#### Abstract

Two ellipsoidal ultimate boundary regions of a special three-dimensional (3D) chaotic system are proposed. To this chaotic system, the linear coefficient of the th state variable in the th state equation has the same sign; it also has two one-order terms and one quadratic cross-product term in each equation. A numerical solution and an analytical expression of the ultimate bounds are received. To get the analytical expression of the ultimate boundary region, a new result of one maximum optimization question is proved. The corresponding ultimate boundary regions are demonstrated through numerical simulations. Utilizing the bounds obtained, a linear controller is proposed to achieve the complete chaos synchronization. Numerical simulation exhibits the feasibility of the designed scheme.

#### 1. Introduction

Bounded chaotic systems and their ultimate bounds are important for chaos synchronization and chaos control [1–3]. But it is generally difficult to obtain the ultimate bound of a chaotic system or the analytical expression of the bound even if the chaotic system has simple dynamic differential equations. The well-known Lorenz chaotic system was presented in 1963 [4]. It is a 3D autonomous system with only two quadratic terms. In 1987, a cylindrical bound and a spherical bound for the globally attractive and positive invariant sets of Lorenz system were proposed by Leonov et al. [5, 6]. Since then, several ultimate boundaries of Lorenz system have been obtained, like another cylindrical bound [7], the improved spherical bound [8], the ellipsoidal bounds [9–11], the butterfly bound [12], and so on [13–15]. References [10, 11] also discussed the ellipsoidal ultimate bounds of the unified Lorenz system [16]. The ultimate boundaries for other well-known chaotic attractors, such as Chen attractor [17], Lü attractor [18], and Qi attractor [19], were also proposed [20–22].

Since the research for the ultimate bounds set of chaotic systems is restricted by the region of the coefficients of the systems, in [20, 21], the ultimate boundary regions of the chaotic systems were researched only in several designated parameters regions. The ultimate boundaries of many existing chaotic systems are still not presented. So, it is also a challenging work to search the ultimate bounds of some new 3D chaotic systems [1, 2, 23–26] and hyperchaotic systems [27–29]. Recently, using the optimization idea and the Lyapunov method, which are often applied to estimate the boundaries of chaotic systems [1, 8, 10, 22, 27, 28], Wang et al. [30] constructed a special method to find the ultimate boundaries of a class of high dimensional autonomous quadratic chaotic systems. In the following parts, this method is called the unified method. Wang et al. [30] solved the ultimate boundary problem of more existing chaotic attractors and hyperchaotic attractors and got the numerical solutions of corresponding bounds. But the unified method is not applied successfully to every existing chaotic system.

In this paper, the following 3D chaotic system which was introduced by Tang et al. [31] in 2012 is considered: where , , are state variables and , , , , , , and . Every state equation has two one-order terms and one quadratic cross-product term. System (1) has complex dynamic behaviors and several larger chaotic coefficient’s regions. It has a typical chaotic attractor when , , , , , , and .

To system (1), the method used in [1, 8, 10, 22, 27, 28] to find the boundary of chaotic attractor does not seem very suitable. One can notice that the coefficients of the th state variable in the th equation have the same sign and they are negative. Under this special condition, the unified method [30] to find the boundary of chaotic attractor can be applied to system (1). In this paper, the unified method [30] is used to get the numerical solution of the ultimate bound of system (1) with , , , , , , and . Moreover, to get the analytical expression of the ellipsoidal ultimate boundary of system (1), a new conclusion about a designated maximum optimization question is proved. Utilizing this result, an analysis expression of the ellipsoidal ultimate boundary is given when the coefficients of the chaotic system . The boundary is useful in the control or synchronization of chaos. Using the boundary set gained, one can realize the complete chaos synchronization.

The rest of the paper includes four sections. Section 2 introduces the unified approach [30] and proposes a new theorem about an interesting analytic solution of a maximum optimization problem. Utilizing the new theorem above and the unified method, Section 3 estimates the ellipsoidal ultimate boundary regions of system (1). Some numerical simulations about the boundary regions are exhibited. Section 4 applies the bound in chaos synchronization. Section 5 provides the conclusions.

#### 2. Some Preliminaries and Notations

The unified method constructed in [30] to estimate the ultimate boundary of chaotic attractor is introduced firstly.

The considered autonomous system is described as where , : . Let be the solution satisfying with the initial time and initial state and let be a compact set. The distance between and is defined by Denote . Obviously, .

*Definition 1 (see [10, 30]). *Suppose that there exists a compact set satisfying
for all . It means that, for any , there exists satisfying for all . Then, the set is called an ultimate bound of system (2).

Consider the HDQADS [30, 32], described by where , , , and . Also, all elements of satisfy .

Construct a general quadratic function candidate [30] where , , and are real parameters to be determined.

Calculating the derivative of (6) along with system (5) [30], one can get where , .

Hereafter, the meaning of is that the matrix is positive definite and of is that is negative definite.

Lemma 2 (see [30]). *If there exists a and a such that
**
for any , then the boundness of system (5) is proved and the ultimate boundary region is
**
where which can be determined by solving the optimization problem:
*

The conditions (8) are sufficient but not necessary [30].

Since the symmetry of , then can be transformed into a positive definite radially unbounded Lyapunov function via .

For simplification, let . One has , . After a simple calculation, one can rewrite Lemma 2 as follows.

