Ultimate Bound of a 3D Chaotic System and Its Application in Chaos Synchronization

Two ellipsoidal ultimate boundary regions of a special three-dimensional (3D) chaotic system are proposed. To this chaotic system, the linear coefficient of the 𝑖 th state variable in the 𝑖 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.


Introduction
Bounded chaotic systems and their ultimate bounds are important for chaos synchronization and chaos control [1][2][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][10][11], the butterfly bound [12], and so on [13][14][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][21][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][24][25][26] and hyperchaotic systems [27][28][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.
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 ( = 1, 2, 3) 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 > 0, > 0, > 0, > 0, > 0, > 0, and > 0. 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.
Lemma 2 (see [30]). If there exists a ∈ × > 0 and a ∈ such that for any = ( 1 , 2 , . . . , ) ∈ , then the boundness of system (5) is proved and the ultimate boundary region is where max ∈ which can be determined by solving the optimization problem: The conditions (8) are sufficient but not necessary [30].
For simplifying , let that is, From Lemma 3, the next theorem is achieved.
which is demonstrated clearly in Figure 2.
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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.