12a13a21a22a23a31a32a33];C=[c1c2c3];f(x)=[xTB1xxTB2xxTB3x];Bi=[bi11bi12bi13bi21bi22bi23bi31bi32bi33],i=1,2,3
with aij,bijk,ci∈R,  i,j,k=1,2,3. This second-order dynamic system may be regarded as the most general Lorenz system. For such system, we can choose Lyapunov function:
(3)V(X(t))=12[(λ1x1-d1)2+(λ2x2-d2)2+(λ3x3-d3)2],
which is obviously positive definite and radially unbounded, where di,λi,  i=1,2,3 are undetermined parameters. In this paper, we will study this more general Lorenz system (1) than the classical system and the generalized Lorenz system. The result obtained contains earlier results as its special cases.This paper is organized as follows. In Section2, we define the globally exponentially attractive set and positive invariant set and the globally conditional exponentially attractive set and positive invariant set of the three-dimensional chaotic systems with cross-product nonlinearities. In Section 3, the qualitative analysis of the exponentially attractive set and positive invariant set of the chaotic systems has been done. In Section 4, we also suggest an idea to construct chaotic systems, and some new chaotic systems and switched chaotic systems are illustrated.


## 2. Preliminaries

In this section, we present some basic definitions which are needed for proving all theorems in the next section. For convenience, denoteX:=(x1,x2,x3) and X(t):=X(t,t0,X0).Definition 1.
For the three-dimensional autonomous systems with cross-product nonlinearities (1), if there exists compact (bounded and closed) set Ω∈R3 such that for all X0∈R3, the following condition: ρ(X(t),Ω):=infY∈Ω∥X(t)-Y∥→0 as t→+∞, holds, then the set Ω is said to be globally attractive. That is, system (1) is ultimately bounded; namely, system (1) is globally stable in the sense of Lagrange or dissipative with ultimate bound.Furthermore, if for allX0∈Ω0⊆Ω⊂R3,  X(t,t0,X0)⊆Ω0, then Ω0 for t≥0 is called the positive invariant set of the system (1).Definition 2.
For the three-dimensional autonomous systems with cross-product nonlinearities (1), if there exist compact set Ω⊂R3 such that for all X0∈R3 and constants M>0,  α>0 such that ρ(X(t),Ω)≤Me-α(t-t0), then the three-dimensional autonomous systems with cross-product nonlinearities system (1) are said to have globally exponentially attractive set, or the system (1) is globally exponentially stable in the sense of Lagrange, and Ω is called the globally exponentially attractive set.Definition 3.
For the three-dimensional autonomous systems with cross-product nonlinearities (1), if there exist compact set Ω⊂R3, a constant α>0, and a bounded function M(x0,t)>0 on t, as t≥t0, such that ρ(X(t),Ω)≤M(x0)e-α(t-t0), where M(x0)=supM(x0,t),  t≥t0, then the system (1) is said to have globally conditional exponentially attractive set, or the system (1) is globally conditional exponentially stable in the sense of Lagrange, and Ω is called the globally conditional exponentially attractive set.In general, from the definition we see that a globally exponential attractive set is not necessarily a positive invariant set. But our results obtained in the next section indeed show that a globally exponentially attractive set is a positive invariant set.Note that it is difficult to verify the existence ofΩ in Definition 2. Since the Lyapunov direct method is still a powerful tool in the study of asymptotic behaviour of nonlinear dynamical systems, the following definition is more useful in applications.Definition 4.
For three-dimensional autonomous systems with cross-product nonlinearities (1), if there exist a positive definite and radially unbounded Lyapunov function V(X(t)) and positive numbers L>0,  α>0 such that the following inequality
(4)V(X(t))-L≤(V(X0)-L)e-α(t-t0)
is valid for V(X(t))>L(t≥t0), then the system (1) is said to be globally exponentially attractive or globally exponentially stable in the sense of Lagrange, and Ω:={X∣V(X(t))≤L,t≥t0} is called the globally exponentially attractive set.Definition 5.
For the three-dimensional autonomous systems with cross-product nonlinearities (1), if there exist a positive definite and radially unbounded Lyapunov function V(X(t)) and a bounded function L(x0,t)>0, on t, as t≥t0, and α>0 such that the following inequality
(5)V(X(t))-L(x0)≤(V(X0)-L(x0))e-α(t-t0)
is valid for V(X(t))>L(x0),  t≥t0, where L(x0)=supL(x0,t),  t≥t0, then the system (1) is said to be globally conditional exponentially attractive or globally conditional exponentially stable in the sense of Lagrange, and Ω:={X∣V(X(t))≤L(x0),t≥t0} is called the globally conditional exponentially attractive set.


