Commun Nonlinear Sci Numer Simulat

Commun Nonlinear Sci Numer Simulat 4 (9) 6 96 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns A constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family P. Yu a, *, X.X. Liao b, S.L. Xie c, Y.L. Fu c a Department of Applied Mathematics, The University of Western Ontario, London, Ontario, Canada N6A 5B7 b Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 474, PR China c School of Electronic and Information Engineering, South China University of Technology, Guangzhou, Guangdong 564, PR China article info abstract Article history: Received 9 October Accepted October Available online November PACS:..Hq Keywords: General Lorenz family Globally exponentially attractive set Positive invariant set Generalized Lyapunov function Ultimate boundedness of chaotic system In this paper, we give a constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family, which contains four independent parameters and is more general than any Lorenz systems studied so far in the literature. The system considered in this paper not only contains the classical Lorenz system and the generalized Lorenz family as special cases, but also provides three new Lorenz systems, which do not belong to the generalized Lorenz system, but the general Lorenz system. The results presented in this paper contain all the existing relative results as special cases. Ó Elsevier B.V. All rights reserved.. Introduction Since the discovery of the Lorenz chaotic attractor [] in 96, many other chaotic systems have been found, including the well-known Rössler system [], Chua s circuit [], which have been served as models for study of chaos. In late 99 s, a new chaotic system was found, which is a dual system to the Lorenz system, and now known as the Chen system [4]. Due to its close relation to the Lorenz system and importance, the Chen system has been widely studied (e.g., see [5 7] and references therein). The ultimate boundedness of a chaotic system is very important for the study of the qualitative behaviour of a chaotic system. If one can show that a chaotic system under consideration has a globally attractive set, then one knows that the system cannot have equilibrium points, periodic or quasi-periodic solutions, or other chaotic attractors existing outside the attractive set. This greatly simplifies the analysis of dynamics of the system. The ultimate boundedness also plays a very important role in the designs of chaos control and chaos synchronization. The ultimate boundedness property of the Lorenz system has been investigated by many researchers (e.g., see [,9, 5]). In particular, in [5] we generalized the boundedness or ultimate boundedness to the concept of globally exponentially attractive set and positive invariant set, and proved the existence of such set for a class of Lorenz family. It, however, has been noticed that so far very little has been achieved on other chaotic systems, regarding the property of ultimate boundedness. A smooth Chua s circuit has been studied and estimation on its ultimate boundedness has been * Corresponding author. Fax: + 59 66 5. E-mail address: pyu@pyu.uwo.ca (P. Yu). 7-574/$ - see front matter Ó Elsevier B.V. All rights reserved. doi:.6/j.cnsns...

P. Yu et al. / Commun Nonlinear Sci Numer Simulat 4 (9) 6 96 7 obtained [6]. For the Chen system, a recent article [7] investigated its property 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 system: _x ¼ rðy xþ; _y ¼ qx xz cy; _z ¼ xy bz; ðþ where the dot denotes differentiation with respect to time t; r >, b >, c > and q ð ; þþ are parameters. When r ¼ ; b ¼ ; c ¼ ; q ¼ ; ðþ system () is the classical Lorenz system; and when r ¼ 5a þ ; b ¼ þ a ; c ¼ 9a; q ¼ 5a; ðþ where a ½; Þ, system () becomes the generalized Lorenz system. 9 As shown in Fig., three new Lorenz systems are found, which do not belong to the classical Lorenz system, nor the generalized Lorenz system. z 5 4 a - -5 - -5 5 x 5 - -- y z 4 6 4 b -6-4 - x 4 6-75 -5-5 75 5 5 y z 5 5 5 c -5-9 -45 45 x 9 5-5 -9-45 5 9 45 y Fig.. The chaotic attractors of system () projected on the x z plane: (a) the Lorenz attractor for r ¼ 5; b ¼ ; c ¼ ; q ¼ ; (b) a chaotic attractor for r ¼ ; b ¼ 5; c ¼ 6; q ¼ ; and (c) a chaotic attractor for r ¼ 4; b ¼ 5; c ¼ ; q ¼ 5.

