1.Introduction XinquanZhao, FengJiang, ZhigangZhang, andJunhaoHu ANewSeriesofThree-DimensionalChaoticSystemswithCross-ProductNonlinearitiesandTheirSwitching ResearchArticle

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Volume 2013, Article ID 590421,14pages http://dx.doi.org/10.1155/2013/590421

Research Article

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

Xinquan Zhao,

¹

Feng Jiang,

¹

Zhigang Zhang,

²

and Junhao Hu

³

1School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China

2Department of Statistics and Applied Mathematics, Hubei University of Economics, Wuhan 430205, China

3School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China

Correspondence should be addressed to Xinquan Zhao; [email protected] Received 31 October 2012; Accepted 25 December 2012

Academic Editor: Xinzhi Liu

Copyright © 2013 Xinquan Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Since Lorenz discovered the well-known Lorenz chaotic system, many other chaotic systems have been found, including the well-known R¨ossler 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:

̇𝑥 = 𝐴𝑥 + 𝑓 (𝑥) + 𝐶, (1)

where𝑥^𝑇= (𝑥₁, 𝑥₂, 𝑥₃)and 𝐴 = [

[

𝑎₁₁ 𝑎₁₂ 𝑎₁₃ 𝑎₂₁ 𝑎₂₂ 𝑎₂₃ 𝑎₃₁ 𝑎₃₂ 𝑎₃₃ ] ]

; 𝐶 = [ [ 𝑐₁ 𝑐₂ 𝑐₃ ] ]

; 𝑓 (𝑥) = [ [

𝑥^𝑇𝐵₁𝑥 𝑥^𝑇𝐵₂𝑥 𝑥^𝑇𝐵₃𝑥 ] ]

;

𝐵_𝑖= [ [

𝑏_𝑖11 𝑏_𝑖12 𝑏_𝑖13 𝑏_𝑖21 𝑏_𝑖22 𝑏_𝑖23 𝑏_𝑖31 𝑏_𝑖32 𝑏_𝑖33 ] ]

, 𝑖 = 1, 2, 3

(2) with𝑎_𝑖𝑗, 𝑏_𝑖𝑗𝑘, 𝑐_𝑖∈ 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:

𝑉 (𝑋 (𝑡)) = 1

2[(𝜆₁𝑥₁− 𝑑₁)²+ (𝜆₂𝑥₂− 𝑑₂)² +(𝜆₃𝑥₃− 𝑑₃)²] ,

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which is obviously positive definite and radially unbounded, where𝑑_𝑖, 𝜆_𝑖, 𝑖 = 1, 2, 3are 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. InSection 2, we define the globally exponentially attractive set and positive invariant

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set and the globally conditional exponentially attractive set and positive invariant set of the three-dimensional chaotic systems with cross-product nonlinearities. InSection 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, denote 𝑋 := (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) setΩ ∈ 𝑅³such that for all𝑋₀ ∈ 𝑅³, the following condition:𝜌(𝑋(𝑡), Ω) :=inf_𝑌∈Ω‖𝑋(𝑡)−𝑌‖ → 0 as𝑡 → +∞, holds, then the setΩ 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, thenΩ₀for𝑡 ≥ 0is 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 setΩ ⊂ 𝑅³ such that for all𝑋₀ ∈ 𝑅³ and constants𝑀 >

0,𝛼 > 0such that𝜌(𝑋(𝑡), Ω) ≤ 𝑀𝑒^{−𝛼(𝑡−𝑡}⁰⁾, 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, andΩis called the globally exponentially attractive set.

Definition 3. For the three-dimensional autonomous systems with cross-product nonlinearities (1), if there exist compact setΩ ⊂ 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−𝑡}⁰⁾, where𝑀(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, andΩis 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 of Ω in Definition 2. Since the Lyapunov direct method is still a pow- erful 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 function𝑉(𝑋(𝑡)) and positive numbers𝐿 > 0, 𝛼 > 0such that the following inequality

𝑉 (𝑋 (𝑡)) − 𝐿 ≤ (𝑉 (𝑋₀) − 𝐿) 𝑒^{−𝛼(𝑡−𝑡}⁰⁾ (4) is valid for𝑉(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, andΩ := {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 function𝑉(𝑋(𝑡)) and a bounded function𝐿(𝑥₀, 𝑡) > 0, on𝑡, as𝑡 ≥ 𝑡₀, and𝛼 > 0 such that the following inequality

𝑉 (𝑋 (𝑡)) − 𝐿 (𝑥₀) ≤ (𝑉 (𝑋₀) − 𝐿 (𝑥₀)) 𝑒^{−𝛼(𝑡−𝑡}⁰⁾ (5) 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, andΩ := {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

𝜆²₁𝑏₁₁₁= 0, 𝜆²₁(𝑏₁₁₂+ 𝑏₁₂₁) + 𝜆²₂𝑏₂₁₁= 0, 𝜆²₂𝑏₂₂₂= 0, 𝜆²₂(𝑏₂₂₁+ 𝑏₂₁₂) + 𝜆²₁𝑏₁₂₂= 0, 𝜆²₃𝑏₃₃₃= 0, 𝜆²₃(𝑏₃₁₃+ 𝑏₃₃₁) + 𝜆²₁𝑏₁₃₃= 0,

𝜆²₁(𝑏₁₁₃+ 𝑏₁₃₁) + 𝜆²₃𝑏₃₁₁= 0, 𝜆²₂(𝑏₂₂₃+ 𝑏₂₃₂) + 𝜆²₃𝑏₃₂₂= 0, 𝜆²₃(𝑏₃₂₃+ 𝑏₃₃₂) + 𝜆²₂𝑏₂₃₃= 0,

𝜆²₁(𝑏₁₂₃+ 𝑏₁₃₂) + 𝜆²₂(𝑏₂₁₃+ 𝑏₂₃₁) + 𝜆²₃(𝑏₃₂₁+ 𝑏₃₁₂) = 0.

