On the study of globally exponentially attractive set of a general chaotic system

Transcription

1 Available online at Communications in Nonlinear Science and Numerical Simulation 1 (008) On the stud of globall exponentiall attractive set of a general chaotic sstem P. Yu a, *, X.X. Liao b,1 a Department of Applied Mathematics, The Universit of Western Ontario, London, Ont., Canada N6A 5B7 b Department of Control Science and Engineering, Huazhong Universit of Science and Technolog, Wuhan, Hubei 40074, PR China Received November 006; accepted November 006 Available online 19 December 006 Abstract In this paper, we prove that there exists globall exponential attractive and positive invariant set for a general chaotic sstem, which does not belong to the known Lorenz sstem, or the Chen sstem, or the Lorenz famil. We show that all the solution orbits of the chaotic sstem are ultimatel bounded with exponential convergent rates and the convergent rates are explicitl estimated. The method given in this paper can be applied to stud other chaotic sstems. Ó 006 Elsevier B.V. All rights reserved. PACS: 0.0.Hq Kewords: Chaotic attractor; Globall exponentiall attractive set; Ultimate boundedness of chaos 1. Introduction Since the discover of the Lorenz chaotic attractor [1] in 196, man other chaotic sstems have been found, including the well-known Rössler sstem [], Chua s circuit [], which have been served as models for stud of chaos. In late 1990s, a new chaotic sstem was found, which is a dual sstem to the Lorenz sstem, and now known as the Chen sstem [4]. Due to its close relation to the Lorenz sstem and importance, the Chen sstem has been widel studied (e.g., see [5 7] and references therein). The ultimate boundedness of a chaotic sstem is ver important for the stud of the qualitative behaviour of a chaotic sstem. If one can show that a chaotic sstem under consideration has a globall attractive set, then one knows that the sstem cannot have equilibrium points, periodic or quasi-periodic solutions, or other chaotic attractors existing outside the attractive set. This greatl simplifies the analsis of dnamics of the sstem. The ultimate boundedness also plas a ver important role in the designs of chaos control and chaos * Corresponding author. Fax: address: pu@pu1.apmaths.uwo.ca (P. Yu). 1 Currentl, the author is also with School of Automatics, Wuhan The Universit of Science and Technolog, Wuhan, Hubei 40070, China /$ - see front matter Ó 006 Elsevier B.V. All rights reserved. doi: /j.cnsns

2 1496 P. Yu, X.X. Liao / Communications in Nonlinear Science and Numerical Simulation 1 (008) snchronization. The ultimate boundedness propert of the Lorenz sstem has been investigated b man researchers (e.g., see [8 14]). However, so far ver little has been achieved on other chaotic sstems, regarding the propert of ultimate boundedness. A smooth Chua s circuit has been studied and estimation on its ultimate boundedness has been obtained [15]. For the Chen sstem, a recent article [16] investigated its propert of ultimate boundedness, but the parameter values considered in this article does not cover the most interesting case of the Chen s chaotic attractor. Consider the following general sstem: _x ¼ að xþ; _ ¼ dx xz þ c; _z ¼ x bz; where the dot denotes differentiation with respect to time t; a, b, c and d are parameters. This general sstem contains man famous chaotic sstems, including the Lorenz sstem (c = 1), the Chen sstem (d = c a) and the Lü sstem (d = 0). In this paper, we generall assume that a >0,c > 0 and d ( 1,1). Since for the known chaotic attractors, the values of b are b ¼ 8 for the Lorenz attractor, b = for the Chen attractor, b ¼ 44 for the Lü attractor, and b 15 ½8 ; Š for the Lorenz famil, we pa particular attention in this paper to ð1þ z x z x z x Fig. 1. The chaotic attractors of sstem (1) for a = 40, b = 0.5, c = 5, projected on the x z plane, when: (a) d = 10; (b) d = 0; and (c) d = 10.

