Global structure for a class of dynamical systems

Chaos, Solitons and Fractals 11 (2000) 735±741

www.elsevier.nl/locate/chaos

Global structure for a class of dynamical systems Suochun Zhang, Zuo-Huan Zheng * Institute of Applied Mathematics, Academia Sinica, Beijing 100080, People's Republic of China Accepted 7 July 1998

Abstract In this paper, we shall consider the global structure of positive bounded systems on the plane which have m singular points, but not any closed orbits and singular closed orbits. We shall prove that these systems have at least m ÿ 1 connecting orbits; and all the connecting orbits, homoclinic orbits and singular points constitute a compact simply connected set. Each of other orbits tends to a singular point as t ! ‡1, and approaches to the in®nity as t ! ÿ1: Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction We consider the ordinary di€erential equations on the plane dx ˆ X …x; y†; dt

dy ˆ Y …x; y†: dt

…1†

We suppose that X and Y are continuous, and Eq. (1) satis®es the existence and uniqueness of solutions for the initial value problem. Let the vector ®eld V ˆ …X ; Y † de®nes a ¯ow u…p; t†. De®nition 1. Suppose that p1 ; p2 2 R2 are two isolated singular points of system (1). If there exists a point p0 2 R2 such that limt!ÿ1 u…p0 ; t† ˆ p1 and limt!‡1 u…p0 ; t† ˆ p2 , then the set u…p0 ; R† ˆ fu…p0 ; t†jt 2 Rg is called a connecting orbit joining singular points p1 and p2 . Many beautiful results about dynamical systems on the plane have been obtained. Some results study connective trajectories (see [1±6] and [9,10]), the existence and nonexistence of closed orbits and singular closed orbits and trajectory behaviors around singular points (see [11±15]). The global structure for some special dynamical systems on the plane had been obtained (see [7,8]). In this paper, our purpose is to investigate global structure of positive bounded systems on the plane which have ®nite singular points, but not any closed orbits and singular closed orbits. We shall prove that these systems have connecting orbits joining singular points; and all the connecting orbits, homoclinic orbits and singular points constitute a compact simply connected set, each of other orbits tends to a singular point as t ! ‡1, but approaches to the in®nity as t ! ÿ1. We shall use the following notations. For p 2 R2 and A  R, let u…p; A† ˆ fu…p; t†jt 2 Ag. Let q is a metric on the plane R2 , for …x1 ; y1 †; …x2 ; y2 † 2 R2 ,

*

Corresponding author. E-mail address: [email protected] (Z.-H. Zheng).

0960-0779/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 9 8 ) 0 0 1 8 4 - 2

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q……x1 ; y1 †; …x2 ; y2 †† ˆ

q 2 2 …x1 ÿ x2 † ‡ …y1 ÿ y2 † ;

O is the origin of R2 . For p ˆ …x; y† 2 R2 and d > 0, let kpk ˆ q……x; y†; O†; and Bd …p† ˆ fq 2 R2 jq…q; p† < dg; Ap and Xp are a-limit and x-limit sets of trajectory u…p; R†, respectively. 2. Main results De®nition 2 [13]. A simply closed curve C is a singular closed orbit if it is made up of some nonclosed trajectories and singular points alternately, and it is contained in a x-limit set or a a-limit set of a trajectory. De®nition 3 [6]. If every trajectory of the system (1) is positive (or negative) bounded, i.e., for arbitrary point p 2 R2 , there exists M > 0 such that u…p; ‰0; ‡1††  BM …O† (or u…p; …ÿ1; 0Š†  BM …O††, then the system (1) is called a positive (or negative) bounded system. Lemma 1. If the system (1) is a positive bounded system, which has finite singular points, but not any closed orbits and singular closed orbits, then every trajectory tends to a singular point as t ! ‡1. Proof. For a point p 2 R2 , without loss of generality we suppose that p is not a singular point. Because u…p; ‰0; ‡1†† is bounded, by Poincare±Bendixson theorem we get that Xp is a singular point, or a closed orbit, or a connected set constituted by some singular points and some trajectories whose positive semitrajectory and negative semi-trajectory are tend to a singular point, respectively. From the condition of Lemma 1 we know that Xp is not a closed orbit. Hence, the second case never take place. If the last situation happens, since Xp has ®nite singular points and it is a chain recurrent set, then Xp contains a simple closed curve C which is constituted by some nonclosed trajectories and singular points alternately. Because C  Xp ; then C is a singular closed orbit. This contradicts to the condition of Lemma 1. Thus Xp is a singular point. Lemma 1 is proved. Lemma 2. If the system (1) is a positive bounded system, which has finite singular points but not any closed orbits and singular closed orbits, then negative semi-trajectory of every trajectory tends to a singular point, or approaches to the infinity. Proof. Suppose that system (1) has m singular points qi ; i ˆ 1; 2; . . . ; m. For a point p 2 R2 , without loss of generality we suppose that p is not a singular point. If u…p; …ÿ1; 0Š† is a bounded set, by Poincare± Bendixson theorm, Ap is a singular point, or a closed orbit, or a connected set constituted by some singular points and some trajectories whose positive semi-trajectory and negative semi-trajectory tend to a singular point, respectively. Similar to the proof of Lemma 1, we can prove that Ap is a singular point. Then the negative semi-trajectory of trajectory through the point p tends to a singular point. If the negative semi-trajectory of trajectory which through p is unbounded, then limt!ÿ1 ku…p; t†k ˆ ‡1, or there exists a point q 2 R2 ; T > 0 and f0 > t10 > t100 > t20 > t200 >