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Condition (6) is satisfied by some known three-dimensional quadratic autonomous chaotic systems, the well- known Lorenz system [1–3], the R¨ossler 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

̇𝑥 = 𝜎 (𝑦 − 𝑥) ,

̇𝑦 = 𝜌𝑥 − 𝛾𝑦 − 𝑥𝑧,

̇𝑧 = 𝑥𝑦 − 𝛽𝑧,

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and the general Lorenz systems

̇𝑥 = −𝑎𝑥 + 𝑏𝑦 + 𝑦𝑧,

̇𝑦 = 𝑐𝑥 − 𝑦 − 𝑥𝑧,

̇𝑧 = 𝑑𝑦 − 𝑧 + 𝑥𝑦,

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̇𝑥 = −𝑧,

̇𝑦 = −𝑦 − 𝑥²,

̇𝑧 = 1.7𝑥 + 𝑦 + 1.7.

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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,𝑏_𝑖𝑖𝑗 = −𝑏_𝑖𝑗𝑖,𝑖, 𝑗 = 1, 2, 3, ∃𝑏_𝑖𝑗𝑘 ̸= − 𝑏_𝑖𝑘𝑗, 𝑖 ̸= 𝑗 ̸= 𝑘, the dynamic system (1) can be rewritten as

̇𝑥1= ³∑

𝑗=1

𝑎_1𝑗𝑥_𝑗+ (𝑏₁₂₃+ 𝑏₁₃₂) 𝑥₂𝑥₃+ 𝑐₁,

̇𝑥2= ³∑

𝑗=1

𝑎_2𝑗𝑥_𝑗+ (𝑏₂₁₃+ 𝑏₂₃₁) 𝑥₁𝑥₃+ 𝑐₂,

̇𝑥3= ³∑

𝑗=1

𝑎_3𝑗𝑥_𝑗+ (𝑏₃₁₂+ 𝑏₃₂₁) 𝑥₁𝑥₂+ 𝑐₃.

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The construction techniques of this kind of Lorenz systems are to pay attention to satisfing formula

𝜆²₁(𝑏₁₂₃+ 𝑏₁₃₂) + 𝜆²₂(𝑏₂₁₃+ 𝑏₂₃₁) + 𝜆²₃(𝑏₃₂₁+ 𝑏₃₁₂) = 0, (11) where𝜆_𝑖, 𝑖 = 1, 2, 3are parameters and

𝑓 (𝜇, 𝑋) =∑³

𝑖=1

𝜆²_𝑖(𝑎_𝑖𝑖+ 𝜇_𝑖) 𝑥²_𝑖

+∑³

𝑖=1

(𝜆²_𝑖𝑐_𝑖+ 𝜇_𝑖𝑑_𝑖𝜆_𝑖− 𝜇_𝑖𝜆²_𝑖𝑑_𝑖−∑³

𝑗=1

𝜆_𝑗𝑑_𝑗𝑎_𝑗𝑖) 𝑥_𝑖

+∑³

𝑖=1

(𝜆_𝑖𝑑_𝑖𝑐_𝑖− 𝜆_𝑖𝑑²_𝑖) ,

(12) where𝜇_𝑖, 𝑖 = 1, 2, 3are undetermined parameters. And we always assume that the supremum 𝑓(𝜇, 𝑋) < +∞ in the paper.

Lemma 6. Suppose𝜆_𝑖> 0,𝑖 = 1, 2, 3,

𝜆²₁𝑎₁₂+ 𝜆²₂𝑎₂₁+ 𝜆₃𝑑₃(𝑏₃₁₂+ 𝑏₃₂₁) = 0, 𝜆²₁𝑎₁₃+ 𝜆²₃𝑎₃₁+ 𝜆₂𝑑₂(𝑏₂₁₃+ 𝑏₂₃₁) = 0, 𝜆²₂𝑎₂₃+ 𝜆²₃𝑎₃₂+ 𝜆₁𝑑₁(𝑏₁₂₃+ 𝑏₁₃₂) = 0,

𝑎₁₁+ 𝜇₁< 0, 𝑎₂₂+ 𝜇₂< 0, 𝑎₃₃+ 𝜇₃< 0.

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The function(12)has maximum.

