A Note on “Uniqueness of Limit Cycles in a Liénard-Type System”

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

208, 260]276 Ž1997.

AY975284

NOTE

A Note on ‘‘Uniqueness of Limit Cycles in a Lienard-Type System’’ ´ Robert E. Kooij Faculty of Technical Mathematics and Informatics, Uni¨ ersity of Technology Delft, Delft, The Netherlands

and Sun Jianhua Department of Mathematics, Nanjing Uni¨ ersity, Nanjing, People’s Republic of China Received June 26, 1995

Huang and Sun w J. Math. Anal. Appl. 184 Ž1994., 348]359x proposed a theorem to guarantee the uniqueness of limit cycles for the generalized Lienard system ´ dxrdt s hŽ y . y F Ž x ., dyrdt s yg Ž x .. We will give a counterexample to their theorem. It will be shown that their theorem is valid only if F Ž x . is monotone on certain intervals. For this case we give a shorter proof and we also show that the limit cycle is hyperbolic. If the condition on the monotonicity of F Ž x . is violated then we need an additional condition to guarantee the uniqueness of the limit cycle. Examples are provided to illustrate our results. Q 1997 Academic Press

1. INTRODUCTION Huang and Sun w5x proposed a theorem to guarantee the uniqueness of limit cyles for the generalized Lienard system ´ dx dy s hŽ y . y F Ž x . , s yg Ž x . , Ž 1.1. dt dt where the functions in Ž1.1. are assumed to be continuous and such that uniqueness for solutions of initial value problems is guaranteed. 260 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

NOTE

261

If we define, as usual, GŽ x . s H0x g Ž j . d j and H Ž y . s H0y hŽt . dt , then the conditions stated by Huang and Sun are as follows: Ži. hŽ0. s 0, hŽ y . is strictly increasing, and hŽ"`. s "`; Žii. xg Ž x . ) 0 when x / 0 and GŽ"`. s `; Žiii. there exist constants x 1 , x 2 with x 1 - 0 - x 2 such that F Ž x 1 . s F Ž0. s F Ž x 2 . s 0 and xF Ž x . - 0 for x g Ž x 1 , x 2 . _  04 ; Živ. there exist constants M ) 0, K, K 0 with K ) K 0 , such that F Ž x . ) K for x G M and F Ž x . - K 0 for x F yM; Žv. one of the following holds: Ža. GŽ x 1 . s GŽ x 2 ., or Žb. GŽyx . G GŽ x . for x ) 0. Furthermore let W Ž x . s hy1 Ž F Ž x .. Ž . H0 h y dy, where hy1 is the inverse function of h. Then Ž a . if x 2 F < x 1 < then max 0 F x F x  GŽ x . q W Ž x .4 G GŽ x 1 ., 2 Ž b . if 0 - < x 1 < - x 2 then max x F x F 0 GŽ x . q W Ž x .4 G GŽ x 2 .. 1 Conditions Ži. ] Žiii. imply that system Ž1.1. has a unique singularity, which will be an antisaddle, i.e., a critical point at which the product of the eigenvalues of the Jacobian matrix is positive. It has been proved by Huang and Sun w5x that it follows from conditions Ži. ] Žiii. that this singularity is located at O Ž0,0. and is unstable. v

If Živ. holds then there exists a closed curve G such that every trajectory intersecting it crosses it in the exterior-to-interior direction, hence implying the existence of at least one stable limit cycle, by the Poincare]Bendixson theorem; see, for instance, Andronov et al. w2x. ´ v

Finally, condition Žv. ensures that all closed trajectories of system Ž1.1. have to intersect both x s x 1 and x s x 2 . v

Huang and Sun claimed that if conditions Ži. ] Žv. hold then system Ž1.1. has exactly one limit cycle. We will point out that this claim is incorrect. In fact, in their proof Huang and Sun compare the values of the differential of the function GŽ x . q H Ž y . integrated along two limit cycles. However, this comparison is valid only if the following condition is added: F Ž x . is nondecreasing for x g Ž y`, x 1 . D Ž x 2 , ` . .

Ž 1.2.

