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

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.

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: 𝑥̇=𝐴𝑥+𝑓(𝑥)+𝐶, (1) where 𝑥^𝑇=(𝑥₁,𝑥₂,𝑥₃) and 𝑎𝐴=¹¹𝑎₁₂𝑎₁₃𝑎₂₁𝑎₂₂𝑎₂₃𝑎₃₁𝑎₃₂𝑎₃₃𝑐;𝐶=𝑐¹₂𝑐₃(𝑥)=𝑥;𝑓^𝑇𝐵₁𝑥𝑥^𝑇𝐵₂𝑥𝑥^𝑇𝐵₃𝑥;𝐵_𝑖=𝑏_𝑖11𝑏_𝑖12𝑏_𝑖13𝑏_𝑖21𝑏_𝑖22𝑏_𝑖23𝑏_𝑖31𝑏_𝑖32𝑏_𝑖33,𝑖=1,2,3 (2) with 𝑎_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))2𝜆₁𝑥₁−𝑑₁²+𝜆₂𝑥₂−𝑑₂²+𝜆₃𝑥₃−𝑑₃², (3) which is obviously positive definite and radially unbounded, where 𝑑_𝑖,𝜆_𝑖,  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 Section 2, 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, 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) set Ω∈𝑅³ such that for all 𝑋₀∈𝑅³, the following condition: 𝜌(𝑋(𝑡),Ω)∶=inf_𝑌∈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 all 𝑋₀∈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 of Ω 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 𝑉(𝑋(𝑡))−𝐿≤𝑉𝑋₀−𝐿𝑒^{−𝛼(𝑡−𝑡0)} (4) 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 𝑉(𝑋(𝑡))−𝐿𝑥₀≤𝑉𝑋₀−𝐿𝑥₀𝑒^{−𝛼(𝑡−𝑡0)} (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, 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 𝜆²₁𝑏₁₁₁=0,𝜆²₁𝑏₁₁₂+𝑏₁₂₁+𝜆²₂𝑏₂₁₁𝜆=0,₂₂𝑏₂₂₂=0,𝜆²₂𝑏₂₂₁+𝑏₂₁₂+𝜆²₁𝑏₁₂₂𝜆=0,₂₃𝑏₃₃₃=0,𝜆²₃𝑏₃₁₃+𝑏₃₃₁+𝜆²₁𝑏₁₃₃𝜆=0,₂₁𝑏₁₁₃+𝑏₁₃₁+𝜆²₃𝑏₃₁₁𝜆=0,₂₂𝑏₂₂₃+𝑏₂₃₂+𝜆²₃𝑏₃₂₂𝜆=0,₂₃𝑏₃₂₃+𝑏₃₃₂+𝜆²₂𝑏₂₃₃𝜆=0,₂₁𝑏₁₂₃+𝑏₁₃₂+𝜆²₂𝑏₂₁₃+𝑏₂₃₁+𝜆²₃𝑏₃₂₁+𝑏₃₁₂=0. (6)

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 ̇,̇̇𝑥=𝜎𝑦−𝑥𝑦=𝜌𝑥−𝛾𝑦−𝑥𝑧,𝑧=𝑥𝑦−𝛽𝑧, (7) and the general Lorenz systems ̇̇̇̇̇𝑥=−𝑎𝑥+𝑏𝑦+𝑦𝑧,𝑦=𝑐𝑥−𝑦−𝑥𝑧,𝑧=𝑑𝑦−𝑧+𝑥𝑦,𝑥=−𝑧,𝑦=−𝑦−𝑥²,𝑧̇=1.7𝑥+𝑦+1.7. (8)(9)

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, supposing 𝑏₁₁₁=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 𝑥̇₁=³_𝑗=1𝑎_1𝑗𝑥_𝑗+𝑏₁₂₃+𝑏₁₃₂𝑥₂𝑥₃+𝑐₁,𝑥̇₂=³_𝑗=1𝑎_2𝑗𝑥_𝑗+𝑏₂₁₃+𝑏₂₃₁𝑥₁𝑥₃+𝑐₂,𝑥̇₃=³_𝑗=1𝑎_3𝑗𝑥_𝑗+𝑏₃₁₂+𝑏₃₂₁𝑥₁𝑥₂+𝑐₃. (10)