Proof. Consider

𝑓_𝑥^󸀠₁(𝜇, 𝑋) = 2 (𝜆²₁𝑎₁₁+ 𝜇₁𝜆²₁) 𝑥₁

+ (𝜆²₁𝑐₁+ 𝜇₁𝑑₁𝜆₁− 𝜇₁𝜆²₁𝑑₁− 𝜆₁𝑑₁𝑎₁₁

−𝜆₂𝑑₂𝑎₂₁− 𝜆₃𝑑₃𝑎₃₁) , 𝑓_𝑥^󸀠₂(𝜇, 𝑋) = 2 (𝜆²₂𝑎₂₂+ 𝜇₂𝜆²₂) 𝑥₂

+ (𝜆²₂𝑐₂+ 𝜇₂𝑑₂𝜆₂− 𝜇₂𝜆²₂𝑑₂− 𝜆₁𝑑₁𝑎₁₂

−𝜆₂𝑑₂𝑎₂₂− 𝜆₃𝑑₃𝑎₃₂) , 𝑓_𝑥^󸀠₃(𝜇, 𝑋) = 2 (𝜆²₃𝑎₃₃+ 𝜇₃𝜆²₃) 𝑥₃

+ (𝜆²₃𝑐₃+ 𝜇₃𝑑₃𝜆₃− 𝜇₃𝜆²₃𝑑₃− 𝜆₁𝑑₁𝑎₁₃

−𝜆₂𝑑₂𝑎₂₃− 𝜆₃𝑑₃𝑎₃₃) , 𝑓_𝑥^󸀠󸀠2

1(𝜇, 𝑋) = 2𝜆²₁(𝑎₁₁+ 𝜇₁) , 𝑓_𝑥^󸀠󸀠2

2(𝜇, 𝑋) = 2𝜆²₂(𝑎₂₂+ 𝜇₂) , 𝑓_𝑥^󸀠󸀠2

3(𝜇, 𝑋) = 2𝜆²₃(𝑎₃₃+ 𝜇₃) , 𝑓_𝑥^󸀠󸀠₁_𝑥₂(𝜇, 𝑋) = 𝑓_𝑥^󸀠󸀠₂_𝑥₁(𝜇, 𝑋) = 𝑓_𝑥^󸀠󸀠₁_𝑥₃(𝜇, 𝑋)

= 𝑓_𝑥^󸀠󸀠₂_𝑥₃(𝜇, 𝑋) = 𝑓_𝑥^󸀠󸀠₃_𝑥₂(𝜇, 𝑋) = 0.

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Thus the Hesse matrix 𝐻_𝑓 of the 𝑓(𝜇, 𝑋)is a negative definite matrix; furthermore max_𝑋∈𝑅3𝑓(𝜇, 𝑋)exists.

These parameters𝑑_𝑖, 𝜇_𝑖, 𝜆_𝑖,𝑖 = 1, 2, 3will be determined by solving the maximum of𝑓₁(𝜇, 𝑋)and formula (12), and let

𝑀⁺ = {𝑀, 𝑀 > 0,

0, 𝑀 ≤ 0. (15)

Theorem 7. If condition(6)exists,𝜂 =min{𝜇₁, 𝜇₂, 𝜇₃},𝑀 = max_𝑋∈𝑅3𝑓(𝜇, 𝑋),𝜆_𝑖> 0,𝑖 = 1, 2, 3, then the estimation

[𝑉 (𝑋 (𝑡)) − 1

2𝜂𝑀⁺] ≤ [𝑉 (𝑋 (𝑡₀)) − 1

2𝜂𝑀⁺] 𝑒^{−2𝜂(𝑡−𝑡}⁰⁾ (16)

(4)

holds, and the set

Ω = {𝑋 | 𝑉 (𝑋 (𝑡)) ≤ 1 2𝜂𝑀⁺}

= {𝑋 | [(𝜆₁𝑥₁− 𝑑₁)²+ (𝜆₂𝑥₂− 𝑑₂)²+ (𝜆₃𝑥₃− 𝑑₃)²]

≤1 𝜂𝑀⁺}

(17) is the globally exponentially attractive set and positive invariant set of system(10); that is,

𝑡 → ∞lim𝑉 (𝑋 (𝑡)) ≤ 1

2𝜂𝑀⁺. (18)

Proof. Differentiating the Lyapunov function𝑉(𝑋(𝑡))in (3) with respect to time𝑡along the trajectory of system (10) yields

̇𝑉 (𝑋 (𝑡))󵄨󵄨󵄨󵄨󵄨(10)= 𝜆₁(𝜆₁𝑥₁− 𝑑₁) ̇𝑥₁+ 𝜆₂(𝜆₂𝑥₂− 𝑑₂) ̇𝑥₂ + 𝜆₃(𝜆₃𝑥₃− 𝑑₃) ̇𝑥₃

= −∑³

𝑗=1

𝜇_𝑗(𝜆_𝑗𝑥_𝑗− 𝑑_𝑗)²+ 𝜆₁(𝜆₁𝑥₁− 𝑑₁)

× (∑³

𝑗=1

𝑎_1𝑗𝑥_𝑗+ 𝜇₁𝜆⁻¹₁ (𝜆₁𝑥₁− 𝑑₁)

+ (𝑏₁₂₃+ 𝑏₁₃₂) 𝑥₂𝑥₃+ 𝑐₁) + 𝜆₂(𝜆₂𝑥₂− 𝑑₂)

× (∑³

𝑗=1

𝑎_2𝑗𝑥_𝑗+ 𝜇₂𝜆⁻¹₂ (𝜆₂𝑥₂− 𝑑₂)

+ (𝑏₂₁₃+ 𝑏₂₃₁) 𝑥₁𝑥₃+ 𝑐₂) + 𝜆₃(𝜆₃𝑥₃− 𝑑₃)

× (∑³

𝑗=1

𝑎_3𝑗𝑥_𝑗+ 𝜇₃𝜆⁻¹₃ (𝜆₃𝑥₃− 𝑑₃)

+ (𝑏₃₁₂+ 𝑏₃₂₁) 𝑥₁𝑥₂+ 𝑐₃)

≤ −𝜂∑³

𝑗=1(𝜆_𝑗𝑥_𝑗− 𝑑_𝑗)²+ 𝑓 (𝜇, 𝑋)

≤ −𝜂∑³

𝑗=1(𝜆_𝑗𝑥_𝑗− 𝑑_𝑗)²+ 𝑀⁺

≤ −2𝜂 [𝑉 (𝑋 (𝑡)) − 1

2𝜂𝑀⁺] ≤ 0, when𝑉 (𝑋 (𝑡)) > 1

2𝜂𝑀⁺,

(19) where𝜇_𝑗> 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) as𝑡 → +∞results in inequality (18).

