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A Class of Cubic Systems with Two Centers or Two Foci
Journal of Mathematical Analysis and Applications 242, 154᎐163 Ž2000. doi:10.1006rjmaa.1999.6630, available online at http:rrwww.idealibrary.com on
A Class of Cubic Systems with Two Centers or Two Foci Lansun Chen Institute of Mathematics, Academia Sinica, Beijing 100080, China
Zhengyi Lu Department of Mathematics, Sichuan Uni¨ ersity, Chengdu 610064, China
and Dongming Wang Laboratoire LEIBNIZ, Institut IMAG, 46, a¨ enue Felix ´ Viallet, 38031 Grenoble, France Submitted by Raul ´ Manase ´ ¨ ich Received February 11, 1999
In this paper, the necessary and sufficient conditions for a class of cubic differential systems to possess two centers are given. Some conditions for the systems to have two weak foci are also derived by using the computer algebra system Maple. 䊚 2000 Academic Press Key Words: Kukles system; center; focus.
1. INTRODUCTION We consider the cubic system of center and focus type
˙x s y,
˙y s yx q Q Ž x, y . ,
Ž 1.
together with its associated system
˙x s y,
˙y s yx q y q Q Ž x, y . ,
where Q Ž x, y . s a1 x 2 q a2 xy q a3 y 2 q a4 x 3 q a5 x 2 y q a6 xy 2 q a7 y 3 . 154 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
Ž 2.
CUBIC SYSTEMS WITH TWO CENTERS
155
The conditions for E0 s Ž0, 0. to be a center of Ž1. were given by Kukles w6x in 1944. Two subclass systems of Ž1., i.e., the cases in which a2 a7 s 0, were considered recently in w2, 5, 8x. For a2 s 0, an example given in w5x suggests that Kukles’ conditions are incomplete; it is then proved in w1x that the conditions are indeed incomplete by showing that the origin is a center in this case. Although Kukles’ conditions are not necessary, w1, 6x proved that they are sufficient. Especially, we have THEOREM A. Conditions Žk4. are sufficient for E0 to be a center of system Ž1., a22 a4 q a5 k s 0,
Ž 3a7 k q k 2 q a6 a22 . a5 y 3a7 k 2 y a6 a22 k s 0, k q a1 a2 q a5 s 0,
Ž k4.
9 a6 a22 q 2 a42 q 9k 2 q 27a7 k s 0, where k s a2 a3 q 3a7 . In w8x, Sadovskii considered Ž1. with a2 s 0 and proved the following THEOREM B. For system Ž1. with a2 s 0, the origin is a center if and only if one of the following conditions holds: Ži. Žii. 3'2 a1.
a5 s a7 s 0; a3 s y2 a1 , a6 s 0, a4 s y1r3a12 , a5 s .1r3'2 a12 , a7 s "1r
It is also shown in w8x that if E0 is a focus of Ž1. with a2 s 0, then the maximal order of E0 is 6 and six small amplitude limit cycles can be constructed for its perturbation system Ž2. with a2 s 0. The following result is established in w2x. THEOREM C. For system Ž1. with a7 s 0, the origin is a center if and only if one of the following conditions holds: Ži. a2 s a5 s 0; Žii. a1 s a3 s a5 s 0; Žiii. a4 s a5 s a6 s 0, a1 q a3 s 0; Živ. a4 s Ž a1 q a3 . a3 , a5 s yŽ a1 q a3 . a2 , a6 s yŽ a1 q a3 . a23 Ž a1 q y1 2 a3 . . In this case, the maximal order for E0 to be a focus of Ž1. with a7 s 0 is 5 and five small amplitude limit cycles can be constructed for system Ž2. with a7 s 0.
156
CHEN, LU, AND WANG
As pointed out in w1x, determining the necessary and sufficient conditions for E0 to be a center of Kukles’ system Ž1. is still a problem that remains open. On the other hand, the problem of finding conditions for cubic systems to have two or more centers has also been considered in the literature w3, 4, 7x. In the present paper, we are concerned with this problem for the Kukles system and derive necessary and sufficient conditions for the system to have two centers or two foci. The two-center conditions obtained are given in Section 2. The sufficient and necessary conditions for Kukles’ system to have two weak foci are established in Section 3 for the case a2 a7 s 0. In Section 4, the two weak foci problem is considered with the help of Maple in dealing with large polynomials. The paper is concluded with some remarks.
2. TWO CENTERS Clearly, system Ž1. has at most two fixed points E x 1 s Ž x 1 , 0. and E x 2 s Ž x 2 , 0. other than the origin. All the fixed points are located on x-axis, which is an isoline. By an index theorem, if x 1 s x 2 , then E x 1 cannot be of center-focus type; if x 1 / x 2 , then at most one of them, say, E x 2 has a center-focus type. Note that the fixed points along the x-axis are alternatively saddles and anti-saddles, therefore if x 1 - 0 - x 2 , then system Ž1. has two saddles. In the sequel, we consider two cases 0 - x 1 - x 2 and 0 ) x 1 ) x 2 . Clearly, x 1 and x 2 are distinct solutions of the quadratic equation y1 q a1 x 2 q a4 x 22 s 0,
Ž 3.
and x 1 x 2 ) 0 if and only if ya12r4 - a4 - 0.