g such that limt!1 u…p; tn0 † ˆ q and ku…p; tn00 †k ˆ T ‡ n; n ˆ 1; 2; . . . Because the system (1) has ®nite singular points, then there exists a positive number T1 such that fqi ji ˆ 1; . . . ; mg  BT1 …O†. Suppose that the second situation take place. If q is a singular point, without loss of generality we suppose that q ˆ q1 . We take N and e1 > 0 such that T ‡ N > T1 and fqi ji ˆ 2; 3; . . . ; mg \ Be1 …q† ˆ ;;

Be1 …q†  BT1 …O†:

Because limn!1 u…p; tn0 † ˆ q, there exists N1 such that fu…p; tn0 †jn ˆ N1 ; N1 ‡ 1; . . .g  Be1 …q†. Let sn ˆ infft > 0jq…u…u…p; tn0 †; t†; q† ˆ e1 g; n ˆ N1 ; N1 ‡ 1; . . . Thus u…u…p; tn0 †; ‰0; sn ††  Be1 …q†; n ˆ N1 ; N1 ‡ 1; . . .

…2†

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Because q is a singular point, the set fsn jn ˆ N1 ; N1 ‡ 1; . . .g is unbounded. Because fu…u…p; tn0 †; sn †jn ˆ N1 ; N1 ‡ 1; . . .g  oBe1 …q†; and oBe1 …q† is a compact set, fu…u…p; tn0 †; sn †jn ˆ N1 ; N1 ‡ 1; . . .g has a convergent subsequence. Without loss of generality we suppose limn!1 u…u…p; tn0 †; sn † ˆ p0 . Then p0 2 oBe1 …q†. Suppose that there exists s > 0 such that u…p0 ; ÿs† 62 Be1 …q†. Because Be1 …q† is a compact set, there exists d > 0 such that Bd …u…p0 ; ÿs†† \ Be1 …q† ˆ ;:

…3†

In view of the continuity of the ¯ow, there exists e2 > 0 such that for an arbitrary point p00 2 Be2 …p0 † and t 2 ‰0; sŠ we have ku…p0 ; ÿt† ÿ u…p00 ; ÿt†k < d:

…4†

Take n1 > N1 such that u…u…p; tn0 1 †; sn1 † 2 Be2 …p0 † and sn1 > s: From inequality (4) we have u…u…p; tn0 †; sn1 ÿ s† 2 Bd …u…p0 ; ÿs††: From equality (3) we have u…u…p; tn0 1 †; sn1 ÿ s† 62 Be1 …q†: This contradicts to Eq. (2). Hence u…p0 ; …ÿ1; 0Š†  Be1 …q†. Using the same method we can prove that limt!ÿ1 u…p0 ; t† ˆ q ˆ q1 : From Lemma 1, we know that the positive semi-part of u…p0 ; R† tends to a singular point. Because limn!1 u…u…p; tn0 †; sn † ˆ limn!1 u…p; tn0 ‡ sn † ˆ p0 and Suppose limt!‡1 u…p0 ; t† ˆ q1 . 00 ku…p; tn †k ˆ T ‡ n > T1 ; …n ˆ N ; N ‡ 1; . . .†, from the de®nition of sn we have tn0 ‡ sn < 00 ; n ˆ N ‡ N1 ‡ 1; N ‡ N1 ‡ 2; . . . Because q is a singular point, limn!1 tn00 ˆ ÿ1. So limn!1 tn0 ‡ tnÿ1 sn ˆ ÿ1: Thus p0 2 Ap . Hence …u…p0 ; R† [ fq1 g†  Ap : So q1 and u…p0 ; R† constitute a singular closed orbit. This contradicts to the condition of Lemma 2. Then there exists i 2 f2; 3; . . . ; mg such that limt!‡1 u…p0 ; t† ˆ qi ; Without loss of generality we suppose i ˆ 2, it is limt!‡1 u…p0 ; t† ˆ q2 . By the continuity of the ¯ow, there exists a sequence ftn000 g such that 0 > t10 > t100 > t1000 > t20 >


