#### 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: where and with , . This second-order dynamic system may be regarded as the most general Lorenz system. For such system, we can choose Lyapunov function: which is obviously positive definite and radially unbounded, where , 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 and .

*Definition 1. *For the three-dimensional autonomous systems with cross-product nonlinearities (1), if there exists compact (bounded and closed) set such that for all , the following condition: as , holds, then the set 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 , , then for 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 such that for all and constants , such that , 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 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 , a constant , and a bounded function on , as , such that , where , , 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 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 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 and positive numbers , such that the following inequality
is valid for , then the system (1) is said to be globally exponentially attractive or globally exponentially stable in the sense of Lagrange, and 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 and a bounded function , on , as , and such that the following inequality
is valid for , , where , , then the system (1) is said to be globally conditional exponentially attractive or globally conditional exponentially stable in the sense of Lagrange, and 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 so as to satisfy conditions

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 and the general Lorenz systems

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 , , , , , the dynamic system (1) can be rewritten as

The construction techniques of this kind of Lorenz systems are to pay attention to satisfing formula where , are parameters and where , are undetermined parameters. And we always assume that the supremum in the paper.

Lemma 6. *Suppose , ,
**
The function (12) has maximum. *

*Proof. *Consider

Thus the Hesse matrix of the is a negative definite matrix; furthermore exists.

These parameters , will be determined by solving the maximum of and formula (12), and let

Theorem 7. *If condition (6) exists, , , , , then the estimation
**
holds, and the set
**
is the globally exponentially attractive set and positive invariant set of system (10); that is,
*

*Proof. *Differentiating the Lyapunov function in (3) with respect to time along the trajectory of system (10) yields
where , . 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 results in inequality (18).

Now, the characters of some of the chaotic systems known are analysed by condition (6). When , , , , , , , else , , , and , , , , , , , , , system (10) can be rewritten as system (7):
We have . Thus
is the globally exponentially attractive set and positive invariant set of system (7).

*Example 1. *Further, taking ito accout , , , , the estimate
holds and that
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 along the trajectory of system (16) lead to
The conclusion of Example 2 is obtained.

*Example 2. *Furthermore, choose , , , , . Get
Then, the estimate
holds and that
is the globally exponentially attractive set and positive invariant set of system (7).

*Example 3. *Taking , , , , , , , else , , , and , , , , , system (6), , and can be rewritten as system (8):
Thus
We have
then
is the estimation of the globally exponentially attractive and positive invariant sets of system (8).

If , , , the dynamic system (1) is shown as
In this case, we can take into account
where denotes modulo-3.

Theorem 8. *Suppose that is the stable point of the defined by (33). If the Hesse matrix of the is a negative definite matrix, the has maximum and the estimation
**
holds; that is,
**
and the set
**
is the globally exponentially attractive set and positive invariant set of system (32).*

*Proof. *If is the stable point of the , that is,
and the Hesse matrix of the is a negative definite matrix, namely,
The has the maximum . Differentiating the Lyapunov function in (3) with respect to time along the trajectory of system (32) yields

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:
*Solution*. Here
The Hesse matrix of the is a negative definite matrix, . The set
is the globally exponentially attractive set and positive invariant set of system (40).*Note*. (a) If the Hesse matrix of the is not a negative definite matrix, the has no maximum .

(b) If , , 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, is a cubic polynomial and there is not maximum if we choose energy function (3) differentiating this Lyapunov function with respect to along the trajectory of system (1). It is very useful to research these problems.

*Example 5. *The new chaotic system shown in Figure 2 is

*Example 6. *The chaotic system shown in Figure 3 is

*Example 7. *The chaotic system shown in Figure 4 is

*Example 8. *The chaotic system shown in Figure 5 is

*Example 9. *The chaotic system shown in Figure 6 is

*Note*. When we analyse Examples 5 to 9 by the previous means, for , 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 where , is the switching law, and with , . Each pair of 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)**

**(a) System (43)–(40)**

**(b) System (43)-(44)**

**(c) System (43)–(45)**

**(d) System (43)–(46)**

**(e) System (43)–(47)**

**(a) System (44)–(40)**

**(b) System (44)-(43)**

**(c) System (44)-(45)**

**(d) System (44)–(46)**

**(e) System (44)–(47)**

**(a) System (45)–(40)**

**(b) System (45)–(43)**

**(c) System (45)-(44)**

**(d) System (45)-(46)**

**(e) System (45)–(47)**

**(a) System (46)–(40)**

**(b) System (46)–(40)**

**(c) System (46)–(44)**

**(d) System (46)-(45)**

**(e) System (46)-(47)**

**(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 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.