## 3. Qualitative Analysis

We call the dynamic system (1) the first class three-dimensional chaotic system with cross-product nonlinearities (1), if there are some nonzero numbers {λ1,λ2,λ3} so as to satisfy conditions (6)λ12b111=0,λ12(b112+b121)+λ22b211=0,λ22b222=0,λ22(b221+b212)+λ12b122=0,λ32b333=0,λ32(b313+b331)+λ12b133=0,λ12(b113+b131)+λ32b311=0,λ22(b223+b232)+λ32b322=0,λ32(b323+b332)+λ22b233=0,λ12(b123+b132)+λ22(b213+b231)+λ32(b321+b312)=0.Condition (6) is satisfied by some known three-dimensional quadratic autonomous chaotic systems, the well-known Lorenz system [1–3], the Rössler system [5], the Rucklidge system [6], and the Chen system [7, 8]. Lorenz systems are widely studied and the references therein [9–12, 19–21]. For example, consider the classical Lorenz system
(7)x˙=σ(y-x),y˙=ρx-γy-xz,z˙=xy-βz,
and the general Lorenz systems
(8)x˙=-ax+by+yz,y˙=cx-y-xz,z˙=dy-z+xy,(9)x˙=-z,y˙=-y-x2,z˙=1.7x+y+1.7.Thus it can be seen that condition (6) is very important in qualitative analysis of the exponentially attractive set and positive invariant set of Lorenz systems.We will research this dynamic system in two cases.First, supposingb111=b122=b133=b211=b222=b233=b311=b322=b333=0,  biij=-biji,  i,j=1,2,3,  ∃bijk≠-bikj,  i≠j≠k, the dynamic system (1) can be rewritten as
(10)x˙1=∑j=13‍a1jxj+(b123+b132)x2x3+c1,x˙2=∑j=13‍a2jxj+(b213+b231)x1x3+c2,x˙3=∑j=13‍a3jxj+(b312+b321)x1x2+c3.The construction techniques of this kind of Lorenz systems are to pay attention to satisfing formula(11)λ12(b123+b132)+λ22(b213+b231)+λ32(b321+b312)=0,
where λi,  i=1,2,3 are parameters and
(12)f(μ,X)=∑i=13‍λi2(aii+μi)xi2+∑i=13‍(λi2ci+μidiλi-μiλi2di-∑j=13‍λjdjaji)xi+∑i=13‍(λidici-λidi2),
where μi,  i=1,2,3 are undetermined parameters. And we always assume that the supremum f(μ,X)<+∞ in the paper.Lemma 6.
Supposeλi>0,  i=1,2,3,
(13)λ12a12+λ22a21+λ3d3(b312+b321)=0,λ12a13+λ32a31+λ2d2(b213+b231)=0,λ22a23+λ32a32+λ1d1(b123+b132)=0,a11+μ1<0,a22+μ2<0,a33+μ3<0.
The function (12) has maximum.Proof.
Consider(14)fx1′(μ,X)=2(λ12a11+μ1λ12)x1f′x1(μ,X)=+(λ12c1+μ1d1λ1-μ1λ12d1-λ1d1a11f′x1(μ,X)=-λ2d2a21-λ3d3a31),fx2′(μ,X)=2(λ22a22+μ2λ22)x2f′x1(μ,X)=+(λ22c2+μ2d2λ2-μ2λ22d2-λ1d1a12f′x1(μ,X)=-λ2d2a22-λ3d3a32),fx3′(μ,X)=2(λ32a33+μ3λ32)x3f′x1(μ,X)=+(λ32c3+μ3d3λ3-μ3λ32d3-λ1d1a13f′x1(μ,X)=-λ2d2a23-λ3d3a33),fx12′′(μ,X)=2λ12(a11+μ1),fx22′′(μ,X)=2λ22(a22+μ2),fx32′′(μ,X)=2λ32(a33+μ3),fx1x2′′(μ,X)=fx2x1′′(μ,X)=fx1x3′′(μ,X)=fx2x3′′(μ,X)=fx3x2′′(μ,X)=0.
Thus the Hesse matrixHf of the f(μ,X) is a negative definite matrix; furthermore maxX∈R3f(μ,X) exists.