; and limn!1 u…p; tn000 † ˆ q2 : Using the same method we can prove that there exists a point p00 2 R2 such that limt!ÿ1 u…p00 ; t† ˆ q2 : If limt!‡1 u…q00 ; t† 2 fq1 ; q2 g; we can prove that there exists a singular closed orbit. This contradicts to the condition of Lemma 2. Thus there exists i 2 f3; 4; . . . ; mg such that limt!‡1 u…p00 ; t† ˆ qi : Repeat this step, we can prove that the system (1) has a singular closed orbit. This contradicts to the condition of Lemma 2. If q is not a singular point, from Lemma 1 we know that there exists i 2 f1; 2; . . . ; mg such that limt!‡1 u…q; t† ˆ qi : Using the same method we can prove that the system (1) has a singular closed orbit. This contradicts to the condition of Lemma 2. Hence limt!ÿ1 ku…p; t†k ˆ ‡1: Lemma 2 is proved. Lemma 3. If the system (1) is a positive bounded system, which has finite singular points, but not any closed orbits and singular closed orbits, then all the connecting orbits, singular points and homoclinic orbits constitute a bounded closed set. Proof. Let all the singular points of system (1) constitute a set G1 and let all the singular points, connecting orbits and homoclinic orbits of system (1) constitute a set G2 . Because the system (1) has ®nite singular points, G1 is bounded, that is, there exists T1 > 0 such that G1  BT1 …O†: Suppose that G2 is unbounded. Then there exists a sequence fqn g  G2 such that for all n 2 f1; 2; . . .g we have kqn k ˆ T1 ‡ n; the positive semi-trajectory and the negative semi-trajectory of u…qn ; R† tends to a singular point, respectively. For an arbitrary natural number n, let sn ˆ sup ft < 0ju…qn ; t† 2 oBT1 …O†g: Then fu…qn ; sn †jn ˆ 1; 2; . . .g  oBT1 …O†; and u…qn ; …sn ; 0Š† \ BT1 …O† ˆ ;;

n ˆ 1; 2; . . .

…5†

Since oBT1 …O† is a compact set, fu…qn ; sn †jn ˆ 1; 2; . . .g has a convergent subsequence. Without loss of generality we suppose limn!1 u…qn ; sn † ˆ q0 : From Lemma 1, the positive semi-trajectory of u…q0 ; R† tends to a singular point. Hence there exists s > 0 such that u…q0 ; s† 2 BT1 …O†: So there exists d > 0 such that Bd …u…q0 ; s††  BT1 …O†: In view of the continuity of the ¯ow, there exists e > 0 such that for an arbitrary point p 2 Be …q0 † and t 2 ‰0; sŠ we have ku…p; t† ÿ u…q0 ; t†k < d:

…6†

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Because u…q0 ; ‰0; ‡1†† is bounded, there exists T2 > T1 such that u…q0 ; ‰0; ‡1††  BT2 …O†. Take n1 2 f1; 2; . . .g such that u…qn1 ; sn1 † 2 Be …q0 †, and T1 ‡ n1 > T2 ‡ d ‡ 1: From inequality (6) we have u…u…qn1 ; sn1 †; s† 2 Bd …u…q0 ; s††  BT1 …O†: From equality (5) we have s > jsn1 j: Hence qn1 2 Bd …u…q0 ; ÿsn1 ††: Because ÿsn1 P 0; kqn1 k < T2 ‡ d: This contradicts to kqn1 k ˆ T1 ‡ n1 : Thus G2 is bounded. Suppose that G2 is not closed. Then there exists a point q 2 R2 n G2 and a sequence fqn g  G2 such that limn!1 qn ˆ q: From Lemmas 1 and 2 we have limt!ÿ1 ku…q; t†k ˆ ‡1: Since G2 is bounded, there exists T > 0 such that G2  BT …O†, and s > 0 such that ku…q; ÿs†k > T ‡ 1: In view of the continuity of the ¯ow, there exists e > 0 such that for an arbitrary point p 2 Be …q† and t 2 ‰ÿs; 0Š we have 1 ku…p; t† ÿ u…q; t†k < : 2