Now, the characters of some of the chaotic systems known are analysed by condition (6). When𝑎₁₁= −𝜎, 𝑎₁₂ = 𝜎,𝑎₂₁= 𝜌,𝑎₂₂ = −𝛾,𝑎₃₃= −𝛽,𝑏₂₁₃ = −1, 𝑏₃₁₂ = 1, else𝑎_𝑖𝑗= 0,𝑏_𝑖𝑗𝑘= 0,𝑐₁ = 𝑐₂ = 𝑐₃ = 0, and𝜆₁ = √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):

𝑉 (𝑋 (𝑡)) = 1

2[𝜆𝑥²₁+ 𝑥²₂+ (𝑥₃− 𝜆𝜎 − 𝜌)²] , 𝑓 (𝜇, 𝑋) = − (𝛽 − 𝜂₁) 𝑥²₃+ (𝛽 − 2𝜂₁) (𝜆𝜎 + 𝜌) 𝑥₃

+ 𝜂₁(𝜆𝜎 + 𝜌)².

(20)

We have𝑀 = 𝛽²(𝜆𝜎 + 𝜌)²/4(𝛽 − 𝜂₁). Thus

[𝑉 (𝑋 (𝑡)) − 𝛽²(𝜆𝜎 + 𝜌)² 8 (𝛽 − 𝜂₁) 𝜂₁]

≤ [𝑉 (𝑋 (𝑡₀)) −𝛽²(𝜆𝜎 + 𝜌)²

8 (𝛽 − 𝜂₁) 𝜂₁] 𝑒^−2𝜂¹^(𝑡−𝑡⁰⁾, Ω₁= {𝑋 | 𝑉 (𝑋) ≤ 𝛽²(𝜆𝜎 + 𝜌)²

8 (𝛽 − 𝜂₁) 𝜂₁}

= {𝑋 | 𝜆𝑥²₁+ 𝑥²₂+ (𝑥₃− 𝜆𝜎 − 𝜌)²≤ 𝛽²(𝜆𝜎 + 𝜌)² 4 (𝛽 − 𝜂₁) 𝜂₁}

(21) is the globally exponentially attractive set and positive invariant set of system (7).

Example 8. Further, taking ito accout𝜇₁ = 𝜎,𝜇₂ = 𝛾, 𝜇₃ = 𝛽/2,𝜂 = 𝜂₂=min{𝜎, 𝛾, 𝛽/2}, the estimate

[𝑉 (𝑋 (𝑡)) −𝛽(𝜆𝜎 + 𝜌)² 4𝜂₂ ]

≤ [𝑉 (𝑋 (𝑡₀)) −𝛽(𝜆𝜎 + 𝜌)²

4𝜂₂ ] 𝑒^−2𝜂²^(𝑡−𝑡⁰⁾

(22)

(5)

holds and that

Ω₂= {𝑋 | 𝑉 (𝑋) ≤ 𝛽(𝜆𝜎 + 𝜌)² 4𝜂₂ }

= {𝑋 | 𝜆𝑥²₁+ 𝑥₂²+ (𝑥₃− 𝜆𝜎 − 𝜌)²≤ 𝛽(𝜆𝜎 + 𝜌)² 2𝜂₂ }

(23)

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 of𝑑𝑉₁/𝑑𝑡along the trajectory of system (16) lead to

𝑓 (𝜇, 𝑋) = −𝑥²₃𝛽

2 +(𝜆𝜎 + 𝜌)²𝛽

2 ,

𝑀 =𝛽(𝜆𝜎 + 𝜌)²

2 .

(24)

The conclusion ofExample 9is obtained.

Example 9. Furthermore, choose𝜇₁= 𝜎,𝜇₂= 𝛾,𝜇₃= 𝛽,0 <

𝜉₁< 𝛽,𝜂 = 𝜂₃=min{𝜎, 𝛾, 𝜉₁}. Get

𝑓 (𝜇, 𝑋) = − (𝛽 − 𝜂₃) 𝑥²₃+ (𝛽 − 2𝜂₃) (𝜆𝜎 + 𝜌) 𝑥₃ + 𝜂₂(𝜆𝜎 + 𝜌)²,

𝑀 = 𝛽²(𝜆𝜎 + 𝜌)² 4 (𝛽 − 𝜂₃) .

(25)

Then, the estimate

[𝑉 (𝑋 (𝑡)) − 𝛽²(𝜆𝜎 + 𝜌)² 8 (𝛽 − 𝜂₃) 𝜂₃]

≤ [𝑉 (𝑋 (𝑡₀)) −𝛽²(𝜆𝜎 + 𝜌)²

8 (𝛽 − 𝜂₃) 𝜂₃] 𝑒^−2𝜂³^(𝑡−𝑡⁰⁾

(26)

holds and that

Ω₃= {𝑋 | 𝑉 (𝑋) ≤ 𝛽²(𝜆𝜎 + 𝜌)² 8 (𝛽 − 𝜂₃) 𝜂₃}

= {𝑋 | 𝜆𝑥²₁+ 𝑥²₂+ (𝑥₃− 𝜆𝜎 − 𝜌)²≤ 𝛽²(𝜆𝜎 + 𝜌)² 4 (𝛽 − 𝜂₃) 𝜂₃}

(27) is the globally exponentially attractive set and positive invariant set of system (7).