We will give an example due to Zhang and Shi w12x, which satisfies conditions Ži. ] Žv. but violates Ž1.2., which has three limit cycles. If Ži. ] Žv. and Ž1.2. do hold we will give a short proof that system Ž1.1. has exactly one limit cycle, not by using a comparison method but by

262

NOTE

estimating the divergence of system Ž1.1. integrated along a limit cycle. By this we can show that the limit cycle is hyperbolic. A limit cycle is hyperbolic, or simple, if for any arbitrarily small analytic perturbation of the system there is no other limit cycle in a sufficiently small neighbourhood of the limit cycle; see, for instance, w2x. Next, we will state an additional condition to guarantee the uniqueness of the limit cycle in case Ž1.2. is violated. If the functions in Ž1.1. are all odd then system Ž1.1. exhibits symmetry with respect to the origin and the conditions of our theorem can be weakened. Finally, we provide some examples that illustrate our results.

2. THREE UNIQUENESS THEOREMS FOR SYSTEM Ž1.1. We will first state a theorem in case both Ži. ] Žv. and Ž1.2. hold. THEOREM 2.1. If conditions Ž i . ] Ž ¨ . and Ž1.2. hold then system Ž1.1. has exactly one closed orbit, a hyperbolic stable limit cycle. This theorem will be proved by showing that if g is a closed orbit then its characteristic exponent H s Hg y f Ž x . dt satisfies H - 0, where f Ž x . s Ž drdx . F Ž x .. This shows that g is hyperbolic and stable; see, for instance, Andronov et al. w2x. Because two adjacent limit cycles cannot both be stable, the uniqueness of g follows. In order to estimate the characteristic exponent the following lemma by Zeng et al. w10x appears to be useful. LEMMA 2.2. Let g be an arc of an orbit of the system Ž1.1., described by y Ž x ., a F x F b . Then

Hg y f Ž x . dt s sgn Ž h Ž y Ž a . . y F Ž a . . q

b

Ha

ln

F Ž b . y hŽ y Ž a . . F Ž a . y hŽ y Ž a . .

Ž F Ž b . y F Ž x . . g Ž x . Ž dhrdy . dx . 2 Ž F Ž b . y hŽ y Ž x . . . Ž F Ž x . y hŽ y Ž x . . .

Proof of Theorem 2.1. It was shown by Huang and Sun w5x that it follows from conditions Ži. ] Žv. that system Ž1.1. has at least one limit cycle G and it intersects both x s x 1 and x s x 2 ; see Figure 2.1. Denote the intersection point of G with the positive y-axis by A. Let B and C be the intersection points of G with x s x 2 in the first and fourth quadrant, respectively. If we denote the arc of G between A and B by g 1 , then

263

NOTE

F Ž x . satisfying Theorem 2.1.

FIG. 2.1.

applying Lemma 2.2 with a s 0 and b s x 2 yields

Hg y f Ž x . dt s H0 1

F Ž x . g Ž x . Ž dhrdy .

x2

hŽ y Ž x . . Ž F Ž x . y hŽ y Ž x . . .

2

dx.

This integral is negative because the integrand is negative by virtue of Ži. ] Žiii.. Thus have we proved

Hg y f Ž x . dt - 0. 1

For g 2 , the arc of G between B and C, we obtain by Ž1.2. and f Ž x . s Ž drdx . F Ž x .

Hg y f Ž x . dt - 0. 2

Proceeding in this way we can prove that HG y f Ž x . dt - 0. This completes the proof. If the monotonicity of F Ž x . is assumed only on the intervals Ž ˜ x 1 , x 1 . and Ž x2 , ˜ x 2 . then we can obtain the following: COROLLARY 2.3. If conditions Ž i . ] Ž ¨ . hold and F Ž x . is nondecreasing on Ž˜ x 1 , x 1 . and Ž x 2 , ˜ x 2 . then in the strip ˜ x1 F x F ˜ x 2 system Ž1.1. has at most one closed orbit, a hyperbolic stable limit cycle.

264

NOTE

Proof. If system Ž1.1. has a closed orbit then its uniqueness can be proved as in Theorem 2.1. However, in the strip ˜ x1 F x F ˜ x 2 the existence of a closed orbit is no longer guaranteed. Next we discuss the example of Zhang and Shi w12x which shows that if the conditions Ži. ] Žv. hold but Ž1.2. does not, then system Ž1.1. can have more than one limit cycle. Consider the following system dx dt

s y y «FŽ x. ,

dy dt

s yx,

Ž 2.1.