…7†

Take n1 2 f1; 2; . . .g such that qn1 2 Be …q†: Thus we have 1 ku…q; ÿs†k 6 ku…qn1 ; ÿs†k ‡ ku…q; ÿs† ÿ u…qn1 ; ÿs†k < T ‡ : 2 This contradicts to ku…q; ÿs†k > T ‡ 1: So G2 is closed. Lemma 3 is proved. Lemma 4. If the system (1) is a positive bounded system, and has m singular points, but not any closed orbits and singular closed orbits, then system (1) has at least m ÿ 1 connecting orbits, which together with m singular points constitute a connected set, and all the singular points, connecting orbits and homoclinic orbits constitute a bounded closed simply connected region. Proof. When m ˆ 1; the conclusion is obvious. When m P 2; let fqi ji ˆ 1; 2; . . . ; m:g is the singular points set of the system (1),   G1 ˆ x 2 R2 j lim u…x; t† ˆ q1 ; t!‡1

H1 ˆ fx 2 R2 j There exists i 2 f2; 3; . . . ; mg such that limt!‡1 u…x; t† ˆ qi g. Then G1 \ H1 ˆ ;; G1 6ˆ ; and H1 6ˆ ;: From Lemma 1 we have H1 [ G1 ˆ R2 : Since R2 is connected, at least one of G1 and H1 is not closed. Without loss of generality we suppose that H1 is not closed. Then there exists a point q 2 G1 is a condensation point of H1 , that is there exists a sequence fpn g  H1 such that limt!1 pn ˆ q: Because limt!‡1 u…q; t† ˆ q1 ; there exists a sequence ftn g  …0; ‡1† such that limn!1 u…pn ; tn † ˆ q1 : Denote pn0 ˆ u…pn ; tn †;

n ˆ 1; 2; . . .

Take e1 > 0 such that if i 6ˆ j, Be1 …qi † \ Be1 …qj † ˆ ;: Let en ˆ 12 enÿ1 ; n ˆ 2; 3; . . . : Take n1 2 f1; 2;


:g such that fpn0 n ˆ n1 ; n1 ‡ 1;


:g  Be1 …q1 †, for n 2 fn1 ; n1 ‡ 1; . . . :g, let m [ Ben …qi †; Bn ˆ iˆ2

rn;1 ˆ inf ft > 0ju…pn0 ; t† 2 oBe1 …q1 †g. Thus fu…pn0 ; rn;1 †jn ˆ n1 ; n1 ‡ 1; . . .g  oBe1 …q1 †; and 1 [ u…pn0 ; ‰0; rn;1 ††  Be1 …q1 †: nˆn1

Because oBe1 …q1 † is compact, fu…pn0 ; rn;1 †jn ˆ n1 ; n1 ‡ 1; . . .g has a convergent subsequence. Without loss of generality we suppose limn!1 u…pn0 ; rn;1 † ˆ q01 : In view of the continuity of the ¯ow and limn!1 pn0 ˆ q1 we have u…q01 ; …ÿ1; 0††  Be1 …q1 †. From Lemma 2 we have limt!ÿ1 u…q01 ; t† ˆ q1 : From Lemma 1 we have limt!‡1 u…q01 ; t† ˆ q1 ; or lim u…q01 ; t† 2 fqi ji ˆ 2; . . . ; m:g:

t!‡1

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Suppose that limt!‡1 u…q01 ; t† ˆ q1 . In view of the continuity of the ¯ow, there exists two sequences fs0n g  …0; ‡1†; ftn0 g  …0; ‡1†; tn0 > s0n ; n2 P n1 such that for n 2 fn2 ; n2 ‡ 1; . . .g we have u…pn0 ; rn;1 ‡ s0n † 2 oBe1 …q1 †; u…pn0 ; …rn;1 ‡ s0n ; rn;1 ‡ tn0 Š†  Be1 …q1 †; lim u…pn0 ; rn;1 ‡ tn0 † ˆ q1 ;