P. Yu et al. / Commun Nonlinear Sci Numer Simulat 4 (9) 6 96 Let Pðr; b; c; qþ be a point in the subspace e R 4 # R 4, defined as er 4 ¼ R þ R þ R þ R ¼ð; þþ ð; þþ ð; þþ ð ; þþ; then the classical Lorenz system corresponds to only one point Pðr; b; c; qþ ¼Pð; ; ; Þ in R e4, while the generalized Lorenz system or the Lorenz family corresponds to a small line segment in R e4, implying that the four parameters given in () are not independent (linearly related). In this paper, we remove the restriction on the parameters, given by () and assume generally that the four parameters r; b; c and q are free to change in the sub-space R e4. We want to prove that there exist globally exponentially attractive set and positive invariant set for such system described by (), and thus confirm the conjecture that a chaotic system has a global asymptotic stability in Lagrange sense, i.e., there exists a compact set X # R which is globally exponentially attractive and positive invariant. The earliest work on ultimate boundedness of chaotic systems goes back to Leonov et al. [] who proved that the classical Lorenz system (when r ¼ ; b ¼ ; c ¼ ; q ¼ ) is ultimately bounded (i.e., the system is asymptotically stable in the sense of Lagrange). However, they did not obtain the globally exponentially Lagrange stability. Later, Liao [] used the same generalized Lyapunov function, but different approaches to further investigate the ultimate boundedness of the classical Lorenz system, a and improved the results. Recently, Yu and Liao [,4] simplified the proofs and studied the globally attractive set and positive invariant set of the classical Lorenz system and the generalized Lorenz system [4]. Very recently, Liao et al. [5] proposed globally exponentially attractive set and positive invariant set for the classical Lorenz system and the generalized system, and provided as constructive proof for the existence of such sets. In this paper, we consider system () which is more general than the classical Lorenz system and the generalized Lorenz system. The main goals are as follows: () The methodology developed in [5] will be extended to study the general Lorenz system () for which the parameter values have been generalized from finite region to infinite region. We shall prove that system () has a generic property: it is always globally exponentially Lagrange stable regardless whether it is chaotic or not. In other words, there always exist globally exponentially attractive set and positive invariant set X # R for such a system. The result contains all existing results as its special cases. () When studying chaotic systems, people usually assume that the system under consideration is ultimately bounded. However, there does not exist a general proof for the ultimate boundedness of chaotic systems. Now, for the general Lorenz system (), we have shown that such system has globally exponentially attractive set and positive invariant set. () As is well-known, one of the most popular and useful approach used in chaos control and chaos synchronization is feedback control, such as linear feedback control, particularly diagonal linear feedback control. Such controls are easy to realize in practice or experiment. However, The control strategy is based on the assumption that the system is ultimately bounded. Such analysis based on an assumption may have some theoretical important, but is hard to realize in practice. With the property