299 3 99 with 0 - « < 1 and F Ž x . s 351 x 7 y 403 x 5 q 4800 x y 6400 x. For « s 0 all trajectories of Ž2.1. are closed and satisfy H Ž x, y . s x 2 q y 2 s r 2 . To find the closed orbits for 0 - « < 1 we have to study Hx 2qy 2sr 2 dH s y2 «H02 p r cos tF Ž r cos t . dt q O Ž « 2 ., whose zeros correspond with limit cycles for system Ž2.1.; see Pontryagin w7x. An elementary calculation reveals that

I Ž r . s H02 p r cos tF Ž r cos t . dt s

p 64

r 2 Ž r 2 y 1. r 2 y

ž

9 10

/ž

r2 y

11 10

/

.

Because the nonzero roots of I Ž r . are simple it follows that Ž2.1. has three hyperbolic limit cycles, located in the vicinity of the circles, x 2 q y 2 s R, with R s Ž12 y i .r10, i s 1, 2, 3. It is easy to check that conditions Ži. ] Žv. hold but Ž1.2. is not satisfied, as can be seen by plotting the graph of F Ž x .. If Ž1.2. is violated then we need an additional condition to guarantee the uniqueness of the limit cycle. In order to formulate this condition we will use a lemma by Zhang et al. w11x. LEMMA 2.4. Let F1Ž x . s F Ž x . and F2 Ž x . s F Žyx ., both for 0 F x F d, where either d g Rq or d s `. Suppose that conditions Ž i . ] Ž iii . are satisfied and in addition assume that the following holds: ŽI. g Žyx . s yg Ž x . and g Ž x . is nondecreasing as x increases; ŽII. y s F1Ž x . intersects y s F2 Ž x . at two points, Ž0, 0. and Ž a, b . with 0 - a - d; ŽIII. F2 Ž x . G F1Ž x . for x g Ž0, a.; ŽIV. For j s 1, 2 there exist t j , j j , g w a, d x with t j F j j such that Ža . Žy 1 . jF jŽ x . F 0 fo r x g wt j , r x ; w a , d x, where r s max js1, 2 t j q j j 4 ; Žb. Žy1. j Fj Ž x . q Žy1. 3y j F3yj Ž x q j j . F 0,k 0, for x g w0, t j x; Žc. F1Ž x . ) 0 and F2 Ž x . - 0 for x g  r, d x.

NOTE

265

Then for all x 0 g w r, d x the backward and forward orbits passing through Ž x 0 , hy1 Ž F Ž x 0 ... cross the y-axis in A and B, respectively. Similarly, the forward and backward orbits passing through Žyx 0 , hy1 Ž F Žyx 0 ... cross the y-axis in C and D, respectively, where yA ) yC and yB ) yD ; see Fig. 2.2. The proof of Lemma 2.4 can be found in Zhang et al. w11, Theorem 7.9, Chap. 4x, where conditions that guarantee the existence of at least n limit cycles are derived. In fact, Lemma 2.4 corresponds with the case n s 1. For a geometrical interpretation of F Ž x . in Lemma 2.4 we refer to Fig. 2.3. We will now show an important implication of Lemma 2.4. COROLLARY 2.5. Lemma 2.4 implies that for all x 0 g w r, d x system Ž1.1. has no closed orbits in the strip < x < F d which cross x s x 0 or x s yx 0 . Proof. First the nonexistence of closed orbits intersecting both x s yx 0 and x s x 0 is shown. Assume by contradiction that system Ž1.1. has a closed orbit G1 intersecting y s hy1 Ž F Ž x .. in SŽ x s , hy1 Ž F Ž x s ... and T Ž x t , hy1 Ž F Ž x t ..., with x s ) x 0 and x t - yx 0 ; see Fig. 2.4. First assume that x s ) yx t . Let R denote the intersection of G1 with the positive y-axis. Then by Lemma 2.4 the forward orbit g passing through Žyx s , hy1 Ž F Žyx s ... will cross the positive y-axis, say in U, such that yU - yR . This is impossible because obviously g cannot intersect G1. The case x s F yx t can be dealt with in a similar way.

FIG. 2.2. The implication of Lemma 2.4.

266

NOTE

FIG. 2.3.

F Ž x . satisfying Lemma 2.4.