3 b (0, 1]. To illustrate that the sstem can exhibit chaos for b (0,1] and for different values of d, three chaotic attractors of sstem (1) are shown in Fig. 1 when a = 40, b = 0.5, c = 5 while d = 10, 0, 10. This clearl shows that these three chaotic attractors do not belong to an of the Lorenz, Chen and Lü sstems, and thus it is necessar to investigate the dnamical propert of this sstem. Some definitions and lemmas are given in Section which will be used for proving the main theorem in Section. The conclusion is drawn in Section 4.. Preliminaries P. Yu, X.X. Liao / Communications in Nonlinear Science and Numerical Simulation 1 (008) In this section, we present some basic definitions and two lemmas which are needed for proving the main theorem in the next section. For convenience, let X :¼ (x,,z) andx(t) :¼ X(t,t 0,X 0 ). Definition 1. For the general chaotic sstem (1), if there exists compact (bounded and closed) set X R such that "X 0 R, the following condition: qðx ðtþ; XÞ :¼ inf kx ðtþ k!0 as t!þ1 X holds, then the set X is said to be globall attractive. That is, sstem (1) is ultimatel bounded, namel, sstem (1) is globall stable in the sense of Lagrange or dissipative with ultimate bound. Further, if "X 0 X 0 X R, X(t,t 0,X 0 ) X 0, then X 0 is called the positive invariant set of sstem (1). Definition. For the general chaotic sstem (1), if there exists compact set X R such that "X 0 R,and constants M(X 0 )>0,a > 0 such that qðx ðtþ; XÞ 6 MðX ðt 0 ÞÞe aðt t0þ ; then sstem (1) is said to have globall exponentiall attractive set, or sstem (1) is globall exponentiall stable in the sense of Lagrange, and X is called the globall exponentiall attractive set. Both globall attractive set and globall exponentiall attractive set are positive invariant sets. Note that it is difficult to verif the existence of X in Definition. Since the Lapunov direct method is still a powerful tool in the stud of asmptotic behaviour of nonlinear dnamical sstems, the following definition is more useful in applications. Definition. For the general chaotic sstem (1), if there exists a positive definite and radiall unbounded Lapunov function V(x), and positive numbers L > 0, a > 0 such that the following inequalit ðv ðxðtþ LÞ 6 ðv ðx 0 Þ LÞe aðt t 0Þ is valid for V(X(t)>L (t P t 0 ), then sstem (1) is said to be globall exponentiall attractive or globall exponentiall stable in the sense of Lagrange, and X :¼ {XjV(t) 6 L, t P t 0 } is called the globall exponentiall attractive set. In order to give a complete proof (in the next section) for the globall exponential boundedness of the general chaotic sstem, we need the following lemmas. Lemma 1. Let X(t) = (x(t), (t), z(t)) be an solution of sstem (1), and be an arbitrar small positive number (0 < 1). Then, we have the following results. (i) The boundedness of (t) implies the boundedness of x(t), i.e., if lim t!1 j(t)j 6 M(M> 0) but jx(t)j > M, then jx(t)j monotonicall decreases as t increases, and jxðtþj 6 M þ ðþ holds for sufficientl large t. (ii) If j(t)j P M > 0 but jx(t)j < M, then jx(t)j monotonicall increases as t increases, and jxðtþj P M ðþ holds for sufficientl large t.