The construction techniques of this kind of Lorenz systems are to pay attention to satisfing formula 𝜆²₁𝑏₁₂₃+𝑏₁₃₂+𝜆²₂𝑏₂₁₃+𝑏₂₃₁+𝜆²₃𝑏₃₂₁+𝑏₃₁₂=0, (11) where 𝜆_𝑖,  𝑖=1,2,3 are parameters and 𝑓𝜇,𝑋=³_𝑖=1𝜆²_𝑖𝑎_𝑖𝑖+𝜇_𝑖𝑥²_𝑖+³_𝑖=1𝜆²_𝑖𝑐_𝑖+𝜇_𝑖𝑑_𝑖𝜆_𝑖−𝜇_𝑖𝜆²_𝑖𝑑_𝑖−³_𝑗=1𝜆_𝑗𝑑_𝑗𝑎_𝑗𝑖𝑥_𝑖+³_𝑖=1𝜆_𝑖𝑑_𝑖𝑐_𝑖−𝜆_𝑖𝑑²_𝑖, (12) where 𝜇_𝑖,  𝑖=1,2,3 are undetermined parameters. And we always assume that the supremum 𝑓(𝜇,𝑋)<+∞ in the paper.

Lemma 6. Suppose 𝜆_𝑖>0,  𝑖=1,2,3, 𝜆²₁𝑎₁₂+𝜆²₂𝑎₂₁+𝜆₃𝑑₃𝑏₃₁₂+𝑏₃₂₁𝜆=0,₂₁𝑎₁₃+𝜆²₃𝑎₃₁+𝜆₂𝑑₂𝑏₂₁₃+𝑏₂₃₁𝜆=0,₂₂𝑎₂₃+𝜆²₃𝑎₃₂+𝜆₁𝑑₁𝑏₁₂₃+𝑏₁₃₂𝑎=0,₁₁+𝜇₁𝑎<0,₂₂+𝜇₂𝑎<0,₃₃+𝜇₃<0. (13) The function (12) has maximum.

Proof. Consider 𝑓^′_𝑥1𝜇,𝑋=2𝜆²₁𝑎₁₁+𝜇₁𝜆²₁𝑥₁+𝜆²₁𝑐₁+𝜇₁𝑑₁𝜆₁−𝜇₁𝜆²₁𝑑₁−𝜆₁𝑑₁𝑎₁₁−𝜆₂𝑑₂𝑎₂₁−𝜆₃𝑑₃𝑎₃₁,𝑓^′_𝑥2𝜇,𝑋=2𝜆²₂𝑎₂₂+𝜇₂𝜆²₂𝑥₂+𝜆²₂𝑐₂+𝜇₂𝑑₂𝜆₂−𝜇₂𝜆²₂𝑑₂−𝜆₁𝑑₁𝑎₁₂−𝜆₂𝑑₂𝑎₂₂−𝜆₃𝑑₃𝑎₃₂,𝑓^′_𝑥3𝜇,𝑋=2𝜆²₃𝑎₃₃+𝜇₃𝜆²₃𝑥₃+𝜆²₃𝑐₃+𝜇₃𝑑₃𝜆₃−𝜇₃𝜆²₃𝑑₃−𝜆₁𝑑₁𝑎₁₃−𝜆₂𝑑₂𝑎₂₃−𝜆₃𝑑₃𝑎₃₃,𝑓^′′_𝑥21𝜇,𝑋=2𝜆²₁𝑎₁₁+𝜇₁,𝑓^′′_𝑥22𝜇,𝑋=2𝜆²₂𝑎₂₂+𝜇₂,𝑓^′′_𝑥23𝜇,𝑋=2𝜆²₃𝑎₃₃+𝜇₃,𝑓^′′_𝑥1𝑥2𝜇,𝑋=𝑓^′′_𝑥2𝑥1𝜇,𝑋=𝑓^′′_𝑥1𝑥3𝜇,𝑋=𝑓^′′_𝑥2𝑥3𝜇,𝑋=𝑓^′′_𝑥3𝑥2𝜇,𝑋=0. (14)
Thus the Hesse matrix 𝐻_𝑓 of the 𝑓(𝜇,𝑋) is a negative definite matrix; furthermore max_𝑋∈𝑅³𝑓(𝜇,𝑋) exists.
These parameters 𝑑_𝑖,𝜇_𝑖,𝜆_𝑖,  𝑖=1,2,3 will be determined by solving the maximum of 𝑓₁(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))𝑀2𝜂⁺≤𝑉𝑋𝑡₀−1𝑀2𝜂⁺𝑒^{−2𝜂(𝑡−𝑡0)} (16) holds, and the set (𝑋(𝑡≤1Ω=𝑋∣𝑉))𝑀2𝜂⁺=𝑋∣𝜆₁𝑥₁−𝑑₁²+𝜆₂𝑥₂−𝑑₂²+𝜆₃𝑥₃−𝑑₃²≤1𝜂𝑀⁺ (17) is the globally exponentially attractive set and positive invariant set of system (10); that is, lim_𝑡→∞𝑉(𝑋(𝑡≤1))𝑀2𝜂⁺. (18)