Finally we exclude the possibility of a closed orbit intersecting only x s yx 0 or x s x 0 . An oscillatory orbit intersecting x s yx 0 but not x s x 0 has to cross the y-axis between A and C; see Fig. 2.2. But then, because yB ) yD , this trajectory cannot intersect x s yx 0 again so it cannot be closed. The same argument holds for trajectories crossing x s x 0 but not x s yx 0 . This completes the proof.

FIG. 2.4. Limit cycle intersecting yx 0 and x 0 .

267

NOTE

Remark 2.1. If g Ž x . does not satisfy condition ŽI. of Lemma 2.4 then we can apply the transformation u s 2G Ž x . sgn x to reduce system Ž1.1. to

'

du dt

dy

s hŽ y . y F Ž x Ž u. . ,

dt

s yu,

Ž 2.2.

where x Ž u. is the inverse function of u s 2G Ž x . sgn x. Now Ž2.2. satisfies condition ŽI. of Lemma 2.4, because g Ž u. s u, but in general it will be quite cumbersome to check the other conditions.

'

THEOREM 2.6. Suppose that system Ž1.1. satisfies conditions Ži. ] Žiii., Žv., and ŽI. ] ŽIV. and in addition assume that F9 Ž x . G 0

for x g Ž yr , x 1 . D Ž x 2 , r . .

Ž 2.3.

Then in the strip < x < F d system Ž1.1. has exactly one closed orbit, a hyperbolic stable limit cycle. Proof. Consider the backward and forward trajectories passing through B0 Ž r, hy1 Ž F Ž r ... and suppose that they cross the y-axis in A 0 and C0 , respectively. Similarly, suppose that the forward and backward trajectories passing through E0 Žyr, hy1 Ž F Žyr ... cross the y-axis in F0 and D 0 , respectively. Then ! # by Lemma " 2.4 every trajectory of Ž1.1. intersecting the curve A 0 B0 C0 D 0 E0 F0 A 0 crosses it in the exterior-to-interior direction, because yA 0 ) yF 0 and yC 0 ) yD 0 ; see Fig. 2.5. Because O Ž0, 0. is an unstable antisaddle it follows from the Poincare]Bendixson theorem that system Ž1.1. has at least one limit cycle ´ in the strip < x < - r. By condition Žv. any such limit cycle will have to intersect both x s x 1 and x s x 2 . Because Ž2.3. holds it follows from Corollary 2.3 that the limit cycle is hyperbolic and stable and hence unique. It follows from applying Corollary 2.5 with x 0 s r that there are no limit cycles in the strip < x < F d that cross x s yr or x s r. This completes the proof. If hŽ y . s y and g Ž x . s x then system Ž1.1. reduces to dx dt

s y y FŽ x. ,

dy dt

s yx.

Ž 2.4.

If an additional condition is satisfied then for system Ž2.4. the conditions of Theorem 2.6 can be weakened.

268

NOTE

FIG. 2.5. Orbits passing through B0 and E0 .

THEOREM 2.7. Suppose that system Ž2.4. satisfies conditions Žiii., Žv., and ŽII. ] ŽIV. and in addition assume that there exist a 2 ) x 2 , a 1 - x 1 such that F9 Ž x . G 0 for x g w x 2 , a 2 x ,

F9 Ž a 2 . s 0,

F0 Ž x . F 0 for x g w a 2 , r x , F9 Ž x . G 0 for x g w a 1 , x 1 x ,

Ž 2.5. F9 Ž a 1 . s 0,

F0 Ž x . 0 G 0 for x g w yr , a 1 x ,

Ž 2.6.

and F Ž a2 . G M q ˆ x

and

F Ž a 1 . F yM y ˆ x,

Ž 2.7.

where ˆ x s max ya 1 , a 2 4 , M s max max 0 F x F x 2ŽyF Ž x .., max x 1 F x F 0 Ž F Ž x ..4 . Then in the strip < x < F d system Ž2.4. has exactly one closed orbit, a hyperbolic stable limit cycle. For the proof of Theorem 2.7 we will use a modification of a result by Rychkov w8x; see also Ye et al. w9, Theorem 7.2x. LEMMA 2.8. Suppose that there exist constants b 1 - a 1 - x 1 - 0 - x 2 a 2 - b 2 such that Ž a. Ž b.