4 1498 P. Yu, X.X. Liao / Communications in Nonlinear Science and Numerical Simulation 1 (008) Proof. (i) For the first equation of sstem (1), construct the positive definite and radiall unbounded Lapunov function V ¼jxj: ð4þ Then, D þ V ¼ D þ jxj 6 ajxjþajj 6 ajxjþam ¼ aðjxj MÞ < 0 when jxj > M; from which we obtain the differential equation: dðjxj MÞ jxj M 6 adt ðx 6¼ 0Þ and its solution is given b jxðtþj 6 M þðjxðt 0 Þj MÞe at ; ð5þ where jx(t 0 )j > M. Hence, jx(t)j 6 M + for sufficientl large t. When x = 0, it is eas to see from the first equation of (1) that an trajector of the sstem cannot sta on the z plane (x = 0) and must cross the plane, as long as 5 0. (ii) Suppose (t) P M > 0, but jx(t)j < M. Then from the first equation of sstem (1) we obtain 6 adt. Thus, the solu- dxðtþ dt which, in turn, results in the differential equation dðm xðtþþ dt tion of x(t) is M xðtþ 6 ðm xðt 0 ÞÞe at ; i.e., xðtþ P M ðm xðt 0 ÞÞe at : ¼ axðtþþaðtþ P axðtþþam ¼ aðm xðtþþ > 0 when jxðtþj < M; 6 aðm xðtþþ or dðm xðtþþ M xðtþ ð6þ This implies that x(t) P M for sufficientl large t, since jx(t 0 )j < M. If, on the other hand, (t) 6 M < 0, but jx(t)j < M. Then, similarl from the first equation of sstem (1) we have dx ¼ axðtþþaðtþ 6 axðtþ am ¼ aðxðtþþmþ < 0 when jxðtþj < M; dt which ields dðxðtþþmþ 6 aðxðtþþmþ, or dðxðtþþmþ 6 adt. The solution is x(t) +M 6 (x(t dt xðtþþm 0)+M)e at, i.e., xðtþ P M ðxðt 0 ÞþMÞe at : ð7þ This clearl indicates that x(t) P M for sufficientl large t. Combining the above two results shows that the conclusion of (ii) is true. h Lemma. jj reaches its maximum value on the ellipse: E : þðz dþ ðz dþ ¼R 1 ; ð8þ as jj max ¼ p ffiffi R 1. Proof. A straightforward calculation shows that þðz dþ ðz dþ ¼ ðz dþ 1 þ 4 ¼ R 1 : Thus, at the two points ð; zþ ¼ p ffiffi R 1 ; d p ffiffi R 1, the maximum value of jj is reached as jj max ¼ p ffiffi R 1. h Remark 1. Based on Lemma 1(i), we onl need to show that (t)andz(t) are ultimatel bounded, and then the boundedness of the variable x(t) follows. B Lemma 1(ii), we know that jx(t)j can be sufficientl large as long as j(t)j is sufficientl large.

5 P. Yu, X.X. Liao / Communications in Nonlinear Science and Numerical Simulation 1 (008) The main theorem and its proof In this section, based on a combination of geometric and algebraic methods, we prove the main theorem for the ultimate boundedness of the general chaotic sstem (1). Theorem 1. When a > 0, 0 < b 6 1, c > 0 and d ( 1,1), the general chaotic sstem (1) has globall exponentiall attractive set, i.e., all solution orbits of sstem (1) are ultimatel bounded. Proof. Based on Lemma 1(i), we onl need to prove that and z are ultimatel bounded. Furthermore, it is noted that sstem (1) is smmetric with respect to (x,), i.e., substituting (x,) with( x, ) does not change the sstem. Therefore, we onl need to consider x > 0 and the case x < 0 can be directl concluded from the smmetr. qffiffiffiffiffiffiffiffiffiffiffiffi c Let H ¼ b þ 1. For convenience, we define five lines: L 1; : z ¼H þjdj; L ;4 : z ¼H jdj; L 5 : z ¼ 1 : ð9þ Note that H > 1 and that the five lines L 1,L,...,L 5, together with the and z axes, divide the z plane into eight regions: (I), (I 0 ), (II), (II 0 ), (III), (III 0 ), (IV) and (IV 0 ). A total of six positive definite and radiall unbounded Lapunov functions are needed for these regions (see Fig. ): V 1 ¼ 1 ½ þðz dþ Š for ðiþ and ði 0 Þ; V ¼ þ 1 z for ðiiþ; V 0 ¼ V ¼ 1 z for ðii0 Þ; ð10þ V ¼ 1 ½ þðz dþ ðz dþš for ðiiiþ and ðiii 0 Þ; V 4 ¼ Hz for ðivþ; V 0 4 ¼ V 4 ¼ þhz for ðiv 0 Þ: In the following, we follow the order of regions given in Eq. (10) to complete the proof. For convenience, denote the points P i ( i,z i ) and P 0 i ð0 i ; z0 i Þ b P i and P 0 i, respectivel. Fig.. Geometric distribution with Lapunov functions.