Example 10. Taking𝑎₁₁ = −𝑎,𝑎₁₂ = 𝑏,𝑎₂₁ = 𝑐,𝑎₂₂ = −1, 𝑎₃₂ = 𝑑,𝑎₃₃ = −1,𝑏₁₂₃ = 𝑏₃₁₂ = 1,𝑏₂₁₃ = −1else𝑎_𝑖𝑗 = 0, 𝑏_𝑖𝑗𝑘= 0,𝑐₁ = 𝑐₂= 𝑐₃ = 0, and𝜆₁= 𝜆₃= 1, 𝜆₂= √2,𝑑₁= 𝑑,

𝑑₂ = 0,𝑑₃= 𝑏 + 2𝑐, system (6),𝑉(𝑋(𝑡)), and𝑓(𝑢, 𝑋)can be rewritten as system (8):

𝑉 (𝑋 (𝑡)) =1

2[(𝑥₁− 𝑑)²+ 2𝑥²₂+ (𝑥₃− 𝑏 − 2𝑐)²] , 𝑓 (𝜇, 𝑋) = − (𝑎 − 𝜇₁) 𝑥²₁+ (𝑎 − 2𝜇₁) 𝑑𝑥₁− 2 (1 − 𝜇₂) 𝑥²₂

− 2 (𝑏 + 𝑐) 𝑑𝑥2− (1 − 𝜇₃) 𝑥²₃

+ (𝑏 + 2𝑐) (1 − 2𝜇₃) 𝑥₃+ 𝜇₁𝑑²+ 𝜇₃(𝑏 + 2𝑐)². (28) Thus

𝑀 =(𝑎 − 2𝜇₁)²𝑑²

4 (𝑎 − 𝜇₁) + (𝑏 + 𝑐)²𝑑²

2 (1 − 𝜇₂)+(𝑏 + 2𝑐)²(1 − 2𝜇₃) 4 (1 − 𝜇₃) + 𝜇₁𝑑²+ 𝜇₃(𝑏 + 2𝑐)².

(29)

We have

𝑉 (𝑥 (𝑡)) − 𝑀 ≤ [𝑉 (𝑥 (𝑡₀)) − 𝑀] 𝑒^{−2𝜂(𝑡−𝑡}⁰⁾, (30)

then

Ω₄= {𝑋 | 𝑉 (𝑥 (𝑡)) − 𝑀}

= {𝑋 | (𝑥₁− 𝑑)²+ 2𝑥²₂+ (𝑥₃− 𝑏 − 2𝑐)²≤ 2𝑀} (31)

is the estimation of the globally exponentially attractive and positive invariant sets of system (8).

If 𝑏₁₁₁ = 𝑏₂₂₂ = 𝑏₃₃₃ = 0,∃𝑏_𝑖𝑗𝑗 ̸= 0,𝑖, 𝑗 = 1, 2, 3, the dynamic system (1) is shown as

̇𝑥1= ³∑

𝑗=1𝑎_1𝑗𝑥_𝑗+ ∑

𝑖=2,3𝑏_1𝑖𝑖𝑥²_𝑖 + ∑³

𝑖 ̸= 𝑗=1(𝑏_1𝑖𝑗+ 𝑏_1𝑗𝑖) 𝑥_𝑖𝑥_𝑗+ 𝑐₁, 𝑐 ̇𝑥₂=∑³

𝑗=1

𝑎_2𝑗𝑥_𝑗+ ∑

𝑖=1,3

𝑏_2𝑖𝑖𝑥²_𝑖 + ∑³

𝑖 ̸= 𝑗=1

(𝑏_2𝑖𝑗+ 𝑏_2𝑗𝑖) 𝑥_𝑖𝑥_𝑗+ 𝑐₂,

̇𝑥3= ³∑

𝑗=1

𝑎_3𝑗𝑥_𝑗+ ∑

𝑖=1,2

𝑏_3𝑖𝑖𝑥²_𝑖 + ∑³

𝑖 ̸= 𝑗=1

(𝑏_3𝑖𝑗+ 𝑏_3𝑗𝑖) 𝑥_𝑖𝑥_𝑗+ 𝑐₃. (32) In this case, we can take into account

𝑓 (𝜇, 𝑋) =∑³

𝑖=1

[𝜆²_𝑖(𝑎_𝑖𝑖+ 𝜇_𝑖) + 𝜆_[𝑖+1]₃𝑑_[𝑖+1]₃𝑏_[𝑖+1]₃_,𝑖𝑖 +𝜆_[𝑖+2]₃𝑑_[𝑖+2]₃𝑏_[𝑖+2]₃_,𝑖𝑖] 𝑥²_𝑖

(6)

+ ∑³

𝑖<𝑗,1(𝜆²_𝑖𝑎_𝑖𝑗+ 𝜆²_𝑗𝑎_𝑗𝑖+ 𝜆₁𝑑₁(𝑏_1𝑖𝑗+ 𝑏_1𝑗𝑖)

+𝜆₂𝑑₂(𝑏_2𝑖𝑗+𝑏_2𝑗𝑖)+𝜆₃𝑑₃(𝑏_3𝑖𝑗+𝑏_3𝑗𝑖)) 𝑥_𝑖𝑥_𝑗 +∑³

𝑖=1

(𝜆²_𝑖(𝑐_𝑖− 𝜇_𝑖𝑑_𝑖) + 𝜇_𝑖𝜆_𝑖𝑑_𝑖− 𝑎_1𝑖𝜆₁𝑑₁

−𝑎_2𝑖𝜆₂𝑑₂− 𝑎_3𝑖𝜆₃𝑑₃) 𝑥₁ +∑³

𝑖=1

𝜆_𝑖𝑑_𝑖(𝑐_𝑖− 𝑑_𝑖) ,

(33) where[⋅]₃denotes modulo-3.