of the ultimate boundedness established with explicit estimation in this paper, it becomes possible to use simple linear feedback controls to reach chaos control and chaos synchronization. In next section, we present some definitions which are needed in the Section. Our main result and its proof for system () to have globally exponentially attractive set and positive invariant set are given in Section. We present two some applications in Section 4 and finally give a summery in Section 5.. Preliminaries In this section, we present some basic definitions and two lemmas which are needed for proving all theorems in the next section. For convenience, let X :¼ ðx; y; zþ and XðtÞ :¼ Xðt; t ; X Þ. Definition. For the general chaotic system (), if there exists compact (bounded and closed) set X R such that X R, the following condition: qðxðtþ; XÞ :¼ inf kxðtþ Yk! as t!þ; YX holds, then the set X is said to be globally attractive. That is, system () is ultimately bounded, namely, system () is globally stable in the sense of Lagrange or dissipative with ultimate bound. Further, if X X # X R, Xðt; t ; X Þ # X, then X for t P is called the positive invariant set of system (). Definition. For the general chaotic system (), if there exists compact set X R MðX Þ >, a > such that such that X R, and constants qðxðtþ; XÞ 6 MðXðt ÞÞe aðt t Þ ;

P. Yu et al. / Commun Nonlinear Sci Numer Simulat 4 (9) 6 96 9 then system () is said to have globally exponentially attractive set, or system () is globally exponentially stable in the sense of Lagrange, and X is called the globally 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 X in Definition. Since the Lyapunov direct method is still a powerful tool in the study of asymptotic behaviour of non-linear dynamical systems, the following definition is more useful in applications. Definition. For the general chaotic system (), if there exists a positive definite and radially unbounded Lyapunov function VðxÞ, and positive numbers L > ; a > such that the following inequality ðvðxðtþ LÞ 6 ðvðx Þ LÞe aðt t Þ is valid for VðXðtÞ > Lðt P t Þ, then system () is said to be globally exponentially attractive or globally exponentially stable in the sense of Lagrange, and X :¼ fxjvðtþ 6 L; t P t g is called the globally exponentially attractive set.. Main result In this section, we present our main result for the general Lorenz system (). We use the same generalized Lyapunov function: V k ¼ ½kx þ y þðz kr qþ Š; ð4þ which is obviously positive definite and radially unbounded. Here, k P is an arbitrary constant. Let X ¼ðx; y; zþ. We have the following result. Theorem. Suppose b > minfr; cg ¼n, and let L k ¼ ðkr þ qþ b ðb nþn : ð5þ Then the estimation ðv k ðxðtþþ L k Þ 6 ðv k ðxðt ÞÞ L k Þe nðt t Þ holds, and thus X k ¼fXjV k ðxþ 6 L k g is the globally exponentially attractive set and positive invariant set of system (), i.e., lim t!þ V k ðxðtþþ 6 L k. ð6þ Proof. Differentiating the Lyapunov function V k in (4) with respect time t along the trajectory of system () yields dv k dt ¼ kx_x þ y _y þðz kr qþ_z ðþ ¼ krx þ krxy þ qxy xyz cy þ xyz krxy qxy bz þ krbz þ qbz ¼ krx cy bz þ bðkr þ qþz 6 knx ny nz þ nðkr þ qþz nðkr þ qþ ðb nþz þðb nþðkr þ qþz þ nðkr þ qþ 6 knx ny nðz kr qþ ðb nþðkr þ qþ ðb nþ z þ ðb nþ ðkr þ qþ þ nðkr þ qþ ðb nþ 4ðb nþ 6 nv k þ ðkr þ qþ ½ðb nþ þ 4ðb nþnš 4ðb nþ ¼ nv k þ ðkr þ qþ b 4ðb nþ 6 nðv k L k Þ: ð7þ and Thus, we have ðv k ðxðtþþ L k Þ 6 ðv k ðxðt ÞÞ L k Þe nðt t Þ lim t!þ V k ðxðtþþ 6 L k ; which clearly shows that X k ¼fXjV k ðxþ 6 L k g is the globally exponentially attractive set and positive invariant set of system (). The proof is complete. h