6 1500 P. Yu, X.X. Liao / Communications in Nonlinear Science and Numerical Simulation 1 (008) Region (I) [ (I 0 ):{z ± H P jdj} [ {z ± H 6 jdj}. The region is shown in Fig.. Consider the positive definite and radiall unbounded Lapunov function V 1 ¼ 1 ½ þðz dþ Š; which represents a famil of circles on the z plane centered at the point (0,d). Then, differentiating V 1 with respect to time t and evaluating the derivative on the solution of sstem (1) ields dv 1 dt ¼ _ þðz dþ_z ¼ dx xz þ c þðx bzþðz dþ ¼c bz þ bdz ð1þ ¼ b ½ þðz dþ Šþ b ½ þðz dþ Šþc bz þ bdz ¼ bv 1 b z c b þ 1 d 6 bv 1 when z c b þ 1 P d : Note that the equation z c b þ 1 ¼ d represents a hperbola with two asmptotes: z = ±H (see Figs. and ). Next, consider the following circle: C : þðz dþ ¼ R P 4d ; ð1þ and let P 1 and P 10 be the intersection points of the circle with the z-axis, and P, P 0, P 9 and P 0 9 be the intersection points of the circle with the four lines L 1, L, L and L 4. Then, the coordinates of the six intersection points are given b P 1 ð 1 ; z 1 Þ¼ð0; d þ RÞ; P 10 ð 10 ; z 10 Þ¼ð0; d RÞ; 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ðjdj dþh þ ð1 þ H ÞR ðjdj dþ P ð ; z Þ¼P ; jdjþdh þ H ð1 þ H ÞR ðjdj A; 1 þ H 1 þ H 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ðjdjþdþh þ ð1 þ H ÞR ðjdjþdþ P 9 ð 9 ; z 9 Þ¼P 9 ; jdjþdh H ð1 þ H ÞR A; 1 þ H 1 þ H P 0 ð0 ; z0 Þ¼P 0 ð ; z Þ; P 0 9 ð0 9 ; z0 9 Þ¼P 0 9 ð 9; z 9 Þ: ð1þ It is obvious that the points P and P 0, the points P 9 and P 0 9 are smmetric, respectivel, about the z-axis (see Fig. ). Thus, in Region (I) [ (I 0 ), when V 1 ðx ðtþþ > R ðt P t 0Þ, there exists exponential estimation given as follows: V 1 ðx ðtþþ 6 V 1 ðx ðt 0 ÞÞe bðt t0þ : ð14þ Now define the set X 1 :¼fð; zþjðz H P jdjþ [ ðz H 6 jdjþ; þðz dþ 6 R g: ð15þ Then an trajector located in Region (I) [ (I 0 ), but outside X 1 must enter either into X 1 or into the neighboring Region (II) [ (II 0 ) [ (IV) [ (IV 0 ), as shown in Fig.. Remark. The case d = 0 becomes simpler since the four half lines z =±H ± jdj emerge into two lines passing through the origin. Then the circle C is reduced to a point (the origin) when R = d, see Fig. (c). ð11þ

7 P. Yu, X.X. Liao / Communications in Nonlinear Science and Numerical Simulation 1 (008) a z= H+ d z P 1 z=h+ d P P Ω 1 P z=/ P P 4 Ω P 5 P 4 Ω P 6 P 5 Ω d d P 7 P 6 P 7 Ω Ω 1 P 8 P 9 P 9 P 10 P 8 z=h d z= H d b z= H+ d z z=h+ d P P P 1 P Ω 1 P Ω P 4 P 5 z=/ P 4 d Ω P 5 Ω d P 6 P 7 P 6 P 7 Ω P 8 Ω 1 P 9 P 8 P 9 z=h d P 10 z= H d c z P = P z=h z=/ P 4 = P 5 P 6 = P 7 Ω Ω P = P P 8= P 9 Ω Ω P 4 = P 5 P 6 = P 7 P 8 = P 9 z=h Fig.. Trapping regions for x > 0: (a) d > 0; (b) d < 0; and (c) d =0.

