Bounds for a new chaotic system and its application in chaos synchronization

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Abstract

This paper has investigated the localization problem of compact invariant sets of a new chaotic system with the help of the iteration theorem and the first order extremum theorem. If there are more iterations, then the estimation for the bound of the system will be more accurate, because the shape of the chaotic attractor is irregular. We establish that all compact invariant sets of this system are located in the intersection of a ball with two frusta and we also compute its parameters. It is a great advantage that we can attain a smaller bound of the chaotic attractor compared with the classical method. One numerical example illustrating a localization of a chaotic attractor is presented as well.

Introduction

Since the discovery of the Lorenz chaotic system in 1963 [1], chaos has been studied extensively. After that, many chaotic systems have been developed such as Rössler system [2], Chen system [3], and Lü system [4]. And these chaotic systems have been widely studied [5], [6], [7], [8], [9]. Efforts of many researchers have aimed towards investigations of bifurcations, chaos and related control problems for new chaotic systems. A noticeable progress has been achieved in spite of substantial complexity of mathematical models of such systems. The paper [10] and [11] has devoted to analysis the pitchfork and Hopf bifurcations based on using bifurcations theory and the central manifold theorem, obtaining an approximate stability boundary and some other topics. Recently, theoretical efforts have been performed to find localization domains containing all compact invariant sets of a nonlinear continuous-time system possessing complex behavior. Mainly, this problem has been examined for a number of mathematical models of different physical processes exhibiting chaos. Here, we recall studies of the system describing dynamics of the nuclear spin generator [12] and the permanent-magnet motor system [13]; bounds for a domain containing all compact invariant sets obtained in these papers in many cases can be used not only for theoretical studies of chaotic attractors, e.g. but also for estimating for the Hausdorff Dimension [14], or for numerical search of attractors. The problem of a localization of compact invariant sets of polynomial with the help of using the first order extremum condition has drawn much attention in the last decade, see [15], more recent papers also talk about it [13], [16], [17], [21]. Here by a localization we mean a description of s set containing all compact invariant sets of the system under consideration in terms of equalities and inequalities that defined in the state space. This paper is devoted to computing the bounds for a compact domain which contains all compact invariant sets of the system defining the behavior of the new chaotic system.

The structure of this paper is as follows. In Section 2, we give some notations and preliminaries. In Sections 3 Ellipsoidal localization with precise bounds, 4 Applications of the iteration theorem, 5 Additional localizing functions, we give an ellipsoidal localization, some polytope Π1 and two-parametric set of parabolic cylinders, and we denote the sign γ(α) the radius for the ellipsoid. At the same time, our work mainly focus in these sections (Sections 3 Ellipsoidal localization with precise bounds, 4 Applications of the iteration theorem, 5 Additional localizing functions). In Section 6, we give one example of the localization of a chaotic attractor for the chaotic system by MATLAB 7.0. In Section 7, we study synchronization between the driver system and the response system. Section 8 gives simulations about synchronization error of the two systems under linear feedback control. Section 9 contains conclusions.

Recently, Tang Liang Rui, Li Jing, Fan Bing, Zhai Ming-Yue introduced the following new system [18]:x˙=-ax+byy˙=cx-xz-dyz˙=xy-e(x+z)where a, b, c, d, e are parameters, when a = 25.6, b = 66.8, c = 39.22, d = 0.2, e = 4, system (1) displays a typical attractor (see [18]).

Some basic dynamical properties were studied in [18]. But many properties of this new system remains to be uncovered. In the following, we will discuss the compact invariant sets of the new chaotic system with the help of the iteration theorem and the first order extremum theorem.

Section snippets

Some notations and preliminaries

Let us introduce a real polynomial right-side systemx˙=f(x).Here x  Rn is the state vector. Let us take a real polynomial h of n real variables which is not a first integral of (2). The function h is used in the solution of the localization problem of compact invariant sets and is called localizing. By hB we denote the restrictions of h on a set B  Rn. By Lfh we denote the Lie derivative of the function h, where Lf(h(x)) is the Lie derivative [19]:Lfh(x)=i=1nfi(x)h(x)xi,(f1(x),f2(x),,fn(x))T=

Ellipsoidal localization with precise bounds

In what follows in this section, we derive two ellipsoidal localizations. The first one is obtained by computing the precise bound for h1,sup, withVλ(x,y,z)=h1(x,y,z)=12(λx2+(y-e)2+(z-bλ-c)2),Lλ=λb2e22d(2a-d)+(e-d)2(bλ+c)22d(2e-d)+e22+(bλ+c)22,where λ is a positive real constant (that is to say λ > 0).