n!1

and lim …tn0 ÿ s0n † ˆ ‡1:

n!1

Let pn00 ˆ u…pn0 ; rn;1 ‡ tn0 †; rn;2 ˆ inf ft > 0ju…pn00 ; t† 2 oBe1 …q1 †g; n ˆ n2 ; n2 ‡ 1; . . . Using the same method we can prove that there exists a point q02 2 oBe1 …q1 † such that /…q02 ; …ÿ1; 0††  Be1 …q1 † and limt!ÿ1 u…q02 ; t† ˆ q1 : From Lemma 1 we have limt!‡1 u…q02 ; t† ˆ q1 ; or limt!‡1 u…q02 ; t† 2 fqi ji ˆ 2; 3; . . . ; mg: If limt!‡1 u…q02 ; t† ˆ q1 ; repeat this step again. After repeating this step ®nite times, we can prove that there exists a point q0j 2 oBe1 …q1 † such that limt!ÿ1 u…q0j ; t† ˆ q1 and limt!‡1 u…q0j ; t† 2 fqi ji ˆ 2; 3; . . . ; mg; or we have to repeat this step in®nite times, to obtain there exists a point q00 2 oBe1 …q1 † and two sequences fbn g; fb0n g  oBe1 …q1 † such that limn!1 bn ˆ limn!1 b0n ˆ q00 and u…bn ; …ÿ1; 0††  Be1 …q1 †;

…8†

u…b0n ; …0; ‡1††  Be1 …q1 †;

…9†

n ˆ 1; 2; . . . : fbn g and fb0n g arrange alternately on oBe1 …q1 †: In view of the continuity of the ¯ow we have u…q00 ; R†  Be1 …q1 †: From Lemmas 1 and 2 we have lim u…q00 ; t† ˆ q1 and lim u…q00 ; t† ˆ q1 :

t!ÿ1

t‡1

Let D1 be the closed region which is enclosed by q1 ; u…b1 ; …ÿ1; 0††; u…q00 ; …ÿ1; 0†† and a part of oBe1 …q1 † containing fu…bn ; …ÿ1; 0††jn ˆ 2; 3; . . . :g [ fu…b0n ; …0; ‡1††jn ˆ 2; 3; . . .g; and D2 be the closed region which is enclosed by q1 ; u…b1 ; …ÿ1; 0††; u…q00 ; …0; ‡1†† and a part of oBe1 …q1 † containing fu…bn ; …ÿ1; 0††jn ˆ 2; 3; . . .g [ fu…b0n ; …0; ‡1††jn ˆ 2; 3; . . .g If u…q00 ; …0; ‡1††  D1 , it contradicts to Eq. (9). Then u…q00 ; …ÿ1; 0††  D2 , which contradicts to Eq. (9). Thus there exists i 2 f2; 3; . . . ; mg such that the system (1) has a trajectory joining q1 and qi . Without loss of generality we suppose i ˆ 2: Let G2 ˆ fg 2 R2 j there exist i 2 f1; 2g such that limt!‡1 u…g; t† ˆ qi g; H2 ˆ fg 2 R2 j there exist j 2 f3; 4; . . . ; mg such that limt!‡1 u…g; t† ˆ qj g: Using the same method we can prove that system (1) has a trajectory joining a point of fq1 ; q2 g and a point of fq3 ; q4 ; . . . ; qm g: Repeat this step m ÿ 1 times, we can ®nd m ÿ 1 connecting orbits. These connecting orbits together with m singular points constitute a connected set. Let G be the set which is constituted by all the singular points, connecting orbits and homoclinic orbits of system (1), C be a simply closed curve of G, G1 be the region which is enclosed by C. From Lemma 1 and Lemma 2, every trajectory through a point of G1 is a connecting orbit, a singular point or a homoclinic orbit. From Lemma 3, G is a bounded closed simply connected region. Lemma 4 is proved. From Lemmas 1±4 we obtain the following result. Theorem 1. If the system (1) is a positive bounded system, and has m singular points but not any closed orbits and singular closed orbits, then (i) system (1) has at least m ÿ 1 connecting orbits; (ii) these connecting orbits together with all singular points constitute a connected set;