8 150 P. Yu, X.X. Liao / Communications in Nonlinear Science and Numerical Simulation 1 (008) Next, we consider region (II) [ (II). To simplif the analsis for this region and the region (IV) [ (IV 0 ), define þ1 for the regions ðiiþ and ðivþ; d ¼ ð16þ 1 for the regions ðii 0 Þ and ðiv 0 Þ: Region (II) [ (II 0 ): z P 0; 1 jj 6 jzj 6 Hjjþjdj. Let n 1 ¼ ½ðc þ 1 þjdjþ þ ðh þ 1Þð1 bþš and g 1 ¼ maxfn 1 þ 1; jdjg: ð17þ So, b Lemma 1(ii) we know that when jj P g 1, x P n 1 as t 1. Consider the positive definite and radiall unbounded Lapunov function V ¼ d þ 1 z ; from which we obtain dv dt ¼ d _ þ 1 ð1þ _z ¼ d þ 1 z þ ¼ d þ 1 z þ 1 z þ dx xz þ c þ 1 ðx bzþ þ ddx xdz þðcþ1þd þ 1 xd þ 1 ð1 bþdz ¼ V þ ddx xjzjþðcþ1þjjþ 1 xjjþ1 ð1 bþjzj 6 V þjdjx 1 xjjþðcþ1þjjþ1 xjjþ1 ð1 bþðhjjþjdjþ 6 V þ 1 6 x þjdj x þðcþ1þþ 1 jj Hð1 bþþ1 jdj ð1 bþ jj jj 6 V 1 6 x ðcþ1þjdjþ 1 ðh þ 1Þð1 bþ jj ¼ V 1 6 ðx n 1Þjj 6 V when x P n 1 : ð18þ This implies that when jj P g 1 (and so x P n 1 ), V (X(t))>0(tP t 0 ), there exists exponential estimation for Region (II) [ (II 0 ): V ðx ðtþþ 6 V ðx ðt 0 ÞÞe ðt t0þ : ð19þ Denote the intersection points of the two lines dð þ 1 zþ¼k 1 ðk 1 > 0Þ with the lines z = H± jdj and z ¼ 1 as P, P 4, P 0 7 and P 0 8, which are given b ¼ k 1 jdj þ H > 0; 4 ¼ 6 7 k 1; 0 7 ¼ 4 ¼ 6 7 k 1; 0 8 ¼ ¼ k 1 jdj þ H : It is clearl shown in Fig. that the minimal value of j j, j 4 j, j 0 7 j and j0 8 j,isj j¼j 0 8 j¼k 1 jdj. Letting ¼ g þh 1 leads to k 1 ¼ 1 þ 1 H g 1 þ 1 jdj; ð0þ k 1 jdj þh and thus the four intersection points are P ð ; z Þ¼P ðg 1 ; Hg 1 þjdjþ; P 4 ð 4 ; z 4 Þ¼P 4 7 ðð þ HÞg þjdjþ; 1 7 ðð þ HÞg þjdjþ ; P 0 7 ð0 7 ; z0 7 Þ¼P ðð þ HÞg þjdjþ; 1 7 ðð þ HÞg þjdjþ ; P 0 8 ð0 8 ; z0 8 Þ¼P 0 8ð g 1; Hg 1 jdjþ: ð1þ

9 P. Yu, X.X. Liao / Communications in Nonlinear Science and Numerical Simulation 1 (008) Then, define the set X :¼ ð; zþjz P 0; 1 jj 6 jzj 6 Hjjþjdj; d þ 1 z 6 k 1 ðþ for Region (II) [ (II 0 ). Therefore, an trajector located in Region (II) [ (II 0 ), but outside X must enter either into X or into the neighboring Region (I) [ (I 0 ) [ (III) [ (III 0 ) (see Fig. ). Region (III) [ (III): fz P 0; jzj 6 1 jjg. Let n ¼ ð6c b þ 1 þ 4jdjÞ and g ¼ max n þ 1; jdj; R pffiffi : ðþ Then, it follows