Lemma 3. *If there exists a real symmetric matrix and a vector such that
**
for any , then the boundness of system (5) is proved and the ultimate boundary region is
**
where which can be determined by solving the optimization problem:
**
where .*

Theorem 4. *Denote the set
**
and , . Then
*

*Proof. *Let = + + . Notice that = , = if and only if .

Now, define
Let
From (17), or . From (18), , = . From (19), , = . From (18), (19), and , one gets . From (20), .

(i) When , substituting into (20), one obtains + = = ; that is, . Then, one gets or and two equilibria and . Since , obviously,

(ii) When and , (18)–(20) have the following solutions: , , , and
Notice that is not able to be zero. In fact, if , by , one has ; this is a contradiction.

When , one has

Since is a closed set and is continuous on , the extreme values of can be attained on . Then, from (i) and (ii), one can achieve

The proof is complete.

#### 3. The Ultimate Bound Set of Chaotic System (1)

In the following, Lemma 3 and Theorem 4 are applied to estimate the ultimate bounds of the 3D chaotic system (1).

Rewrite system (1) into the form of system (5); then, one has

Let , . According to (11) in Lemma 2, calculate Since holds for any , letting one gets For simplifying , let that is, then So, one has

From Lemma 3, the next theorem is achieved.

Theorem 5. *Suppose that , , , , , , , , and . Denote
**
where . Then, is the ultimate bound set of system (1). can be found by calculating the maximum optimization question:
*

*Proof. *Since , , , one gets , . According to (13), one obtains the Lyapunov-like quadratic function
and its derivative along with system (1)
From Lemma 3, the above conclusion holds.

*Remark 6. *It is generally difficult to get the analytic solution of the optimization problem (35). But, by using Lingo, it is very easy to solve the optimization problem (35) numerically for the fixed system parameters. For example, obviously, one has , for , , , , , , , , , and . With the appointed parameters, utilizing Lingo to deal with the optimization problem (35), one gets the corresponding ultimate boundary region of system (1) as follows:
Figure 1 exhibits the ultimate boundary set of the chaotic strange attractor of system (1) under , , and .

Furthermore, to simplify the constraint condition of the maximum optimization problem (35), let the coefficient of be equal to 0. That is, . Then, if , one can solve the maximum problem (35) analytically.

Theorem 7. *Suppose that , , , , , , and , , and . Then, system (1) possesses following ultimate bound:
*

*Proof. *When
one has
According to (44) and Theorem 5, one obtains
and the following maximum problem
The above optimization problem is rewritten by
Denote , , , , and . By , the corresponding maximum problem is described by
Set . Since , then . Then, with the new result in Theorem 4 and (42) in problem (46), has the maximum . According to Lemma 3, Theorem 5, (41)–(43), and , system (1) gets the ellipsoidal ultimate boundary region as follows:
The proof is complete.

*Remark 8. *If one lets , system (1) still possesses a large range of chaos. Through Theorem 7, the analytic expression of the ultimate bound can be acquired easily. For example, when , , , , , and , the corresponding ellipsoidal ultimate boundary set of (1) is gained as
which is demonstrated clearly in Figure 2.

#### 4. Application in Chaos Synchronization

Consider two nonlinear autonomous systems
where , , , *∈*, and . means the nonnegative real set. Let (51) be the drive system and let (52) be the response system. means the controller function. , are the initial values of (51), (52).

*Definition 9. *The driver system (51) and the response system (52) are called to achieve global complete synchronization, if for any initial values , .

Next, let system (1) be the driver system. Design the controller . So, the response system to system (1) is described as follows: where , , are state variables and are all controller parameters which can be adjusted.

Theorem 10. *The driver system (1) and the response system (53) are globally complete synchronization when
**
where , , , , , , and .*

*Proof. *Let . From Theorem 5, one has , , and . Let the state errors be , , and , then the error dynamics of system (1) and system (53) is

Noticing the formula (28), one has . Let ; then its time derivative along the orbit of system (57) is
where ,
with
When , . One can draw that the origin of the error system (57) is asymptotically stable, which implies that the driver system (1) and the response system (53) achieve globally complete synchronization.

*Remark 11. *The numerical simulations are studied by MATLAB 7.6.0. Take and as the values of the initial condition of system (1) and system (53), respectively. When , , , , , , , , , and , from Remark 6, one gets , , , and . By Theorem 10, one can choose the three feedback control coefficients as , , and . Figure 3 proves that the response system realizes synchronization with the driver system through a short time.

#### 5. Conclusion

In this paper, the ultimate boundary regions of a special 3D chaotic system are studied through a unified method for the ultimate boundary set estimating of chaotic systems. In this unified way, to get the analytical expression of the ultimate boundary region, the key is to calculate the analytical solution of the maximum optimization problem. Furthermore, an interesting result about the analytic solution of the corresponding maximum optimization problem is proposed to obtain the analytic ellipsoidal ultimate boundary regions of the chaotic system. The ultimate bounds which are useful in chaos synchronization are demonstrated through numerical simulations.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was partially supported by the National Natural Science Foundation of China through Grant no. 11102226, the Fundamental Research Funds for the Central Universities through Grant nos. ZXH2010D011, ZXH2012B003, and ZXH2012K002, and the Scientific Research Foundation of Civil Aviation University of China through Grant no. 07QD05X.