Proof. Differentiating the Lyapunov function 𝑉(𝑋(𝑡)) in (3) with respect to time 𝑡 along the trajectory of system (10) yields 𝑉̇(𝑋(𝑡))₍₁₀₎=𝜆₁𝜆₁𝑥₁−𝑑₁𝑥̇₁+𝜆₂𝜆₂𝑥₂−𝑑₂𝑥̇₂+𝜆₃𝜆₃𝑥₃−𝑑₃𝑥̇₃=−³_𝑗=1𝜇_𝑗𝜆_𝑗𝑥_𝑗−𝑑_𝑗²+𝜆₁𝜆₁𝑥₁−𝑑₁׳_𝑗=1𝑎_1𝑗𝑥_𝑗+𝜇₁𝜆₁⁻¹𝜆₁𝑥₁−𝑑₁+𝑏₁₂₃+𝑏₁₃₂𝑥₂𝑥₃+𝑐₁+𝜆₂𝜆₂𝑥₂−𝑑₂׳_𝑗=1𝑎_2𝑗𝑥_𝑗+𝜇₂𝜆₂⁻¹𝜆₂𝑥₂−𝑑₂+𝑏₂₁₃+𝑏₂₃₁𝑥₁𝑥₃+𝑐₂+𝜆₃𝜆₃𝑥₃−𝑑₃׳_𝑗=1𝑎_3𝑗𝑥_𝑗+𝜇₃𝜆₃⁻¹𝜆₃𝑥₃−𝑑₃+𝑏₃₁₂+𝑏₃₂₁𝑥₁𝑥₂+𝑐₃≤−𝜂³_𝑗=1𝜆_𝑗𝑥_𝑗−𝑑_𝑗²+𝑓𝜇,𝑋≤−𝜂³_𝑗=1𝜆_𝑗𝑥_𝑗−𝑑_𝑗²+𝑀⁺𝑉(𝑋(𝑡−1≤−2𝜂))𝑀2𝜂⁺(𝑋(𝑡>1≤0,when𝑉))𝑀2𝜂⁺, (19) where 𝜇_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) 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  𝑎_𝑖𝑗=0, 𝑏_𝑖𝑗𝑘=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): 𝑉(𝑋(𝑡=1))2𝜆𝑥²₁+𝑥²₂+𝑥₃−𝜆𝜎−𝜌²,𝑓𝜇,𝑋=−𝛽−𝜂₁𝑥²₃+𝛽−2𝜂₁𝜆𝜎+𝜌𝑥₃+𝜂₁𝜆𝜎+𝜌². (20) We have 𝑀=𝛽²(𝜆𝜎+𝜌)²/4(𝛽−𝜂₁). Thus 𝑉(𝑋(𝑡−𝛽))²𝜆𝜎+𝜌²8𝛽−𝜂₁𝜂₁≤𝑉𝑋𝑡₀−𝛽²𝜆𝜎+𝜌²8𝛽−𝜂₁𝜂₁𝑒^{−2𝜂1(𝑡−𝑡0)},Ω₁=(𝑋)≤𝛽𝑋∣𝑉²𝜆𝜎+𝜌²8𝛽−𝜂₁𝜂₁=𝑋∣𝜆𝑥²₁+𝑥²₂+𝑥₃−𝜆𝜎−𝜌²≤𝛽²𝜆𝜎+𝜌²4𝛽−𝜂₁𝜂₁ (21) is the globally exponentially attractive set and positive invariant set of system (7).