from Lemma 1(ii) that x P n when jj P g as t 1. For Region (III) [ (III 0 ), construct the positive definite and radiall unbounded Lapunov function V ¼ 1 ½ þðz dþ ðz dþš; which denotes a famil of ellipses on the z plane centered at the point (0, d). Then, we have dv dt ¼ _ þðz dþ_z 1 ð1þ ðz dþ_ 1 _z ¼ c bz þ bdz 1 ðz dþðdx xz þ cþ 1 ðx bzþ ¼ ½ þðz dþ ðz dþš þ d þ½ þðz dþ ðz dþš d þ c þ bdz 1 x þ 1 xz þ d x dxz þ cd þ b z bz c z ¼ V þ d 1 x þ 1 xz þ d ðc þ Þd x dxz þ þ b z ð bþdz þðcþ1þ þð1 bþz c þ z 6 V þ d 1 x þ 1 xz þ d þ Þ x þjdjxjzjþðc jdjjj þ b z þð bþjdjjzjþðcþ1þ þð1 bþz 6 V þ d þ 1 x þ 1 8 x þ d x þ jdj x þ jj jj jj jdj þð bþ þðcþ1þþ 1 b jj 4 6 V þ d þ 8 x þ jdj þ jdj ðc þ Þ þ þ b 4 þ b 6 V þ d 8 x ð6c b þ 1 þ 4jdjÞ ðc þ Þjdj þ b jj 4 þ c þ 1 þ 1 b 4 ¼ V þ d 8 ðx n Þ 6 V 1 d when x P n : ð4þ Therefore, in Region (III) [ (III 0 ), when jj P g (and so x P n ), V ðx ðtþþ > d ðt P t 0Þ, we obtain the exponential estimation: V ðx ðtþþ d 6 V ðx ðt 0 ÞÞ d e ðt t 0Þ : Now we need to determine the four intersection points of the ellipse þðz dþ ðz dþ ¼k P d with the line z ¼ 1 and the -axis: P 5( 5,z 5 ), P 6 ( 6,z 6 ), P 0 5 ð0 5 ; z0 5 Þ and P 0 6 ð0 6 ; z0 6Þ such that the minimum of the absolute values of the four -coordinates is not less than g. It is eas to see that this ellipse has two intersection ð5þ

10 1504 P. Yu, X.X. Liao / Communications in Nonlinear Science and Numerical Simulation 1 (008) points with the -axis (one positive and one negative) since k P d, and thus it also has two intersection points with the line z ¼ 1, with one in the first quadrant and the other in the third quadrant. The coordinates of these four intersection points are 5 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi k d; 6 ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d þ 4k d ; 0 5 ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 4k d ; 0 6 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi k d: Then choosing the smallest one as g ields qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi min p ffiffiffi k d; 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k d jdj ¼ g ; from which we obtain k ¼ max g þ d þ g jdj; 4 g þ d ¼ g þ d þ g jdj: ð6þ Hence, the four intersection points