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(iii) all the singular points, connecting orbits and homoclinic orbits constitute a bounded closed simply connected region G; (iv) for an arbitrary point p 2 R2 n G, the positive semi-trajectory of u…p; R† tends to a singular point, while its negative semi-trajectory approaches to the infinity. Proposition 1. If the system (1) has just two singular points, which all simply singular points, and has not any closed orbits and singular closed orbits, then there exists a point q 2 R2 such that u…q; ‰0; ‡1†† is unbounded. Proof. Let fq1 ; q2 g be a singular point set of the system (1). Suppose that the system (1) is a positive bounded system. From Theorem 2.1 in [6] we have that there is a connecting orbit joining q1 and q2 . Thus q1 and q2 are not centers. Moreover, they are not sources or sinks at the same time. If there exists a saddle point in fq1 ; q2 g, without loss of generality we suppose that q1 is a saddle point. If q2 is a saddle point, then R2 n fq1 ; q2 g has at most four trajectories whose positive semi-trajectories tend to a singular point, respectively. This contradicts to Lemma 1. Use the same method, we can prove that q2 can not be a source. Suppose that q2 is a sink. Take fp1 ; p2 ; p3 ; p4 g  R2 n fq1 ; q2 g such that limt!‡1 u…p1 ; t† ˆ q1 ; limt!ÿ1 u…q2 ; t† ˆ q1 ; limt!‡1 u…p3 ; t† ˆ q1 ; limt!ÿ1 u…p4 ; t† ˆ q1 , and fu…pi ; R†ji ˆ 1; 2; 3; 4g are di€erent each other. Because the system (1) has connecting orbits, without loss of generality we suppose lim u…p2 ; t† ˆ q2 :

t!‡1

From Lemma 1 we have the positive semi-trajectory of u…p4 ; R† tends to a singular point. Suppose that limt!‡1 u…p4 ; t† ˆ q1 : Let G be a open region which is enclosed by q1 and u…p4 ; R†: Then G has in®nite trajectories whose negative semi-trajectories tend to a singular point. This contradicts to that q1 is a saddle point and q2 is a sink. Hence limt!‡1 u…p4 ; t† ˆ q2 : Let G1 be a open region which is enclosed by q1 ; q2 ; u…p2 ; R† and u…p4 ; R†. Then G1 has in®nite trajectories whose negative semi-trajectories tend to a singular point. This contradicts whose q1 is a saddle point and q2 is a sink. Hence there are not any saddle points in fq1 ; q2 g: So one of fq1 ; q2 g is a source, the other is a sink. Without loss of generality we suppose q1 is a source and q2 is a sink. Then there exists open neighbourhoods B…q1 † of q1 and B…q2 † of q2 such that B…q1 † \ B…q2 † ˆ ;; oB…q1 † and oB…q2 † are simple closed curves, every point of oB…q1 † is a strictly exit point of B…q1 † and every point of oB…q2 † is a strictly enter point of B…q2 † (see [13] and [14]). From Lemma 1, for an arbitrary point p 2 oB…q1 †; u…p; ‰0; ‡1†† must intersect with oB…q2 † and the intersection point pI is unique. We construct a mapping F: oB…q1 † ! oB…q2 †; F …p† ˆ pI . Then F …oB…q1 †† ˆ oB…q2 †: Let G0 be the region which is constituted by all the connecting orbits, homoclinic orbits and singular points of system (1). From Theorem 1, there exists T > 0 such that G  BT …O†: Take p00 2 R2 n BT …O†. From Lemma 1, u…p00 ; …0; ‡1†† must intersect with oB…q2 †: This contradicts to the uniqueness of the ¯ow. Proposition 1 is proved.

3. Applications In this section, we will give some examples to show the application of the theorems in section 2. Example 1. We consider the di€erential equations dx ˆ ÿx3 ÿ x2 ‡ 6x; dt