Example 1. Further, taking ito accout 𝜇₁=𝜎,  𝜇₂=𝛾,  𝜇₃=𝛽/2,  𝜂=𝜂₂=min{𝜎,𝛾,𝛽/2}, the estimate 𝑉(𝑋(𝑡−𝛽))𝜆𝜎+𝜌²4𝜂₂≤𝑉𝑋𝑡₀−𝛽𝜆𝜎+𝜌²4𝜂₂𝑒^{−2𝜂2(𝑡−𝑡0)} (22) holds and that Ω₂=(𝑋)≤𝛽𝑋∣𝑉𝜆𝜎+𝜌²4𝜂₂=𝑋∣𝜆𝑥²₁+𝑥²₂+𝑥₃−𝜆𝜎−𝜌²≤𝛽𝜆𝜎+𝜌²2𝜂₂ (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 of 𝑑𝑉₁/𝑑𝑡 along the trajectory of system (16) lead to 𝑓𝑥𝜇,𝑋=−²³𝛽2+𝜆𝜎+𝜌²𝛽2,𝛽𝑀=𝜆𝜎+𝜌²2. (24) The conclusion of Example 2 is obtained.

Example 2. Furthermore, choose 𝜇₁=𝜎,  𝜇₂=𝛾,  𝜇₃=𝛽,  0<𝜉₁<𝛽,  𝜂=𝜂₃=min{𝜎,𝛾,𝜉₁}. Get 𝑓𝜇,𝑋=−𝛽−𝜂₃𝑥²₃+𝛽−2𝜂₃𝜆𝜎+𝜌𝑥₃+𝜂₂𝜆𝜎+𝜌²,𝛽𝑀=²𝜆𝜎+𝜌²4𝛽−𝜂₃. (25) Then, the estimate 𝑉(𝑋(𝑡−𝛽))²𝜆𝜎+𝜌²8𝛽−𝜂₃𝜂₃≤𝑉𝑋𝑡₀−𝛽²𝜆𝜎+𝜌²8𝛽−𝜂₃𝜂₃𝑒^{−2𝜂3(𝑡−𝑡0)} (26) holds and that Ω₃=(𝑋)≤𝛽𝑋∣𝑉²𝜆𝜎+𝜌²8𝛽−𝜂₃𝜂₃=𝑋∣𝜆𝑥²₁+𝑥²₂+𝑥₃−𝜆𝜎−𝜌²≤𝛽²𝜆𝜎+𝜌²4𝛽−𝜂₃𝜂₃ (27) is the globally exponentially attractive set and positive invariant set of system (7).