are given b P 5 ð 5 ; z 5 Þ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi g þ g 1 jdj; p ffiffi g þ g jdj ; P 6 ð 6 ; z 6 Þ¼ g þ 1 ðjdj dþ; 0 ; P 0 5 ð0 5 ; z0 5 Þ¼ g 1 ðjdjþdþ; 0 ; P 0 6 ð0 6 ; z0 6 Þ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi g 1 þ g 1jdj; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi g þ g jdj : ð7þ Define a set inside Region (III) [ (III 0 )as X :¼ ð; zþjz P 0; jzj 6 1 jj; þðz dþ ðz dþ 6 k : ð8þ Then, an trajector located in Region (III) [ (III 0 ), but outside X must ultimatel enter either into X or into the neighboring Region (II) [ (II 0 ) [ (IV) [ (IV 0 ). (See Fig..) Region (IV) [ (IV 0 ): { z6 0,jzj 6 Hjj + jdj}. Let c þ 1 þ jdjþhðh þ 1Þð1 bþ n ¼ and g H ¼ maxfn þ 1; jdjg: ð9þ So, when jj P g, x P n holds as t 1 due to Lemma 1(ii). Consider the positive definite and radiall unbounded Lapunov function V 4 ¼ dð HzÞ: Then, we have dv 4 dt ¼ dð_ H _zþ ¼d½ð þ HzÞþð HzÞþdx xz þ c Hx þ HbzŠ ð1þ ¼ dð HzÞþddx xdz þðcþ1þd Hxd Hð1 bþdz 6 V 4 þjdjxþxjzj Hxjjþðcþ1ÞjjþHð1 bþjzj 6 V 4 þjdjxþxðhjjþjdjþ Hxjjþðcþ1ÞjjþHð1 bþðhjjþjdjþ x 6 V 4 þ Hx þ jdj þðcþ1þþh ð1 bþþhð1 bþ jdj jj jj jj 6 V 4 ½Hx jdj ðcþ1þ HðH þ 1Þð1 bþšjj ¼ V 4 Hðx n Þjj 6 V 4 when x P n : ð0þ

11 P. Yu, X.X. Liao / Communications in Nonlinear Science and Numerical Simulation 1 (008) This clearl indicates that when jj P g (and so x P n ), V 4 (X(t)) > 0 (t P t 0 ), there exists estimation for Region (IV) [ (IV 0 ): V 4 ðx ðtþþ 6 V 4 ðx ðt 0 ÞÞe ðt t 0Þ : ð1þ Let the intersection points of the two lines d( H z)=k (k > 0) with the lines z = H ± jdj and the - axis be P 7, P 8, P 0 and P 0 4 (see Fig. ), then a similar discussion as that for Region (II) [ (II0 ) leads to k ¼ð1 þ H Þg þ Hjdj; ðþ and then the four points are described as P 7 ð 7 ; z 7 Þ¼P 7 ðð1 þ H Þg þ Hjdj; 0Þ; P 8 ð 8 ; z 8 Þ¼P 8 ðg ; Hg jdjþ; P 0 ð0 ; z0 Þ¼P 0 ð g ; Hg þjdjþ; P 0 4 ð0 4 ; z0 4 Þ¼P 0 4 ð ð1 þ H Þg Hjdj; 0Þ: ðþ z= H+ d z z=h+ d P P 1 P P Ω 1 z=/ P P 4 Ω P 5 P 4 Ω P 6 P 5 Ω d d P 7 P 7 P 6 Ω P 9 P 8 z=h d P 8 Ω 1 P 9 P 10 z= H d z= H+ d z z=h+ d P P1 P P z=/ Ω1 P Ω P 4 P 5 P 4 P 5 Ω d d Ω P 6 P 7 P 7 Ω P 8 P 6 Ω1 P 9 P 8 P10 P 9 z=h d z= H d Fig. 4. Closed trapping region X for x > 0: (a) d > 0 and (b) d <0.