Theorem 11. Suppose that𝐺₀ = (𝑥⁰₁, 𝑥⁰₂, . . . , 𝑥_𝑛⁰)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 maximum𝑀and the estimation

[𝑉 (𝑋 (𝑡)) − 1 2𝜂𝑀⁺]

≤ [𝑉 (𝑋 (𝑡₀)) − 1

2𝜂𝑀⁺] 𝑒^{−2𝜂(𝑡−𝑡}⁰⁾

(34)

holds; that is,

𝑡 → ∞lim𝑉 (𝑋 (𝑡)) ≤ 1

2𝜂𝑀⁺, (35)

and the set

Ω = {𝑋 | 𝑉 (𝑋 (𝑡)) ≤ 1 2𝜂𝑀⁺}

= {𝑋 | [(𝜆₁𝑥₁− 𝑑₁)²+ (𝜆₂𝑥₂− 𝑑₂)²+ (𝜆₃𝑥₃− 𝑑₃)²]

≤1 𝜂𝑀⁺}

(36) is the globally exponentially attractive set and positive invariant set of system(32).

Proof. If𝐺₀is the stable point of the𝑓(𝜇, 𝑋), that is,

∇(𝑓)_𝐺₀ = (𝑓_𝑥^󸀠₁, 𝑓_𝑥^󸀠₂, 𝑓_𝑥^󸀠₃) = 0, (37) and the Hesse matrix𝐻_𝑓of the𝑓(𝜇, 𝑋)is a negative definite matrix, namely,

𝑓_𝑥^󸀠󸀠₁_𝑥₁ < 0, 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

𝑓_𝑥^󸀠󸀠₁_𝑥₁ 𝑓_𝑥^󸀠󸀠₁_𝑥₂ 𝑓_𝑥^󸀠󸀠₂_𝑥₁ 𝑓_𝑥^󸀠󸀠₂_𝑥₂󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨> 0,

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

𝑓_𝑥^󸀠󸀠₁_𝑥₁ 𝑓_𝑥^󸀠󸀠₁_𝑥₂ 𝑓_𝑥^󸀠󸀠₁_𝑥₃ 𝑓_𝑥^󸀠󸀠₂_𝑥₁ 𝑓_𝑥^󸀠󸀠₂_𝑥₂ 𝑓_𝑥^󸀠󸀠₂_𝑥₃ 𝑓_𝑥^󸀠󸀠₃_𝑥₁ 𝑓_𝑥^󸀠󸀠₃_𝑥₂ 𝑓_𝑥^󸀠󸀠₃_𝑥₃

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨< 0.

(38)

0 5 10 15

0 10 200 20 40 60 80

−10 −5

−10

Figure 1: Simulation of system (40).

0 5 10 15

0.5 0 1.5 1

0 0.5 1

−1.5

−0.5

−1

−2

The𝑓(𝜇, 𝑋)has the maximum𝑀. Differentiating the Lya- punov function𝑉(𝑋(𝑡))in (3) with respect to time𝑡along the trajectory of system (32) yields

̇𝑉 (𝑋 (𝑡))󵄨󵄨󵄨󵄨󵄨(32)= 𝜆₁(𝜆₁𝑥₁− 𝑑₁) ̇𝑥₁+ 𝜆₂(𝜆₂𝑥₂− 𝑑₂) ̇𝑥₂ + 𝜆₃(𝜆₃𝑥₃− 𝑑₃) ̇𝑥₃

≤ −𝜂∑³

𝑗=1

(𝜆_𝑗𝑥_𝑗− 𝑑_𝑗)²+ 𝑓 (𝜇, 𝑋)

≤ −𝜂∑³

𝑗=1

(𝜆_𝑗𝑥_𝑗− 𝑑_𝑗)²+ 𝑀⁺

≤ −2𝜂 [𝑉 (𝑋 (𝑡)) − 1

2𝜂𝑀⁺] ≤ 0, when𝑉 (𝑋 (𝑡)) > 1

2𝜂𝑀⁺.

(39)

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

(7)

0 50 100 50 0

100 0 50 100 150

−100 −50

−50

−100

0 20 40

0 50 1000

50 100 150

−50 −60 −40 −20 Figure 4: Simulation of system (45).

of new chaotic systems that the condition (6) is fulfilled and study the switching system between them.

Example 12. Consider a Lorenz system shown inFigure 1:

̇𝑥1= −12𝑥₁+ 5𝑥₂− 0.8𝑥₁𝑥₃+ 𝑥₂𝑥₃,

̇𝑥2= 28𝑥₁− 𝑥₂− 𝑥₁𝑥₃,

̇𝑥3= −3𝑥₂− 𝑥₃+ 10𝑥²₁+ 𝑥₁𝑥₂.