9 P. Yu et al. / Commun Nonlinear Sci Numer Simulat 4 (9) 6 96 Corollary. When r ¼ a a ¼ 5a þ ; b ¼ b a ¼ þ a ; q ¼ c a ¼ 5a; c ¼ d a ¼ 9a; n ¼ minfr; cg ¼minfa a ; d a g¼d a. Thus, L k ¼ ðkr þ qþ b ðb nþn ¼ ðka a þ c a Þb a ðb a d a Þd a ; and the estimate (6) holds, implying that lim t!þ V k ðxðtþþ 6 L k, i.e., X k ¼fXjV k ðxþ 6 L k g is the globally exponentially attractive set and positive invariant set of system (). Remark. Corollary gives the result of Theorem in [5], implying that Theorem of [5] is a special case of Theorem in this paper. Corollary. When r P, b P, c ¼, n ¼ minfr; cg ¼, and thus b Lk ¼ ðkr þ qþ b : ðb Þ Therefore, the following: holds and ðv k ðxðtþþ b LkÞ 6 ðv k ðxðt ÞÞ b L k Þe ðt t Þ lim t!þ V k ðxðtþþ 6 b L k ; i.e., b X k ¼fXjV k ðxþ 6 b L k g is the globally exponentially attractive set and positive invariant set of system (). Remark. Corollary is the result of Theorem in [5], indicating that Theorem of [5] is a special case of Theorem in this paper. Remark. It is seen from Theorem that larger (smaller) value of n implies larger (smaller) value of L k, and thus fast (slow) speed for trajectories to converge to the globally exponentially attractive set of system (). Remark 4. To illustrate the globally exponentially attractive set and positive invariant set, we use the classical Lorenz system as an example. For this system, the parameter values are given in (), and so L k ¼ ðk þ Þ ð=þ ð5=þ ¼ ð5k þ 4Þ ; 5 which gives the following estimate of the ultimate bound: X k ¼fXjV k ðxþ 6 L k g which is the globally exponentially attractive set and positive invariant set of the classical Lorenz system. Remark 5. Note that the parameter values for the Lorenz attractor shown in Fig. (a) satisfy the condition in Theorem : ¼ b > minfr; cg ¼minf5; g ¼, that for the second and third chaotic attractors shown in Fig. b and c do not satisfy the condition in Theorem. Thus, we need new theorem when this condition does not hold. Now we turn to consider the case that the condition in Theorem is not satisfied. Theorem. Let n ¼ minfr; c; b g, and L k ¼ ðkr þ qþ b : ðþ 4n Then the estimate lim t!þ V k ðxðtþþ 6 L k holds, i.e., X k ¼fXjV k ðxþ 6 L k g is the globally exponentially attractive set and positive invariant set of system (). Proof. Again applying the positive definite and radially unbounded generalized Lyapunov function given in (4) and evaluating the derivative of dv k along the trajectory of system () leads to dt

dv k dt ¼ kx_x þ y _y þðz kr qþ_z ðþ ¼ krx cy bz þ bðcr þ qþz P. Yu et al. / Commun Nonlinear Sci Numer Simulat 4 (9) 6 96 9 ¼ krx cy b z b ½z ðkr þ qþz þðkr þ qþ ðkr þ qþ Š ð9þ which implies that Thus, 6 knx ny nz nðz kr qþ þ b ðkr þ qþ 6 knx ny nðz kr qþ þ n L k ¼ nðv k L k Þ; ðv k ðxðtþþ L k Þ 6 ðv k ðxðt ÞÞ L k Þe nðt t Þ : lim t!