Example 3. Taking 𝑎₁₁=−𝑎,  𝑎₁₂=𝑏,  𝑎₂₁=𝑐,  𝑎₂₂=−1,  𝑎₃₂=𝑑,  𝑎₃₃=−1,  𝑏₁₂₃=𝑏₃₁₂=1,  𝑏₂₁₃=−1 else 𝑎_𝑖𝑗=0,  𝑏_𝑖𝑗𝑘=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): 𝑉(𝑋(𝑡=1))2𝑥₁−𝑑²+2𝑥²₂+𝑥₃−𝑏−2𝑐²,𝑓𝜇,𝑋=−𝑎−𝜇₁𝑥²₁+𝑎−2𝜇₁𝑑𝑥₁−21−𝜇₂𝑥²₂−2(𝑏+𝑐)𝑑𝑥₂−1−𝜇₃𝑥²₃+(𝑏+2𝑐)1−2𝜇₃𝑥₃+𝜇₁𝑑²+𝜇₃(𝑏+2𝑐)². (28) Thus 𝑀=𝑎−2𝜇₁²𝑑²4𝑎−𝜇₁+(𝑏+𝑐)²𝑑²21−𝜇₂+(𝑏+2𝑐)²1−2𝜇₃41−𝜇₃+𝜇₁𝑑²+𝜇₃(𝑏+2𝑐)². (29) We have 𝑉(𝑥(𝑡))−𝑀≤𝑉𝑥𝑡₀−𝑀𝑒^{−2𝜂(𝑡−𝑡0)}, (30) then Ω₄={(𝑥(𝑡}=𝑥𝑋∣𝑉))−𝑀𝑋∣₁−𝑑²+2𝑥²₂+𝑥₃−𝑏−2𝑐²≤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 𝑥̇₁=³_𝑗=1𝑎_1𝑗𝑥_𝑗+_𝑖=2,3𝑏_1𝑖𝑖𝑥²_𝑖+³_{𝑖≠𝑗=1}𝑏_1𝑖𝑗+𝑏_1𝑗𝑖𝑥_𝑖𝑥_𝑗+𝑐₁,𝑥̇₂=³_𝑗=1𝑎_2𝑗𝑥_𝑗+_𝑖=1,3𝑏_2𝑖𝑖𝑥²_𝑖+³_{𝑖≠𝑗=1}𝑏_2𝑖𝑗+𝑏_2𝑗𝑖𝑥_𝑖𝑥_𝑗+𝑐₂,𝑥̇₃=³_𝑗=1𝑎_3𝑗𝑥_𝑗+_𝑖=1,2𝑏_3𝑖𝑖𝑥²_𝑖+³_{𝑖≠𝑗=1}𝑏_3𝑖𝑗+𝑏_3𝑗𝑖𝑥_𝑖𝑥_𝑗+𝑐₃. (32) In this case, we can take into account 𝑓𝜇,𝑋=³_𝑖=1𝜆²_𝑖𝑎_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+³_𝑖<𝑗,1𝜆²_𝑖𝑎_𝑖𝑗+𝜆²_𝑗𝑎_𝑗𝑖+𝜆₁𝑑₁𝑏_1𝑖𝑗+𝑏_1𝑗𝑖+𝜆₂𝑑₂𝑏_2𝑖𝑗+𝑏_2𝑗𝑖+𝜆₃𝑑₃𝑏_3𝑖𝑗+𝑏_3𝑗𝑖𝑥_𝑖𝑥_𝑗+³_𝑖=1𝜆²_𝑖𝑐_𝑖−𝜇_𝑖𝑑_𝑖+𝜇_𝑖𝜆_𝑖𝑑_𝑖−𝑎_1𝑖𝜆₁𝑑₁−𝑎_2𝑖𝜆₂𝑑₂−𝑎_3𝑖𝜆₃𝑑₃𝑥₁+³_𝑖=1𝜆_𝑖𝑑_𝑖𝑐_𝑖−𝑑_𝑖, (33) where [⋅]₃ denotes modulo-3.

Theorem 8. Suppose that 𝐺₀=(𝑥⁰₁,𝑥⁰₂,…,𝑥⁰_𝑛) 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 𝑉(𝑋(𝑡−1))𝑀2𝜂⁺≤𝑉𝑋𝑡₀−1𝑀2𝜂⁺𝑒^{−2𝜂(𝑡−𝑡0)} (34) holds; that is, lim_𝑡→∞𝑉(𝑋(𝑡≤1))𝑀2𝜂⁺, (35) and the set (𝑋(𝑡≤1Ω=𝑋∣𝑉))𝑀2𝜂⁺=𝑋∣𝜆₁𝑥₁−𝑑₁²+𝜆₂𝑥₂−𝑑₂²+𝜆₃𝑥₃−𝑑₃²≤1𝜂𝑀⁺ (36) is the globally exponentially attractive set and positive invariant set of system (32).