F Ž x 1 . s F Ž0. s F Ž x 2 . s 0 and xF Ž x . - 0 for x g Ž x 1 , x 2 . _  04 ; F9Ž x . G 0 for x g Ž a 1 , x 1 . j Ž x 2 , a 2 .;

NOTE

269

Ž c . F0 Ž x . F 0 for x g Ž a 2 , b 2 . and F0 Ž x . G 0 for x g Ž b 1 , a 1 .; Ž d . e¨ ery closed orbit of system Ž2.4. intersects both x s x 1 and x s x 2 ; Ž e . e¨ ery closed orbit of system Ž2.4. that is not inside the strip a 1 F x F a 2 intersects both x s a 1 and x s a 2 . Then system Ž2.4. has at most two limit cycles in the strip b 1 F x F b 2 . If two limit cycles exist then the inner one is stable and the outer one unstable. The method of proof for Lemma 2.8 is exactly the same as in Rychkov’s theorem; see Ye et al. w9, Theorem 7.2x. This is because the limit cycles of Ž2.4. are either intersecting both x s a 1 and x s a 2 , or they are inside the strip a 1 F x F a 2 while intersecting both x s x 1 and x s x 2 . Proof of Theorem 2.7. It was proved by Huang w3x, see also Ye et al. w9, Theorem 5.2x, that it follows from condition Ž2.7. that the backward orbits passing through P2 Ž a 2 , F Ž a 2 .. and P1Ž a 1 , F Ž a 1 .. intersect x s a 1 and x s a 2 in B1 and A1 respectively with yB1 ) F Ž a 1 . and yA 1 - F Ž a 2 .; see Fig. 2.6. An oscillatory orbit intersecting x s a 1 but not x s a 2 has to cross x s a 1 between B1 and P1. But then this trajectory cannot cross x s a 1 again Žbecause it will first intersect y s F Ž x .; see Fig. 2.6.; hence it cannot be closed. The same argument holds for trajectories crossing x s a 2 but not x s a 1. Therefore Ž2.7. implies that condition Že. of Lemma 2.8 is

FIG. 2.6. Backward orbits through P1 and P2 .

270

NOTE

satisfied. Because O Ž0, 0. is an unstable antisaddle the existence of a limit cycle Gin in the strip a 1 F x F a 2 also follows by the Poincare]Bendixson ´ theorem. Condition Žd. is satisfied because Žv. holds; see Huang and Sun w5x. The limit cycle Gin in the strip a 1 F x F a 2 is unique, hyperbolic, and stable by Corollary 2.3. As in Theorem 2.6 there are no limit cycles intersecting x s yr or x s r. Because all conditions of Lemma 2.8 are satisfied it follows that if system Ž2.4. has a limit cycle Gout intersecting x s a 1 and x s a 2 then it has to be unstable and unique. This is impossible because as a result of the Poincare]Bendixson theorem the ´ number of stable and unstable limit cycles in the strip < x < F r intersecting both x s a 1 and x s a 2 has to be equal; see, for instance, w2x. This completes the proof. Remark 2.2. It is easy to see that if we assume that F9Ž x . G 0 on wyr, x 1 x instead of Ž2.6. then we do not need Ž2.7. to prove the uniqueness of the limit cycle. Remark 2.3. If both on w x 1 , 0x and on w0, x 2 x F9Ž x . has only one zero, say a1 and a2 , respectively, then condition Ž2.7. can be weakened, see Huang and Yang w4x, to F Ž a2 . G M q ˆ xy

s 2Mqˆ x

and

F Ž a 1 . F yM y ˆ xq

s

, 2Mqˆ x Ž 2.8.

where M s max yF Ž a2 ., F Ž a1 .4 , ˆ x s max ya 1 , a 2 4 , and s s min 12 a 22 1 2 1 1 2 2 y 2 a2 , 2 a 1 y 2 a1 4 .

3. UNIQUENESS THEOREMS WHEN hŽ y ., g Ž x ., AND F Ž x . ARE ODD If the functions hŽ y ., g Ž x ., and F Ž x . are odd then system Ž1.1. is symmetric with respect to the origin. This means that the conditions of Theorems 2.6 and 2.7 can be weakened. For this case we will not use Lemma 2.4 but a more general result by Alsholm w1, Corollary 3x. LEMMA 3.1. Consider system Ž1.1. and suppose that Ž i . and Ž ii . are satisfied. Furthermore assume that Ž a . hŽyy . s yhŽ y ., g Žyx . s yg Ž x ., and F Žyx . s yF Ž x .; Ž b . there exists x 2 ) 0 such that F Ž0. s F Ž x 2 . s 0 and F Ž x . - 0 for x g Ž0, x 2 .;

271

NOTE

Žg . let I s w0, x 2 x and J s w x 2 , d x with x 2 - d. w : I ª J is weakly increasing, continuous and g Ž w Ž x . . w 9Ž x . G g Ž x . Žd . Ž« .

for

x g I;

with w satisfying Žg . we ha¨ e F Ž w Ž x .. G yF Ž x . for x g I; F Ž x . ) 0 for all x 0 g w x 2 , d x.