These parametersdi,μi,λi,  i=1,2,3 will be determined by solving the maximum of f1(μ,X) and formula (12), and let
(15)M+={M,M>0,0,M≤0.Theorem 7.
If condition (6) exists, η=min{μ1,μ2,μ3},  M=maxX∈R3f(μ,X),  λi>0,  i=1,2,3, then the estimation
(16)[V(X(t))-12ηM+]≤[V(X(t0))-12ηM+]e-2η(t-t0)
holds, and the set
(17)Ω={X∣V(X(t))≤12ηM+}={1ηM+X∣[(λ1x1-d1)2+(λ2x2-d2)2+(λ3x3-d3)2]≤1ηM+}
is the globally exponentially attractive set and positive invariant set of system (10); that is,
(18)lim¯t→∞V(X(t))≤12ηM+.Proof.
Differentiating the Lyapunov functionV(X(t)) in (3) with respect to time t along the trajectory of system (10) yields
(19)V˙(X(t))|(10)=λ1(λ1x1-d1)x˙1+λ2(λ2x2-d2)x˙2+λ3(λ3x3-d3)x˙3=-∑j=13‍μj(λjxj-dj)2+λ1(λ1x1-d1)×(∑j=13‍a1jxj+μ1λ1-1(λ1x1-d1)+(b123+b132)x2x3+c1∑j=13‍)+λ2(λ2x2-d2)×(∑j=13‍a2jxj+μ2λ2-1(λ2x2-d2)+(b213+b231)x1x3+c2∑j=13‍)+λ3(λ3x3-d3)×(∑j=13‍a3jxj+μ3λ3-1(λ3x3-d3)+(b312+b321)x1x2+c3∑j=13‍)≤-η∑j=13‍(λjxj-dj)2+f(μ,X)≤-η∑j=13‍(λjxj-dj)2+M+≤-2η[V(X(t))-12ηM+]≤0,whenV(X(t))>12ηM+,
where μj>0,  j=1,2,3. Integrating both sides of (19) yields (16) and (17). By the definition, taking into account limit on both sides of the above inequality (16) as t→+∞ results in inequality (18).
Now, the characters of some of the chaotic systems known are analysed by condition (6). When a11=-σ, a12=σ, a21=ρ, a22=-γ, a33=-β, b213=-1, b312=1, else  aij=0, bijk=0, c1=c2=c3=0, and λ1=λ, λ2=λ3=1, d1=d2=0, d3=λσ+ρ, μ1=σ, μ2=γ, μ3=min{σ,γ}, η=η1=μ3, β>η1, system (10) can be rewritten as system (7):
(20)V(X(t))=12[λx12+x22+(x3-λσ-ρ)2],f(μ,X)=-(β-η1)x32+(β-2η1)(λσ+ρ)x3+η1(λσ+ρ)2.
We have M=β2(λσ+ρ)2/4(β-η1). Thus
(21)[V(X(t))-β2(λσ+ρ)28(β-η1)η1]≤[V(X(t0))-β2(λσ+ρ)28(β-η1)η1]e-2η1(t-t0),Ω1={X∣V(X)≤β2(λσ+ρ)28(β-η1)η1}={X∣λx12+x22+(x3-λσ-ρ)2≤β2(λσ+ρ)24(β-η1)η1}
is the globally exponentially attractive set and positive invariant set of system (7).Example 1.
Further, taking ito accoutμ1=σ,  μ2=γ,  μ3=β/2,  η=η2=min{σ,γ,β/2}, the estimate
(22)[V(X(t))-β(λσ+ρ)24η2]≤[V(X(t0))-β(λσ+ρ)24η2]e-2η2(t-t0)
holds and that
(23)Ω2={X∣V(X)≤β(λσ+ρ)24η2}={X∣λx12+x22+(x3-λσ-ρ)2≤β(λσ+ρ)22η2}
is the globally uniform exponentially attractive set and positive invariant set of system (7).Proof.
Again applying Lyapunov function given in (19) and evaluating the derivative of dV1/dt along the trajectory of system (16) lead to
(24)f(μ,X)=-x32β2+(λσ+ρ)2β2,M=β(λσ+ρ)22.
The conclusion of Example 2 is obtained.Example 2.
Furthermore, chooseμ1=σ,  μ2=γ,  μ3=β,  0<ξ1<β,  η=η3=min{σ,γ,ξ1}. Get
(25)f(μ,X)=-(β-η3)x32+(β-2η3)(λσ+ρ)x3+η2(λσ+ρ)2,M=β2(λσ+ρ)24(β-η3).
Then, the estimate