dy ˆ ÿy: dt

…10†

System (10) has three singular points q1 ˆ …0; 0†; q2 ˆ …ÿ3; 0† and q3 ˆ …2; 0†: q1 is a saddle point, q2 and q3 are sinks. For y 2 R we have dx=dt…0; y†j…10† ˆ 0; for x 2 R we have dy=dt…x; 0†j…10† ˆ 0: Thus x- and y-axis are all invariant sets of the system (10). For …x; y† 2 R2 ; we have dy=dt…x; y†j…10† < 0; as y > 0; dy=dt…x; y†j…10† > 0; as y < 0; dx=dt…x; y†j…10† < 0; as x > 2; dx=dt…x; y†j…10† > 0; as x < ÿ3. Hence the system (10) is a positive bounded system, which has not any closed orbits and singular closed orbits. From Theorem 1 we know that there exists two connecting orbits. Take x1 2 …0; 2†; x2 2 …ÿ3; 0†; y3 2 …0; ‡1†; y4 2 …ÿ1; 0†; x5 2 …2; ‡1†; x6 2 …ÿ1; ÿ3†; x7 2 …0; ‡1†; y7 2 …ÿ1; 0† [ …0; ‡1†; x8 2 …ÿ1; 0†;

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y8 2 …ÿ1; 0† [ …0; ‡1†: Let p1 ˆ …x1 ; 0†; p2 ˆ …x2 ; 0†; p3 ˆ …0; y3 †; p4 ˆ …0; y4 †; p5 ˆ …x5 ; 0†; p6 ˆ …x6 ; 0†; p7 ˆ …x7 ; y7 †; p8 ˆ …x8 ; y8 †. We have lim u…p1 ; t† ˆ q3 ;

t!‡1

lim u…p2 ; t† ˆ q2 ;

t!‡1

lim u…p3 ; t† ˆ q1 ;

t!‡1

lim u…p4 ; t† ˆ q1 ;

t!‡1

lim u…p5 ; t† ˆ q3 ;

t!‡1

lim u…p6 ; t† ˆ q2 ;

t!‡1

lim u…p7 ; t† ˆ q3 ;

t!‡1

lim u…p8 ; t† ˆ q2 ;

t!‡1

lim u…p1 ; t† ˆ q1 ;

t!ÿ1

lim u…p2 ; t† ˆ q1 ;

t!ÿ1

lim ku…p3 ; tk ˆ ‡1;

t!ÿ1

lim ku…p4 ; t†k ˆ ‡1;

t!ÿ1

lim ku…p5 ; t†k ˆ ‡1;

t!ÿ1

lim ku…p6 ; t†k ˆ ‡1;

t!ÿ1

lim ku…p7 ; t†k ˆ ‡1;

t!ÿ1

lim ku…p8 ; t†k ˆ ‡1:

t!ÿ1

References [1] Yu Shuxiang, The existence of trajectories joining critical points, J. Di€erential Equations 66 (2) (1987) 230±242. [2] Yu Shuxiang, Trajectories joining critical points, Math. Ann. (in chinese) 9A (6) (1988) 671±674. [3] C.C. Conley, J.A. Smoller, Viscosity matrices for two-dimensional nonlinear hyperbolic systems, Comm. Pure Appl. Math. 23 (1970) 867±884. [4] C. Conley, R. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc. 158 (1) (1971) 35±61. [5] C. Conley, Isolated invariant sets and the Morse index, in: Conf. Board Math. Sci. No. 38. Amer. Math. Soc., Providence, 1978.. [6] Huang Tusen, The existence of trajectories joining critical points, Acta Math. Sinica 40 (4) (1997) 551±558. [7] L. Markus, Global structure of ordinary di€erential equations in the plane, Trans. Amer. Math. Soc. 76 (1954) 127±148. [8] D. Neumann, T. O'Brein, Global structure of continuous ¯ow on 2-manifolds, J. Di€erential Equations 22 (1976) 89±110. [9] Jiang Jifa, On the existence and uniqueness of connecting orbits for cooperative systems, Acta Math. Sinica New Series 8 (2) (1992) 184±188. [10] Z. Artstein, M. Slemrod, Trajectories joining critical points, J. Di€erential Equations 44 (1982) 40±62. [11] Zheng Zuohuan, Global semistable systems on a plane, J. Sys. Sci & Math. Scis. (in Chinese) 14 (1) (1991) 21±28. [12] Zheng Zuohuan, On the limit cycles for a class of planar systems, Nonlinear Analysis 24 (4) (1995) 605±614. [13] Wang Huifeng, Yu Shuxiang, Qualitative theory of ordinary di€erential equations, Higher Education Press of Guangdong Province, 1996. [14] Zhang Zhifen et al., Qualitative Theory of Ordinary Di€erential Equations, Scienti®c Press (in China), 1985. [15] Yanqian Ye et al., Theory of Limit Cycles, Shanghai Scienti®c Press, 1984.