Proof. If 𝐺₀ is the stable point of the 𝑓(𝜇,𝑋), that is, ∇𝑓_𝐺0=𝑓^′_𝑥1,𝑓^′_𝑥2,𝑓^′_𝑥3=0, (37) and the Hesse matrix 𝐻_𝑓 of the 𝑓(𝜇,𝑋) is a negative definite matrix, namely, 𝑓^′′_𝑥1𝑥1𝑓<0,^{′′𝑥1𝑥1}𝑓^′′_𝑥1𝑥2𝑓^′′_𝑥2𝑥1𝑓^′′_𝑥2𝑥2𝑓>0,_{′′𝑥1𝑥1}𝑓^′′_𝑥1𝑥2𝑓^′′_𝑥1𝑥3𝑓^′′_𝑥2𝑥1𝑓^′′_𝑥2𝑥2𝑓^′′_𝑥2𝑥3𝑓^′′_𝑥3𝑥1𝑓^′′_𝑥3𝑥2𝑓^′′_𝑥3𝑥3<0. (38) 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 𝑉̇(𝑋(𝑡))₍₃₂₎=𝜆₁𝜆₁𝑥₁−𝑑₁𝑥̇₁+𝜆₂𝜆₂𝑥₂−𝑑₂𝑥̇₂+𝜆₃𝜆₃𝑥₃−𝑑₃𝑥̇₃≤−𝜂³_𝑗=1𝜆_𝑗𝑥_𝑗−𝑑_𝑗²+𝑓𝜇,𝑋≤−𝜂³_𝑗=1𝜆_𝑗𝑥_𝑗−𝑑_𝑗²+𝑀⁺𝑉(𝑋(𝑡−1≤−2𝜂))𝑀2𝜂⁺(𝑋(𝑡>1≤0,when𝑉))𝑀2𝜂⁺. (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 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 Figure 1: 𝑥̇₁=−12𝑥₁+5𝑥₂−0.8𝑥₁𝑥₃+𝑥₂𝑥₃,𝑥̇₂=28𝑥₁−𝑥₂−𝑥₁𝑥₃,𝑥̇₃=−3𝑥₂−𝑥₃+10𝑥²₁+𝑥₁𝑥₂. (40)
Solution. Here 𝑎₁₁=−12,𝑎₁₂=5,𝑎₂₁=28,𝑎₂₂=−1,𝑎₃₂𝑎=−3,₃₃=−1,els𝑎_𝑖𝑗=0,𝑏₁₁₃=𝑏₁₃₁𝑏=−0.4,₁₂₃=𝑏₁₃₂=0.5,𝑏₂₁₃=𝑏₂₃₁=−0.5,𝑏₃₁₁𝑏=10,₃₁₂=𝑏₃₂₁=0.5,els𝑏_𝑖𝑗𝑘=0,𝜆₁=√𝜆12.5,₂=√13.5,𝜆₃=1,𝑑₁6=−25,𝑑₂=4,𝑑45₃=440.5,𝜇₁=2,𝜇₂=𝜇₃=121,𝜂=2,𝑓₂𝜇,𝑋=−125𝑥²₁−6.75𝑥²₂−0.5𝑥²₃+−48√2512.5−112√4513.5𝑥₁+168√2512.5+26432𝑥₂[phone],𝑓405000_′2𝑥1𝜇,𝑋=−250𝑥₁−48√2512.5+112√45,𝑓13.5_′2𝑥1𝜇,𝑋=0,𝑥₁1=−1250485√12.5+1129√𝑓13.5≈0.064,_′2𝑥2𝜇,𝑋=−13.5𝑥₂+168√2512.5+26432,𝑓^′_2𝑥2𝜇,𝑋=0,𝑥₂=3013556√2512.5+8812𝑓≈99.65,_′2𝑥3𝜇,𝑋=−𝑥₃,𝑓^′_2𝑥3𝜇,𝑋=0,𝑥₃𝑓=0,_{′′2𝑥21}𝜇,𝑋=−250,𝑓^′′_2𝑥22𝑓𝜇,𝑋=−13.5,_{′′2𝑥23}𝑓𝜇,𝑋=−1,_{′′2𝑥1𝑥2}𝜇,𝑋=𝑓^′′_2𝑥2𝑥1𝜇,𝑋=𝑓^′′_2𝑥1𝑥3𝑓𝜇,𝑋=0,_{′′2𝑥3𝑥1}𝜇,𝑋=𝑓^′′_2𝑥2𝑥3𝜇,𝑋=𝑓^′′_2𝑥3𝑥2𝜇,𝑋=0. (41) The Hesse matrix of the 𝑓(𝜇,𝑋) is a negative definite matrix,  max𝑓(𝜇,𝑋)≈164045.42. The set Ω=𝑋∣√12.5𝑥₁+526²+√13.5𝑥₂−445²+𝑥₃−440.5²≤328090.84 (42) 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 ∃𝑎_𝑖0𝑖0≥0,  lim_{𝑥𝑖0→∞}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 Figure 2 is 𝑥̇₁=−2𝑥₁+5𝑥₃+5.7𝑥₁𝑥₃+4.7𝑥₂𝑥₃𝑥̇+4,₂=−𝑥₁−2𝑥₂+3𝑥₃𝑥̇+5,₃=−6𝑥₁−2𝑥₂−4𝑥₃+0.2𝑥²₁+3.4𝑥₁𝑥₂+7. (43)

Figure 2 

Simulation of system (43).