(26)[V(X(t))-β2(λσ+ρ)28(β-η3)η3]≤[V(X(t0))-β2(λσ+ρ)28(β-η3)η3]e-2η3(t-t0)
holds and that
(27)Ω3={X∣V(X)≤β2(λσ+ρ)28(β-η3)η3}={X∣λx12+x22+(x3-λσ-ρ)2≤β2(λσ+ρ)24(β-η3)η3}
is the globally exponentially attractive set and positive invariant set of system (7).Example 3.
Takinga11=-a,  a12=b,  a21=c,  a22=-1,  a32=d,  a33=-1,  b123=b312=1,  b213=-1 else aij=0,  bijk=0,  c1=c2=c3=0, and λ1=λ3=1,  λ2=2,  d1=d,  d2=0,  d3=b+2c, system (6), V(X(t)), and f(u,X) can be rewritten as system (8):
(28)V(X(t))=12[(x1-d)2+2x22+(x3-b-2c)2],f(μ,X)=-(a-μ1)x12+(a-2μ1)dx1-2(1-μ2)x22-2(b+c)dx2-(1-μ3)x32+(b+2c)(1-2μ3)x3+μ1d2+μ3(b+2c)2.
Thus
(29)M=(a-2μ1)2d24(a-μ1)+(b+c)2d22(1-μ2)+(b+2c)2(1-2μ3)4(1-μ3)+μ1d2+μ3(b+2c)2.
We have
(30)V(x(t))-M≤[V(x(t0))-M]e-2η(t-t0),
then
(31)Ω4={X∣V(x(t))-M}={X∣(x1-d)2+2x22+(x3-b-2c)2≤2M}
is the estimation of the globally exponentially attractive and positive invariant sets of system (8).
Ifb111=b222=b333=0,  ∃bijj≠0,  i,j=1,2,3, the dynamic system (1) is shown as
(32)x˙1=∑j=13‍a1jxj+∑i=2,3‍b1iixi2+∑i≠j=13‍(b1ij+b1ji)xixj+c1,x˙2=∑j=13‍a2jxj+∑i=1,3‍b2iixi2+∑i≠j=13‍(b2ij+b2ji)xixj+c2,x˙3=∑j=13‍a3jxj+∑i=1,2‍b3iixi2+∑i≠j=13‍(b3ij+b3ji)xixj+c3.
In this case, we can take into account
(33)f(μ,X)=∑i=13‍[λi2(aii+μi)+λ[i+1]3d[i+1]3b[i+1]3,ii+λ[i+2]3d[i+2]3b[i+2]3,ii]xi2+∑i<j,13‍(λi2aij+λj2aji+λ1d1(b1ij+b1ji)+λ2d2(b2ij+b2ji)+λ3d3(b3ij+b3ji))xixj+∑i=13‍(λi2(ci-μidi)+μiλidi-a1iλ1d1-a2iλ2d2-a3iλ3d3λi2(ci-μidi))x1+∑i=13‍λidi(ci-di),
where [·]3 denotes modulo-3.Theorem 8.
Suppose thatG0=(x10,x20,…,xn0) is the stable point of the f(μ,X) defined by (33). If the Hesse matrix of the f(μ,X) is a negative definite matrix, the f(μ,X) has maximum Mand the estimation
(34)[V(X(t))-12ηM+]≤[V(X(t0))-12ηM+]e-2η(t-t0)
holds; that is,
(35)lim¯t→∞V(X(t))≤12ηM+,
and the set
(36)Ω={X∣V(X(t))≤12ηM+}={1ηX∣[(λ1x1-d1)2+(λ2x2-d2)2+(λ3x3-d3)2]≤1ηM+}
is the globally exponentially attractive set and positive invariant set of system (32).Proof.
IfG0 is the stable point of the f(μ,X), that is,
(37)∇(f)G0=(fx1′,fx2′,fx3′)=0,
and the Hesse matrix Hf of the f(μ,X) is a negative definite matrix, namely,
(38)fx1x1′′<0,|fx1x1′′fx1x2′′fx2x1′′fx2x2′′|>0,|fx1x1′′fx1x2′′fx1x3′′fx2x1′′fx2x2′′fx2x3′′fx3x1′′fx3x2′′fx3x3′′|<0.
The f(μ,X) has the maximum M. Differentiating the Lyapunov function V(X(t)) in (3) with respect to time t along the trajectory of system (32) yields
(39)V˙(X(t))|(32)=λ1(λ1x1-d1)x˙1+λ2(λ2x2-d2)x˙2+λ3(λ3x3-d3)x˙3≤-η∑j=13‍(λjxj-dj)2+f(μ,X)≤-η∑j=13‍(λjxj-dj)2+M+≤-2η[V(X(t))-12ηM+]≤0,whenV(X(t))>12ηM+.
The proof is complete.