Then for all x 0 g w w Ž x 2 ., d x the backward and forward orbits passing through Ž x 0 , hy1 Ž F Ž x 0 ... cross the y-axis, in A and B respecti¨ ely and yA ) yyB . Remark 3.1. If the functions in system Ž1.1. are all odd then Lemma 2.4 corresponds to a special case of Lemma 3.1 with w Ž x . s x q j , with j s j 1 s j 2 , by symmetry. THEOREM 3.2. Suppose that system Ž1.1. satisfies conditions Ž i ., Ž ii ., Ž a ., Ž b ., Žg ., Ž d ., and Ž « .. Furthermore assume that F9 Ž x . G 0

for

x g Ž x2 , w Ž x2 . . .

Ž 3.1.

Then in the strip < x < F d system Ž1.1. has exactly one closed orbit, a hyperbolic stable limit cycle. The proof of Theorem 3.2 is basically the same as that of Theorem 2.6 and is therefore omitted. Note that we have dropped condition Žv. because if Ž a . holds then if Ž x Ž t ., y Ž t .. is a solution of Ž1.1. then so is Žyx Ž t ., yy Ž t ... Therefore every closed orbit is symmetric with respect to the origin. For a geometrical interpretation of Theorem 3.2 we refer to Fig. 3.1. Finally we will state a theorem for the case that hŽ y . s y, g Ž x . s x, and F Žyx . s yF Ž x ., hence weakening the conditions of Theorem 2.7. THEOREM 3.3. Suppose that system Ž2.4. with F Žyx . s yF Ž x . satisfies the conditions Ž b ., Žg ., Ž d ., and Ž « .. Furthermore assume that there exists

FIG. 3.1.

F Ž x . satisfying Theorem 3.2.

272

NOTE

a ) x 2 such that F9 Ž x . G 0 for x g w x 2 , a x

and

F0 Ž x . F 0 for x g a , w Ž x 2 . .

Then in the strip < x < F d system Ž2.4. has exactly one closed orbit, a hyperbolic stable limit cycle. The proof is basically the same as the proof of Theorem 2.7 and is therefore omitted. Remark 3.2. We have dropped condition Ž2.7. as condition Že. of Lemma 2.8 is always satisfied because every closed orbit is symmetric with respect to the origin and a 1 s ya 2 . Remark 3.3. Because Ž2.7. does not necessarily hold the limit cycle might not be contained in < x < F a . If this occurs then we cannot prove that the limit cycle is hyperbolic through Theorem 2.1. However, we can use a result by Odani w6, Theorem Bx, to arrive at the same conclusion.

EXAMPLES In this section we will give some examples that illustrate our results. For all systems in this section we will prove that there exists a unique, hyperbolic stable limit cycle. EXAMPLE 4.1. Consider the system dx dt

s

1 5

y y x Ž x y 1 . Ž x q 1.1. ,

dy dt

s yx 3 .

Ž 4.1.

This example was discussed by Huang and Sun w5, Example 3.1x. It satisfies all conditions of Theorem 2.1. EXAMPLE 4.2. Consider the system dy dt

s h Ž y . y kF Ž x . ,

dy dt

s yg Ž x . ,

Ž 4.2.

with k ) 0, F Ž x . s 201 x Ž x 2 y 1.Ž20 x 2 y 140 x q 247., see Fig. 4.1, and where hŽ y . and g Ž x . satisfy Ži., Žii., and ŽI.. It is easy to check that Žiii., Žv., ŽII., and ŽIII. are also satisfied with x 2 s yx 1 s 1 and hence a s 1. In order to apply Theorem 2.6 we only need to ascertain that ŽIV. and Ž2.3. are also satisfied. It can be shown that for t 1 s t 2 s 1, j 1 s j 2 s 1.1 ŽIV. is also satisfied; see Fig. 4.2. Note that F9Ž x . ) 0 for x g w 1, a . with a f 2.27427 and so r s 1 q 1.1 - a and hence Ž2.3. also holds.