12 1506 P. Yu, X.X. Liao / Communications in Nonlinear Science and Numerical Simulation 1 (008) Similarl, we define the set X 4 :¼ fð; zþjz 6 0; jzj 6 Hjjþjdj; dð HzÞ 6 k g ð4þ for Region (IV) [ (IV 0 ). Then, an trajector located in Region (IV) [ (IV 0 ), but outside X must either enter into X or into the neighboring Region (I) [ (I 0 ) [ (III) [ (III 0 ), as shown in Fig.. The remaining task is to obtain a closed trapping region from the above defined regions X k, k = 1,,,4. It should be pointed out that Fig. is onl a general illustration. When the parameters a, b, c and d are varing, the intersection points P n, n =1,,...,10, 0, 0,...,9 0 ma change their relative positions. The boundaries include arcs of circles, arcs of ellipses, and line segments. Since the slopes of the lines P P 4, and P 0 7 P 0 8 are fixed as and that for the lines P 0 P 0 4 and P 7P 8 are fixed as 1 H, in general one cannot adjust them to get all the boundar cures (lines) to be connected. However, it is eas to show that if the two slopes of the lines are allowed to be varied, then one can easil make these non-closed boundaries to be closed b first choosing the farthest point among the 18 points and then determine all the other points accordingl. However, the easiest wa to obtain a closed trapping region is to draw a closed curve which encloses all the ten boundar curves (lines), as shown in Fig. 4 for d < 0 and d > 0. When d = 0, the distribution of the regions are smmetric about the origin, and thus the closed boundar can be easil constructed, as shown in Fig. (c). Define the closed trapping region as X (see Fig. 4). Then, all solution orbits of sstem (1) outside X must move towards X and finall enter X for sufficientl large t. Thus proof of Theorem 1 is complete. h 4. Conclusion In this paper, b combining geometric and algebraic methods, we have shown that the general chaotic sstem (1) is ultimatel bounded, for an interval of b (0, 1] which has not been investigated. Six Lapunov functions are constructed according to different geometric configurations. Thus, for an combination of the sstem parameter values: a >0, 0<b 6 1, c >0 and d ( 1,1), this chaotic sstem has globall exponentiall attractive set. The exponential estimation is explicitl derived. This combination method can be applied to stud other chaotic sstems. The case b > 1 is under investigation and the results will be presented in a forthcoming paper. Acknowledgements This work was supported b the Natural Sciences and Engineering Research Council of Canada (NSERC No. R686A0) and the Natural Science Foundation of China (Grant No ). References [1] Lorenz EN. Deterministic nonperiodic flow. J Atmos Sci 196;0: [] Rössler OE. An equation for continuous chaos. Phs Lett A 1976;57:97 8. [] Chua LO. Chua s circuit: an overview ten ears later. J Circ Sst & Comput 1994;4: [4] Chen G, Ueta GT. Yet another chaotic attractor. Int J Bifurcat & Chaos 1999;9: [5] Čelikovsý S, Chen G. On a generalized Lorenz canonical form of chaotic sstems. Int J Bifurcat & Chaos 00;1: [6] Ueta T, Chen G. Bifurcation analsis of Chen s attractor. Int J Bifurcat & Chaos 000;10: [7] Chen G, Lü JH. Dnamical analsis, control and snchronization of Lorenz families. Beijing: Chinese Science Press; 00. [8] Leonov G, Bunin A, Koksch N. A tractor localization of the Lorenz sstem. ZAMM 1987;67: [9] Leonov G. On estimates of attractors of Lorenz sstem. Vestnik Leningradskogo Universiten Matematika 1988;1: 7. [10] Progromsk Yu, Santoboni G, Nijneijer H. An ultimate bound on the trajectories of the Lorenz sstem and its applications. Nonlinearit 00;16: [11] Yu P, Liao XX. Globall attractive and positive invariant set of the Lorenz sstem. Int J Bifurcat & Chaos 006;16: [1] Li D, Lu JA, Wu X, Chen G. Estimating the bounds for the Lorenz famil of chaotic sstem. Chaos, Solitons & Fractals 005;:59 4. [1] Yu P, Liao XX. New estimations for globall attractive and positive invariant set of the famil of the Lorenz sstems. Int J Bifurcat & Chaos 006;16(11).

13 P. Yu, X.X. Liao / Communications in Nonlinear Science and Numerical Simulation 1 (008) [14] Liao XX, Fu Y, Xie S, Yu P. Globall exponentiall attractive sets of the famil of Lorenz sstems. Sci in China, Ser F: Information Sciences, in press. [15] Liao XX, Yu P, Xie S, Fu Y. Stud on the global propert of the smooth Chua s sstem. Int J Bifurcat & Chaos 006;16(10). [16] Qin WX, Chen G. On the boundedness of solutions of the Chen sstem. J Math Anal Appl 006, doi: /j.jmaa