þ V k ðxðtþþ 6 L k ; suggesting that X k ¼fXjV k ðxþ 6 L k g is the globally exponentially attractive set and positive invariant set of system (). This completes the proof of Theorem. h For the application of Theorem, let us consider the first two chaotic attractors depicted in Fig. a and b. Example. For the first chaotic attractor shown in Fig. a, the parameters are r ¼ ; b ¼ 5; c ¼ 5; q ¼. Thus n ¼ min r; c; b ¼ minf; 6:5g ¼:5: Taking k ¼, we have L ¼ ðr þ qþ b 4n ¼ ð þ Þ 5 ¼ 65; 4 :5 and thus we have the following estimation: ðv ðxðtþþ 65Þ 6 ðv ðxðt ÞÞ 65Þe 5ðt t Þ ; the globally exponentially attractive set and positive invariant set is given by X ¼fXjV ðxþ 6 65g: For the second chaotic attractor shown in Fig. b, the parameters are r ¼ 4; b ¼ ; c ¼ 5; q ¼ 5. Thus n ¼ min r; c; b ¼ minf4; :5g ¼:5: Again taking k ¼, we have L ¼ ðr þ qþ b 4n So the following estimation: ¼ ð4 þ 5Þ 5 ¼ 45: 4 :5 ðv ðxðtþþ 45Þ 6 ðv ðxðt ÞÞ 45Þe 5ðt t Þ holds, and X ¼fXjV ðxþ 6 65g is the globally exponentially attractive set and positive invariant set. Remark 6. Although the proof of Theorem is simple, it is conservative since the term b z is neglected. In the following, we give a theorem which is the result of generalization and improvement of Theorems and. However, this new theorem has a parameter to be determined, while the parameters used in Theorems and are the system parameters. Theorem. Choose g ð; bþ. Let n ¼ minfr; c; gg, and L k ¼ ðkr þ qþ b ðb n Þn : ðþ Then when V k ðxðtþþ > L k and V kðxðt ÞÞ > L k, there exists the estimation ðv k ðxðtþþ L k Þ 6 ðv kðxðt ÞÞ L k Þe n ðt t Þ ; ðþ

6 n ðv k L k Þ: ðþ 9 P. Yu et al. / Commun Nonlinear Sci Numer Simulat 4 (9) 6 96 and thus lim t!þ V k ðxðtþþ 6 L k, i.e., X k ¼fXjV kðxþ 6 L kg is the globally exponentially attractive set and positive invariant set of system (). Proof. Using the V k ðxþ given in (), we obtain dv k dt ¼ kx_x þ y _y þðz kr qþ_z ðþ ¼ krx cy bz þ bðkr þ qþz 6 krx cy n z þ n ðkr þ qþz n ðkr þ qþ ðb n Þz þðb n Þðkr þ qþz þ n ðkr þ qþ 6 kn x n y n ðz kr qþ ðb n Þ z ðb n Þðkr þ qþ ðb n þ ðb n Þ ðkr þ qþ Þ 4ðb n þ n ðkr þ qþ Þ 6 n V k þ ðkr þ qþ b 4ðb n Þ Therefore, when V k ðxðtþþ > L k and V kðxðt ÞÞ > L k, we have ðv k ðxðtþþ L k Þ 6 ðv kðxðt ÞÞ L k Þe n ðt t Þ ; and thus X k ¼fXjV kðxþ 6 L kg is the globally exponentially attractive set and positive invariant set of system (). h Remark 7. When b > minfr; cg, we can take n ¼ minfr; cg, leading to the conclusion of Theorem ; when b 6 minfr; cg, we can choose g ¼ b, and so n ¼ minfr; c; b g¼b, resulting in the result of Theorem. Note that Theorem does not need to discuss b and minfr; cg, and thus is more general and applicable; while the conditions given in Theorems and are explicitly expressed in terms of system parameters, which does not need to determine n, and thus easier to be used in application. In this following, we may separate the variable x form the variables y and z, and obtain another result as follows. Let V ¼ ½y þðz qþ Š and L ¼ b q >< minfr; cg; if r c; where n ¼ c þ e ðþ ðb nþn >: ðe > Þ; if r ¼ c: Theorem 4. If b > minfr; cg ¼n, then ( ðv ðxðtþþ L Þ 6 ðv ðxðt ÞÞ L Þe nðt tþ ; and ðx ðtþ L Þ 6 ðx ðt ÞÞ L Þe nðt tþ ; < lim t!þ ðy ðtþþðzðtþ qþ Þ 6 b q ; 4ðb nþn : lim t!þ x ðtþ 6 b q ; 4ðb nþn < X :¼ X V 6 L < ¼ y þðz qþ 6 b q : x : x 6 b q 4ðb nþn 4ðb nþn ; ð4þ ð5þ ð6þ is the globally exponentially attractive set and positive invariant set of system (). Proof. Take k ¼ intheorem. Similar to the proof for Theorem, we obtain dv dt ¼ yðqz xz cyþþðz qþðxy bzþ ¼ cy bz þ qbz 6 ny nz þ nqz nq ðb nþz þðb nþqzþnq 6 nv þ b q 4ðb nþ 6 nðv L Þ; which, in tern, results in ðv ðxðtþþ L Þ 6 ðv ðxðt ÞÞ L Þe nðt t Þ