273

NOTE

FIG. 4.1.

F Ž x . in system Ž4.2..

EXAMPLE 4.3. Consider the system dx dt

s y y FŽ x. ,

dy dt

s yx,

Ž 4.3.

1 with F Ž x . s 1000 x Ž x 2 y 1.Ž10 x 2 y 46 x q 53.Ž100 x 2 q 460 x q 539.; see Fig. 4.3. We will show that Theorem 2.7 can be applied. Again it is easy to verify conditions Žiii., Žv., ŽII., and ŽIII. with x 2 s yx 1 s 1 and hence a s 1. Let a 2 s min x ) 1 N F9Ž x . s 04 and b 2 s min x ) a 2 N F0 Ž x . s 04 . In a similar way define a 1 and b 1; see also Fig. 4.3. It can be shown that for t 1 s t 2 s 1, j 1 s j 2 s 1, ŽIV. is also satisfied; see Fig. 4.4. Note that r s 1 q 1 s 2. Because b 2 f 2.04569 ) r and b 1 f y2.00427 - yr Ž2.5. and Ž2.6. also hold. Finally we will verify Ž2.7.. Clearly, M s max yF Žg 2 ., F Žg 1 .4 , where F9Žg 2 . s 0, 0 - g 2 - 1 and F9Žg 1 . s 0, y1

FIG. 4.2.

F1Ž x . and F2 Ž x . in system Ž4.2..

274

NOTE

FIG. 4.3.

F Ž x . in system Ž4.3..

- g 1 - 0. In fact M s F Žg 1 . f 9.88062. With ˆ x s max ya 1 , a 2 . s ya 1 f 1.75595 we can check that F Ž a 2 . f 19.3022 ) M q ˆ x and F Ž a 1 . f y23.8457 - yM y ˆ x. Remark 4.1. If we multiply F Ž x . in system Ž4.3. with a constant k ) 0 then all conditions of Theorem 2.7 still hold, except possibly condition Ž2.7.. However, for

k G max

½

yxˆ ˆx , f 0.186375 F Ž a2 . y M F Ž a1 . q M

FIG. 4.4.

5

F1Ž x . and F2 Ž x . in system Ž4.3..

275

NOTE

Ž2.7. is also satisfied. By using Ž2.8. instead of Ž2.7. this lower bound of k can be lowered. We leave this as an exercise for the reader. EXAMPLE 4.4. Consider the system dx dt

s h Ž y . y kF Ž x . ,

dy dt

s yx,

Ž 4.4.

with k ) 0, F Ž x . s 4 x Ž x 2 y 1.rŽ4 q 3 x 4 ., hŽ y . satisfies Ži., and hŽyy . s yhŽ y .. We will show that the conditions of Theorem 3.2 are fulfilled. It is easy to see that Ž a . and Ž b . are satisified with x 2 s 1; see Fig. 4.5. Let a ) 1 satisfy F9Ž a . s 0 , then a f 1.98273 - 2. Therefore we cannot use a function of the form w Ž x . s x q C, C G 1, as in Lemma 2.4 to prove that Žg . holds such that w Ž1. - a . Instead we propose to use w Ž x . s 'x 2 q 2 with x g w0, 1x. Obviously w : w0, 1x ª w'2 , '3 x ; w1, `. and w Ž x . w 9Ž x . s x. Because for x g w0, 1x F Ž w Ž x .. G yF Ž x ., see Fig. 4.6, Žg . is also satisfied. As w Ž1. s '3 - a , Ž3.1. holds and all conditions of Theorem 3.2 are satisfied, with d s `. Remark 4.2. The choice of the function w Ž x . is motivated by Example 1 of Odani w6x.

ACKNOWLEDGMENTS The research which led to this paper was conducted while the first author was visiting the second author in May]July 1995. Financial support for this visit was granted by the Netherlands Organization for Scientific Research ŽNWO., Grant R 61-321.

FIG. 4.5.

F Ž x . in system Ž4.4..

276

NOTE

FIG. 4.6.

F Ž x . and F Ž w Ž x .. in system Ž4.4..

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