(40)

Solution. Here

𝑎₁₁= −12, 𝑎₁₂= 5, 𝑎₂₁= 28, 𝑎₂₂= −1, 𝑎₃₂= −3, 𝑎₃₃= −1, els𝑎_𝑖𝑗= 0, 𝑏₁₁₃= 𝑏₁₃₁= −0.4, 𝑏₁₂₃= 𝑏₁₃₂= 0.5, 𝑏₂₁₃= 𝑏₂₃₁= −0.5, 𝑏₃₁₁ = 10,

𝑏₃₁₂= 𝑏₃₂₁= 0.5, els𝑏_𝑖𝑗𝑘= 0, 𝜆₁= √12.5, 𝜆₂= √13.5, 𝜆₃= 1, 𝑑₁= −6

25, 𝑑₂= 4 45, 𝑑₃= 440.5, 𝜇₁= 2, 𝜇₂= 𝜇₃= 1

2, 𝜂 = 1 2, 𝑓₂(𝜇, 𝑋) = −125𝑥²₁− 6.75𝑥²₂− 0.5𝑥²₃

0 20 40

0 200 50 100 150 200

−40

−20

−40 −20 Figure 5: Simulation of system (46).

0 100

0 100 2000

50 100 150 200

−100 −200

−100

+ (−48

25√12.5 − 11245 √13.5) 𝑥₁ + (168

25 √12.5 + 26432 ) 𝑥₂+39293196661 405000 , 𝑓_2𝑥^󸀠 ₁(𝜇, 𝑋) = −250𝑥₁− (48

25√12.5 + 11245√13.5) , 𝑓_2𝑥^󸀠

1(𝜇, 𝑋) = 0, 𝑥₁= − 1 1250(48

5√12.5 + 1129 √13.5)

≈ 0.064, 𝑓_2𝑥^󸀠₂(𝜇, 𝑋) = −13.5𝑥₂+ (168

25 √12.5 + 26432 ) , 𝑓_2𝑥^󸀠₂(𝜇, 𝑋) = 0, 𝑥₂= 30

135(56

25√12.5 + 8812 ) ≈ 99.65, 𝑓_2𝑥^󸀠 ₃(𝜇, 𝑋) = −𝑥₃, 𝑓_2𝑥^󸀠₃(𝜇, 𝑋) = 0, 𝑥₃= 0,

𝑓_2𝑥^󸀠󸀠2

1(𝜇, 𝑋) = −250, 𝑓_2𝑥^󸀠󸀠2

2(𝜇, 𝑋) = −13.5, 𝑓_2𝑥^󸀠󸀠²₃(𝜇, 𝑋) = −1,

𝑓_2𝑥^󸀠󸀠₁_𝑥₂(𝜇, 𝑋) = 𝑓_2𝑥^󸀠󸀠₂_𝑥₁(𝜇, 𝑋) = 𝑓_2𝑥^󸀠󸀠₁_𝑥₃(𝜇, 𝑋) = 0, 𝑓_2𝑥^󸀠󸀠₃_𝑥₁(𝜇, 𝑋) = 𝑓_2𝑥^󸀠󸀠₂_𝑥₃(𝜇, 𝑋) = 𝑓_2𝑥^󸀠󸀠₃_𝑥₂(𝜇, 𝑋) = 0.

(41)

(8)

0 2 4 6 8 10 0

20 0 5 10 15

−5

−10 −20 −2

×10¹²

×10¹¹

(a) System (40)–(43)

0 50 100

50 0 100

0 50 100 150

−50 −50

−50

−100 −100 (b) System (40)–(44)

0 5

50 0 1000

50 100 150 200 250

−50 −100 −15 −10 −5 ×10¹² (c) System (40)–(45)

0 20 40

0 200 50 100 150 200

−20

−20 −40

−40

(d) System (40)–(46)

0 50

50 0 1000

50 100 150

−50 −50

−100 −100

−150 (e) System (40)–(47)

Figure 7: Switched system between system (40) and others.

The Hesse matrix of the𝑓(𝜇, 𝑋)is a negative definite matrix, max𝑓(𝜇, 𝑋) ≈ 164045.42. The set

Ω = {𝑋 | (√12.5𝑥₁+ 5

26)²+ (√13.5𝑥₂− 4 45)² +(𝑥₃− 440.5)²≤ 328090.84}

(42)

is the globally exponentially attractive set and positive invariant set of system (40).

Note. (a) If the Hesse matrix of the𝑓(𝜇, 𝑋)is not a negative definite matrix, the𝑓(𝜇, 𝑋)has no maximum𝑀.

(b) If∃𝑎_𝑖₀_𝑖₀ ≥ 0, lim_𝑥_𝑖0_{→ ∞}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 to𝑡along the trajectory of system (1). It is very useful to research these problems.

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0 2 4 6 8 0

10 20

0 5 10 15

−5

−10 −2

×10¹¹

×10¹²

(a) System (43)–(40)

0 2

2 0 4 0 2 4

−2 −2

−2

×10¹²

×10¹² ×10¹²

−8 −6 −4

−4

(b) System (43)-(44)

0 5

5 0 15 10

0 5 10 15

−5

×10¹¹

×10¹²

−15 −10 (c) System (43)–(45)

0 2 4 6 8

10 0 20

0 5 10 15

−5

−2

×10¹¹

×10¹²

−10 −20

(d) System (43)–(46)

0 2 4 6 8 10

5 0 150 10

0.5 1 1.5 2

−5

×10¹⁵

(e) System (43)–(47)

Figure 8: Switched system between system (43) and others.