P. Yu et al. / Commun Nonlinear Sci Numer Simulat 4 (9) 6 96 9 and so lim t!þ ðy ðtþþðzðtþ qþ Þ 6 b q 4ðb nþn : Next, for the first equation of (), construct the positive definite and radially unbounded Lyapunov function: ev ¼ x ; and thus d e V dt ¼ rx þ rxy 6 rx þ rjxjjyj from which we have 6 rx þ rx þ ry 6 r e V þ r e V þ rðv ðx Þ L Þe nðt t Þ ð e V ðxðtþþ L Þ 6 ð e V ðx Þ L Þe rðt t Þ þ Z t Now we estimate the integral in the above inequality: Z t rðv ðx Þ L Þe rtþnt t r ¼ðV ðx Þ L Þe rtþnt ½e r ðr nþt e ðr nþt Š n r ¼ðV ðx Þ L Þ ½e r nðt tþ e rðt tþ Š n >< ðv ðx Þ L Þ r e nðt t Þ when r > n; r n 6 >: ðv ðx Þ L Þ e rðt t Þ when r < n: r n r e ðr nþs ds t e rðt sþ rðv ðx Þ L Þe nðs t Þ ds: Hence, lim t!þv e ðxðtþþ 6 e L, i.e., lim t!þ x 6 e L. Therefore, summarizing the above two parts shows that any trajectory of system () globally exponentially converges to X, namely, X is the globally exponentially attractive set and positive invariant set. The proof is complete. h Let n ¼ minfc; b g, e L ¼ q b. Again choose V 4n ðy; zþ ¼ ½y þðz qþ Š. Then we have Theorem 5. When V ðxðtþþ > e L and V ðxðt ÞÞ > e L, the following estimate: holds with i.e., ðv ðxðtþþ e L Þ 6 ðv ðxðt ÞÞ e L Þe ~ n ðt t Þ lim t!þ e V ðxðtþþ 6 e L and lim t!þ x ðtþ 6 e L ; ( ex ¼ X V ðxðtþþ 6 e ) L x 6 e L is the globally exponentially attractive set and positive invariant set of system (). Proof. In Theorem, taking k ¼ yields and thus dv dt ¼ n ½y þðz qþ Šþn e L 6 n ðv e L Þ; ðv ðxðtþþ e L Þ 6 ðv ðxðt ÞÞ e L Þe n ðt t Þ

94 P. Yu et al. / Commun Nonlinear Sci Numer Simulat 4 (9) 6 96 which implies that lim t!þv e ðxðtþþ 6 e L. Further, by Theorem 4, we have lim t!þ x ðtþ 6 e L. Combining these two results shows that ( ex ¼ X V ðxðtþþ 6 e ) L x 6 e L is the globally exponentially attractive set and positive invariant set of system (). h Example. Let us apply the above theorem to re-consider the first two chaotic attractors shown in Figs. a and b. For the first chaotic attractor, r ¼ ; b ¼ 5; c ¼ 5; q ¼, n ¼ min r; c; b ¼ minf; 5; :5g¼:5: Thus e L ¼ q b ¼ 5 ¼ ; 4n giving a better estimation than that obtained in Example. For the second chaotic attractor, r ¼ 4; b ¼ ; c ¼ 5; q ¼ 5. n ¼ minfc; b g¼:5, and thus e L ¼ ðr þ qþ b ¼ ð5þ 5 ¼ 5; 4n which is sharp than that given in Example. Theorem 6. Choose g ½; bš and let minfc; gg ¼n. Let L ¼ q b ðb n Þn : Then, when V ðxðtþþ > L and V ðxðt ÞÞ > L, the following estimate holds with i.e., ðv ðxðtþþ L Þ 6 ðv ðxðt ÞÞ L Þe n ðt t Þ lim t!þ e V ðxðtþþ 6 L and lim t!þ x ðtþ 6 L ; ( ) X ¼ X y þ ðz qþ 6 L x 6 L is the globally exponentially attractive set and positive invariant set of system (). Proof. In Theorem take k ¼, then following the proofs given for Theorems and 5, we can similarly prove this theorem, and the details are omitted for simplicity. h At the end of this section, we give an application of Theorem 6 to globally exponentially stabilize the origin ð; ; Þ using a simple feedback control. 4. Application In this section, we only present two simple examples to illustrate the application of the theoretical results established in the previous section for the globally exponential attracting set and positive invariant set. More applications in chaos control and chaos synchronization will be discussed in other papers. 4.. Application to stability and stabilization Theorem 7. When q 6, the equilibrium point ð; ; Þ of system () is globally exponentially stable, and is thus unique. When q >, there exist many feedback control laws to stabilize the equilibrium point ð; ; Þ, and the simplest one is to add a linear feedback control cx to the second equation of (), where c P q.