## 4. Switched Chaotic Systems

Condition (6) has helpfully provided us with instructions on how to find the new chaotic systems. We construct a series of new chaotic systems that the condition (6) is fulfilled and study the switching system between them.Example 4.
Consider a Lorenz system shown in Figure1:
(40)x˙1=-12x1+5x2-0.8x1x3+x2x3,x˙2=28x1-x2-x1x3,x˙3=-3x2-x3+10x12+x1x2.
Solution. Here
(41)a11=-12,a12=5,a21=28,a22=-1,a32=-3,a33=-1,elsaij=0,b113=b131=-0.4,b123=b132=0.5,b213=b231=-0.5,b311=10,b312=b321=0.5,elsbijk=0,λ1=12.5,λ2=13.5,λ3=1,d1=-625,d2=445,d3=440.5,μ1=2,μ2=μ3=12,η=12,f2(μ,X)=-125x12-6.75x22-0.5x32f2(μ,X)=+(-482512.5-1124513.5)x1f2(μ,X)=+(1682512.5+26432)x2+39293196661405000,f2′x1(μ,X)=-250x1-(482512.5+1124513.5),f2′x1(μ,X)=0,x1=-11250(48512.5+112913.5)f′2x1(μ,X)=0x1≈0.064,f2x2′(μ,X)=-13.5x2+(1682512.5+26432),f2x2′(μ,X)=0,x2=30135(562512.5+8812)≈99.65,f2x3′(μ,X)=-x3,f2x3′(μ,X)=0,x3=0,f2x12′′(μ,X)=-250,f2x22′′(μ,X)=-13.5,f2x32′′(μ,X)=-1,f2x1x2′′(μ,X)=f2x2x1′′(μ,X)=f2x1x3′′(μ,X)=0,f2x3x1′′(μ,X)=f2x2x3′′(μ,X)=f2x3x2′′(μ,X)=0.
The Hesse matrix of the f(μ,X) is a negative definite matrix,  maxf(μ,X)≈164045.42. The set
(42)Ω={X∣(12.5x1+526)2+(13.5x2-445)2+(x3-440.5)2≤328090.84(12.5x1+526)2}
is the globally exponentially attractive set and positive invariant set of system (40).
Note. (a) If the Hesse matrix of the f(μ,X) is not a negative definite matrix, the f(μ,X) has no maximum M.
(b) If∃ai0i0≥0,  limxi0→∞f(μ,X)=+∞, this type of chaotic system needs further research.
(c) We call the dynamic system (1) the second class three-dimensional chaotic system with cross-product nonlinearities, if it does not satisfy condition (6). For this class of chaotic systems, f(μ,X) is a cubic polynomial and there is not maximum if we choose energy function (3) differentiating this Lyapunov function with respect to t along the trajectory of system (1). It is very useful to research these problems.Figure 1
Simulation of system (40).Example 5.
The new chaotic system shown in Figure2 is
(43)x˙1=-2x1+5x3+5.7x1x3+4.7x2x3+4,x˙2=-x1-2x2+3x3+5,x˙3=-6x1-2x2-4x3+0.2x12+3.4x1x2+7.Figure 2
Simulation of system (43).Example 6.
The chaotic system shown in Figure3 is
(44)x˙1=-11x1+0.15x1x2+1.38x2x3+1,x˙2=30x1-x2-x1x3+0.1x2x3,x˙3=15x2-2.5x3+x1x2.Figure 3
Simulation of system (44).Example 7.
The chaotic system shown in Figure4 is
(45)x˙1=30(x2-x1)-0.48x12,x˙2=80x1-6x2-x1x3,x˙3=x1x2-5x3.Figure 4
Simulation of system (45).Example 8.
The chaotic system shown in Figure5 is
(46)x˙1=-12x1+5x2+x2x3,x˙2=28x1-x2-x1x3,x˙3=-3x2-x3+4x12+x1x2.Figure 5
Simulation of system (46).Example 9.
The chaotic system shown in Figure6 is
(47)x˙=(207)x-yz+9,y˙=-10y+xz+0.5z2,z˙=-4z+xy+yz.Figure 6
Simulation of system (47).Note. When we analyse  Examples 5 to  9 by the previous means, for supX∈R3f(μ,X)=+∞, the globally exponentially attractive set and positive invariant set of them have not been obtained. The globally exponentially attractive set and positive invariant set really exist by their trajectories. Particularly,  by Lü et al. chaotic system [11] and Example 9 we conjecture that they have globally conditional exponentially attractive set and positive invariant set, according to preliminary study. These are waiting for us to do further research. Meanwhile, we can compute that the maximum Lyapunov exponents of Examples 4–9 are 1.06, 0.02, 1.84, 0.01, 0.92, and 0.95, respectively.