Example 13. The new chaotic system shown inFigure 2is

̇𝑥1= −2𝑥₁+ 5𝑥₃+ 5.7𝑥₁𝑥₃+ 4.7𝑥₂𝑥₃+ 4,

̇𝑥2= −𝑥₁− 2𝑥₂+ 3𝑥₃+ 5,

̇𝑥3= −6𝑥₁− 2𝑥₂− 4𝑥₃+ 0.2𝑥²₁+ 3.4𝑥₁𝑥₂+ 7.

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Example 14. The chaotic system shown inFigure 3is

̇𝑥1= −11𝑥₁+ 0.15𝑥₁𝑥₂+ 1.38𝑥₂𝑥₃+ 1,

̇𝑥2= 30𝑥₁− 𝑥₂− 𝑥₁𝑥₃+ 0.1𝑥₂𝑥₃,

̇𝑥3= 15𝑥₂− 2.5𝑥₃+ 𝑥₁𝑥₂.

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0 50 100 50 0

100 0 50 100

−100 −50

−50 −100

−50

(a) System (44)–(40)

0 5

50 0 100

0 1 2 3

−5

−1

×10¹²

−20 −15 −10

−50 −100

(b) System (44)-(43)

0 1

50 0 100

0 50 100 150

−2

−50

−50 −100

×10¹³

−3

−1 (c) System (44)-(45)

0 50 100

50 0 100

0 200 400 600 800 1000

−200

−150 −100

−50

−50 −100

(d) System (44)–(46)

0 100 200

50 0 100

0 50 100 150

−200 −100

−50 −100

−50

(e) System (44)–(47)

Figure 9: Switched system between Example (44) and others.

̇𝑥1= 30 (𝑥₂− 𝑥₁) − 0.48𝑥²₁,

̇𝑥2= 80𝑥₁− 6𝑥₂− 𝑥₁𝑥₃,

̇𝑥3= 𝑥₁𝑥₂− 5𝑥₃.

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̇𝑥1= −12𝑥₁+ 5𝑥₂+ 𝑥₂𝑥₃,

̇𝑥2= 28𝑥₁− 𝑥₂− 𝑥₁𝑥₃,

̇𝑥3= −3𝑥₂− 𝑥₃+ 4𝑥²₁+ 𝑥₁𝑥₂.

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0 5 50 0

1000 50 100 150

−50 −10

−100

×10¹²

−15

−5 (a) System (45)–(40)

0 5

0 50 1000

1 2 3 4

−50 −20 −15

×10¹²

−10 −5 (b) System (45)–(43)

0 1

50 0 100

0 50 100 150

−2

−50

−50 −100

×10¹²

−3

−1 (c) System (45)-(44)

0 5

50 0 1000

50 100 150

−50 −10

−100

×10¹²

−15

−5 (d) System (45)-(46)

0 5

0 1000 50

50 100 150

−50 −10

−100

×10¹²

−15

−5

(e) System (45)–(47)

̇𝑥 = (207) 𝑥 − 𝑦𝑧 + 9,

̇𝑦 = −10𝑦 + 𝑥𝑧 + 0.5𝑧²,

̇𝑧 = −4𝑧 + 𝑥𝑦 + 𝑦𝑧.

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Note. When we analyse Examples13 to17 by the previous means, for sup_𝑋∈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 L¨u et al. chaotic system [11] andExample 17we 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 Examples12–17are 1.06, 0.02, 1.84, 0.01, 0.92, and 0.95, respectively.

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0 20 40 0

200 50 100 150 200

−20

−40 −40 ⁻²⁰ (a) System (46)–(40)

0 5 10 15 20

0 200

1 2 3 4

−20

−40 −5

×10¹²

(b) System (46)–(40)

0 50 100

50 0 100

0 200 400 600

−200

−100 −50

−50 −100

(c) System (46)–(44)

0 1

0 500 50 100 150 200

−50

−100 −3 −2 −1 ×10¹³

(d) System (46)-(45)

0 50

50 0 1000

50 100 150 200

−100 −150

−50 −100 −50

(e) System (46)-(47)

5. Simulation of Switched System

In this section, we will show some simulation results of the following switching system

̇𝑥 = 𝐴_𝜎𝑥 + 𝑓_𝜎(𝑥) + 𝐶_𝜎, (48) where𝑥^𝑇= (𝑥₁, 𝑥₂, 𝑥₃),𝜎is the switching law, and

𝐴_𝜎= [[ [

𝑎₁₁^𝜎 𝑎₁₂^𝜎 𝑎₁₃^𝜎 𝑎₂₁^𝜎 𝑎₂₂^𝜎 𝑎₂₃^𝜎 𝑎₃₁^𝜎 𝑎₃₂^𝜎 𝑎₃₃^𝜎 ]] ]

, 𝐶_𝜎= [[ [

𝑐₁^𝜎 𝑐₂^𝜎 𝑐₃^𝜎 ]] ] ,

𝑓_𝜎(𝑥) = [[ [

𝑥^𝑇𝐵^𝜎₁𝑥 𝑥^𝑇𝐵^𝜎₂𝑥 𝑥^𝑇𝐵^𝜎₃𝑥 ]] ]

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with𝑎_𝑖𝑗^𝜎, 𝑏_𝑖𝑗𝑘^𝜎, 𝑐_𝑖^𝜎 ∈ 𝑅, 𝑖, 𝑗, 𝑘 = 1, 2, 3. Each pair of(𝐴_𝜎, 𝐶_𝜎, 𝐵^𝜎₁, 𝐵^𝜎₂, 𝐵₃^𝜎) takes the form fromExample 8to Example 14. 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 Figures7to12that 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.

6. Conclusion

In this paper, the methods in [19–21] have been extended to study the globally exponentially or globally conditional