P. Yu et al. / Commun Nonlinear Sci Numer Simulat 4 (9) 6 96 95 Proof. When q <, in Theorem we choose kr þ q ¼, namely k ¼ q r. Thus the L k in Theorem becomes L k ¼ L q r ¼. This shows that the globally exponential attracting set is reduced to the minimum, i.e., X k ¼ Xq r ¼fg. Therefore, the equilibrium point ð; ; Þ of system () is globally exponentially stable. When q ¼, by using Theorem 4, we have L ¼ b q lim t!þ y ðtþþz ðtþ ¼ and lim x ðtþ ¼: t!þ ðb nþn ¼, which implies that Hence, the minimum globally exponential attracting set X ¼fg, i.e., the equilibrium point ð; ; Þ of system () is globally exponentially stable. When q >, adding the feedback control cx to the second equation of system () yields the following controlled system: _x ¼ rðy xþ; _y ¼ qx xz cy cx ¼ ~qx xz cy; _z ¼ xy bz; where ~q ¼ q c 6. From the above discussions for the cases q < and q ¼, we know that the conclusion is also true for q >. h 4.. Qualitative study on the trajectory of system () in the complementary set of the globally exponential attracting set X The second example is to consider the qualitative behaviour of the trajectories of system () in the complementary set of the globally exponential attracting set X, R =X. Theorem. In the complementary set R =X, (i.e., outside of the globally exponential attracting set X), there do not exist chaotic attractor, or equilibrium point, or periodic solution, or quasi-periodic motion, or any other type of positive invariant sets. Proof. First of all, all trajectories in R =X move according to the exponential decay, converging towards the globally exponential attracting set X. Obviously, there cannot exist any strange attractor like the Lorenz attractor. Thus, in R =X there are no fundamental differences between chaotic and non-chaotic systems. Because the equilibrium points, periodic solutions and quasi-periodic solutions of system () are all positive invariant sets of system (). Thus, we prove for general case of positive invariant set. Suppose this is not true. Without loss of generality, assume Q is a positive invariant set of R =X. Then Q S X ¼ /, where / denotes the empty set. This implies that Q and X do not intersect. So we have inf kx Xk > : XX;XQ From the definition of positive invariant set, we know that Xðt; t ; X ÞQ ðt P t Þ as long as X Q. Thus, inf kx Xðt; t ; X Þk > : XX;Xðt;t ;X ÞQ;tPt On the other hand, since X is a globally exponential attracting set, we have that X R, Xðt; t ; X Þ!X, which implies inf kx Xðt; t ; X Þk ¼ ; XX;Xðt;t ;X ÞQ;tPt leading to a contradiction to the above inequality. This shows that the conclusion of Theorem is true. h Remark. The X used in Theorem is a general notation. The conclusion is always true when X is chosen as any of the X; X ; X e ; X ; X k ; X k and X k, which are used in the previous section. 5. Conclusion In this paper, we have extended the method developed in [5] to study the globally exponentially attractive set and positive invariant set for a more general Lorenz family. It has been shown that such system indeed has globally exponentially attractive set and positive invariant set, and contains all the existing relative results as special cases. Exponential estimation is explicitly derived. The approach presented in this paper may be applied to study other chaotic systems. Acknowledgements This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC No. R66A) and the Natural Science Foundation of China (Grant Nos. 65, 677494 and 6746).

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