## 5. Simulation of Switched System

In this section, we will show some simulation results of the following switching system(48)x˙=Aσx+fσ(x)+Cσ,
where xT=(x1,x2,x3), σ is the switching law, and
(49)Aσ=[a11σa12σa13σa21σa22σa23σa31σa32σa33σ],Cσ=[c1σc2σc3σ],fσ(x)=[xTB1σxxTB2σxxTB3σx]
with aijσ,bijkσ,ciσ∈R,  i,j,k=1,2,3. Each pair of (Aσ,Cσ,B1σ,B2σ,B3σ) takes the form from Example 1 to Example 6. The switching law is that the system will stay in each subsystem for a constant time. In the following, we assume that (a, b) denotes a switched system which switches between system (a) and system (b). It can be seen from Figures 7 to 12 that the switched systems (18,20), (18,22), (18,23), (20,18), (20,22), (20,23), (22,18), (22,20), (22,23), (23,18), (23,20), and (23,22) can also yield chaotic systems.Switched system between system (40) and others.
(a)
System (40)–(43)
(b)
System (40)–(44)
(c)
System (40)–(45)
(d)
System (40)–(46)
(e)
System (40)–(47)Switched system between system (43) and others.
(a)
System (43)–(40)
(b)
System (43)-(44)
(c)
System (43)–(45)
(d)
System (43)–(46)
(e)
System (43)–(47)Switched system between Example (44) and others.
(a)
System (44)–(40)
(b)
System (44)-(43)
(c)
System (44)-(45)
(d)
System (44)–(46)
(e)
System (44)–(47)Switched system between Example (45) and others.
(a)
System (45)–(40)
(b)
System (45)–(43)
(c)
System (45)-(44)
(d)
System (45)-(46)
(e)
System (45)–(47)Switched system between Example (46) and others.
(a)
System (46)–(40)
(b)
System (46)–(40)
(c)
System (46)–(44)
(d)
System (46)-(45)
(e)
System (46)-(47)Switched system between Example (47) and others.
(a)
System (47)–(40)
(b)
System (47)–(43)
(c)
System (47)–(44)
(d)
System (47)–(45)
(e)
System (47)-(46)


## 6. Conclusion

In this paper, the methods in [19–21] have been extended to study the globally exponentially or globally conditional exponentially attractive set and positive invariant set of the three-dimensional chaotic system family with cross-product nonlinearities. We have given two theorems for studying this question and given some examples to show that such system indeed has the globally exponentially or globally conditional exponentially attractive set and positive invariant set, and the exponential estimation is explicitly derived. We have also suggested an idea to construct the chaotic systems, and some new chaotic systems have been illustrated. The simulation results are given for switched system between these new chaotic systems. It is very interesting to further research that the Hesse matrix of the f(μ,X) is not a negative definite matrix, and the dynamic system (1) is a second class three-dimensional chaotic system with cross-product nonlinearities.



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*Source: 590421-2013-02-24.xml*