When a>d2,e>d2,c>0,d>0,e>0, we can get an estimation of the globally exponentially attractive set for the system (1), given byVλ(X(t))-Lλ[Vλ(X(t0))-Lλ]e-d(t-t0).Especially, the setΩλ=X|λx2+(y-e)2+(z

The second step of localization

We note that, K(h1) is contained in the polytope Π1 defined by|x|γ(α);|y|e+γ(α);|z|b+c+γ(α).Let us take h2(x, y, z) = x, then the set S(h2) is given by −ax + by = 0 andh2|S(h2)=bay.Therefore the setK(h1)S(h2)=x2+(y-e)2+(z-b-c)2γ2(α);x=bay=(x,y,z)|a2+b2b2x-abea2+b22+(z-b-c)2γ2(α)-b2e2a2+b2.So we can attain the following result|x|abea2+b2+b2γ2(α)-b2e2a2+b2a2+b2,andK1,2:=|x|abea2+b2+b2γ2(α)-b2e2a2+b2a2+b2.It is clearly that the latter can be refined by the formulaK1,2:=|x|minγ(α);abea2+b2+b2γ2(α)-

Additional localizing functions

Now we can apply additional localizing function, let us takeh4(x,y,z)=y2+z2-2ey-2cz-4px,wherep=14a.Then the set S(h4) is described by the equation-2dy2-2ez2+4apx+(2de-4pb)y+2cez=0.Therefore by using this formula we geth4|S(h4)=a-2day+2ade-b-2ea22a(a-2d)2+a-2eaz+-ca+cea-2e2-(2ade-b-2ea2)24a3(a-2d)-(ca-ce)2a(a-2e).Then we get the following formula for the localizing set:

If a > 2d, a > 2e, then we haveK(h4):=y2+z2-2ey-2cz-1axR(a,c,e),R(a,c,e)=-(2ade-b-2ea2)24a3(a-2d)-(ca-ce)2a(a-2e).

If 0 < a < 2d, 0 < a < 2e

One example of the localization of a chaotic attractor for the system

In this section, we describe one example of the localization of a chaotic attractor which was found in [18]. Parameters of this chaotic system are chosen a = 25.6, b = 66.8, c = 39.22, d = 0.2, e = 4. For convenience, the figure by Theorem 5 is omitted. We depict some ellipsoid K(h1) (according to Theorem 4: ellipsoidal localization with precise bounds) see Fig. 1.

Synchronization of the new chaotic system

Let the system (1) be the driver system, and the response system is:x˙1=-ax1+by1-k1(x1-x)y˙1=cx1-x1z1-dy1-k2(y1-y)z˙1=x1y1-e(x1+z1)System (1) can synchronize the system (10) by adjusting parameter k1, k2. From Theorem 6, then we can get the boundness of y, z that is |y|2L0+|e|,|z|2L0+|c|+|b| and we deonte L0=b2e22d(2a-d)+(e-d)2(b+c)22d(2e-d)+e22+(b+c)22. For convenience, let us denote My=2L0+|e|,2L0+|c|+|b|=Mz, then we have the following theorem.

Theorem 6

System (1) and system (10) are globally

Simulation studies

The numerical simulations are carried out using the MATLAB 7.4. The initial conditions of the driver and response systems are (1, 1, 2) and (2, 1.5, 3.5). When a = 25.6, b = 66.8, c = 39.22, d = 0.2, e = 4 [18], it is easy to obtain L0 = 8665, then the bound of the coefficients of feedback control k1, k2 could be obtained according to the Theorem 6. Choose k1 = 50, k2 = 10. The response system synchronizes with the drive system as shown in Fig. 2.

Conclusions

In this paper, we study the localization problem of compact invariant sets of the new chaotic system with the help of the iteration theorem and the first order extremum theorem. Conclusions about iteration theorem is not obtained from results of this paper, it was obtained in papers of Luis N. Coria and Konstantin E. Starkov. Firstly, we estimate the positively invariant set and globally exponentially attractive set using Definition 1 in Ref. [20]. Secondly, we use the iteration theorem and the

Acknowledgements

Project supported by the Foundation item the National Nature Youth Foundation of China (No. 10 601071) and Natural Science Foundation Project of CSTC (No. 2009BB3185). At the same time, we also thank the reviewers and the editors for the helpful suggestions and criticism that helped in improving this paper.

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