Example 6. The chaotic system shown in Figure 3 is 𝑥̇₁=−11𝑥₁+0.15𝑥₁𝑥₂+1.38𝑥₂𝑥₃𝑥̇+1,₂=30𝑥₁−𝑥₂−𝑥₁𝑥₃+0.1𝑥₂𝑥₃,𝑥̇₃=15𝑥₂−2.5𝑥₃+𝑥₁𝑥₂. (44)

Figure 3 

Simulation of system (44).

Example 7. The chaotic system shown in Figure 4 is 𝑥̇₁=30𝑥₂−𝑥₁−0.48𝑥²₁,𝑥̇₂=80𝑥₁−6𝑥₂−𝑥₁𝑥₃,𝑥̇₃=𝑥₁𝑥₂−5𝑥₃. (45)

Figure 4 

Simulation of system (45).

Example 8. The chaotic system shown in Figure 5 is 𝑥̇₁=−12𝑥₁+5𝑥₂+𝑥₂𝑥₃,𝑥̇₂=28𝑥₁−𝑥₂−𝑥₁𝑥₃,𝑥̇₃=−3𝑥₂−𝑥₃+4𝑥²₁+𝑥₁𝑥₂. (46)

Figure 5 

Simulation of system (46).

Example 9. The chaotic system shown in Figure 6 is 𝑥̇=207̇𝑥−𝑦𝑧+9,𝑦=−10𝑦+𝑥𝑧+0.5𝑧²,𝑧̇=−4𝑧+𝑥𝑦+𝑦𝑧. (47)

Figure 6 

Simulation of system (47).

Note. When we analyse Examples 5 to 9 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 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 𝑥̇=𝐴_𝜎𝑥+𝑓_𝜎(𝑥)+𝐶_𝜎, (48) where 𝑥^𝑇=(𝑥₁,𝑥₂,𝑥₃), 𝜎 is the switching law, and 𝐴_𝜎=𝑎^𝜎₁₁𝑎^𝜎₁₂𝑎^𝜎₁₃𝑎^𝜎₂₁𝑎^𝜎₂₂𝑎^𝜎₂₃𝑎^𝜎₃₁𝑎^𝜎₃₂𝑎^𝜎₃₃,𝐶_𝜎=𝑐^𝜎₁𝑐^𝜎₂𝑐^𝜎₃,𝑓_𝜎(𝑥)=𝑥^𝑇𝐵^𝜎₁𝑥𝑥^𝑇𝐵^𝜎₂𝑥𝑥^𝑇𝐵^𝜎₃𝑥 (49) with 𝑎^𝜎_𝑖𝑗,𝑏^𝜎_𝑖𝑗𝑘,𝑐^𝜎_𝑖∈𝑅,  𝑖,𝑗,𝑘=1,2,3. Each pair of (𝐴_𝜎,𝐶_𝜎,𝐵^𝜎₁,𝐵^𝜎₂,𝐵^𝜎₃) 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.

(a) System (40)–(43)

(b) System (40)–(44)

(c) System (40)–(45)

(d) System (40)–(46)

(e) System (40)–(47)

Figure 7 

Switched system between system (40) and others.

(a) System (43)–(40)

(b) System (43)-(44)

(c) System (43)–(45)

(d) System (43)–(46)

(e) System (43)–(47)

Figure 8 

Switched system between system (43) and others.

(a) System (44)–(40)

(b) System (44)-(43)

(c) System (44)-(45)

(d) System (44)–(46)

(e) System (44)–(47)

Figure 9 

Switched system between Example (44) and others.

(a) System (45)–(40)

(b) System (45)–(43)

(c) System (45)-(44)

(d) System (45)-(46)

(e) System (45)–(47)

Figure 10 

Switched system between Example (45) and others.

(a) System (46)–(40)

(b) System (46)–(40)

(c) System (46)–(44)

(d) System (46)-(45)

(e) System (46)-(47)

Figure 11 

Switched system between Example (46) and others.

(a) System (47)–(40)

(b) System (47)–(43)

(c) System (47)–(44)

(d) System (47)–(45)

(e) System (47)-(46)

Figure 12 

Switched system between Example (47) and others.

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.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities, China Postdoctoral Science Foundation funded project under Grant 2012M511615, National Natural Science Foundation of China under Grants 60474011 and 60904005, and the Hubei Provincial Natural Science Foundation of China under Grant 2009CDB026.