Ž 4.
If a1 - 0, then x2 s
1 2 a4
q 4 a4 .
1r2
/ - 0;
y Ž a12 q 4 a4 .
1r2
/ ) 0.
ž ya q Ž a 1
2 1
if a1 ) 0, then x2 s
1 2 a4
ž ya
1
157
CUBIC SYSTEMS WITH TWO CENTERS
By using the transformation x s x y x2 ,
y s y,
system Ž1. leads to
˙x s y, ˙y s yax q ⌬ y q a1 x 2 q a2 xy q a3 y 2 q a4 x 3 q a5 x 2 y
Ž 5.
q a6 xy 2 q a7 y 3 , where x and y are used instead of x and y, and a s a1 x 2 y 2, ⌬ s Ž a 2 q a5 x 2 . x 2 , a1 s a1 q 3a4 x 2 , a 2 s a 2 q 2 a5 x 2 , a 3 s a 3 q a6 x 2 . Since E x 2 is supposed to be of center-focus type, ⌬ ' 0, that is Ž x 2 / 0. a1 q a5 x 2 s 0.
Ž 6.
It is easy to know from Ž3. that
¡1 2 a4 a s a1 x 2 y 2 s
~
¢
a1 y Ž a12 q 4 a4 .
1r2
Ž a12 q 4 a4 .
1r2
) 0,
a1 - 0, 1 2 a4
ya1 y Ž a12 q 4 a4 .
1r2
Ž a12 q 4 a4 .
1r2
) 0,
a1 ) 0.
From the above discussions, we obtain LEMMA 1.
E x 2 is of center-focus type if and only if Ž3., Ž4., and Ž6. hold.
Clearly, Ž3., Ž4., and Ž6. imply in the case of a5 / 0 that U0 s a22 a4 y a1 a2 a5 y a52 s 0.
Ž 7.
LEMMA 2. System Ž1. with a2 s 0 has two centers if and only if Ž4. holds and a5 s a7 s 0.
158
CHEN, LU, AND WANG
Proof. Suppose that both the origin and E x 2 are centers of Ž1.. Then by Theorem B and Ž6., we have a5 s 0,
a7 s 0.
Ž 8.
Note that x 2 / 0. Observe that a ) 0, and under Ž8., Ž5.< a 2s0 is invariant under the transformation Ž x, y, t . ª Ž x, yy, yt .. It follows that the origin is a center of Ž5.< a 2s0 by the symmetry principle. This proves the lemma. Now consider system Ž1. with a2 a5 / 0. By the transformation x s a1r2 x,
y s y,
s a1r2 t
with a s a1 x 2 y 2 s y2 y a1 a2ra5 , system Ž5. is changed to
˙x s y, ˙y s yx q b1 x 2 q b 2 xy q b 3 y 2 q b4 x 3 q b5 x 2 y q b6 xy 2 q b 7 y 3 ,
Ž 9.
where x and y are used instead of x and y, and b1 s ay3r2 a1 y
ž
b 3 s ay1r2 a3 y
ž
b6 s ay1 a6 ,
3a2 a4 a5 a 2 a6 a5
/
/ ,
b 2 s yay1a2 ,
,
b4 s ay2 a4 ,
b5 s ay3r2 a5 ,
b 7 s ay1r2 a7 .
LEMMA 3. Let Ž4. hold true and let a7 s 0. Then system Ž1. has two centers if and only if Živ. in Theorem C holds. Proof. Clearly, we only need to check the sufficiency. The corresponding values in Živ. of Theorem C for system Ž5. at the origin are c1 s b4 y Ž b1 q b 3 . b 3 , c 2 s b5 y Ž b1 q b 3 . b 2 , c 3 s b6 q Ž b1 q b 3 . b 32 Ž b1 q 2 b 3 .
Ž 10 . y1
.
Substituting each concrete bi Ž i s 1, . . . , 6. which is a function of a i Ž i s 1, . . . , 6. into c j Ž j s 1, 2, 3. and removing the nonzero factors, we obtain a set of three polynomials, denoted by c1 , c 2 , c 3 , in a i Ž i s 1, . . . , 6..
CUBIC SYSTEMS WITH TWO CENTERS
159
By using Wu’s method w10x, it can be proved that Živ. of Theorem C implies that c1 s c 2 s c 3 s 0. This shows that the origin is a center of Ž5.. The proof is complete. Denote the ith focal value of E0 by Vi and of E x 2 by Ui ; V1 , V2 , V3 , V4 , V5 , V6 , and U1 consist of 4, 13, 49, 131, 292, 577, and 11 terms, respectively. How to compute focal values of a system of differential equations of center-focus type is described in w9x. LEMMA 4. V1 s V2 s U0 s U1 s 0 and a2 a7 / 0 if and only if the four equalities Žk4. in Theorem A hold. Lemma 4 means that, when E0 is a center, Žk4. are necessary conditions for E x 2 to be a center. Lemma 4 also implies that if both E0 and E x 2 are weak foci and one of them has order 2, then the other has order at most 1. Now we show that Žk4. are also sufficient to ensure E x 2 to be a center. LEMMA 5. If the equalities in Theorem A hold, then the origin is a center of Ž9., which in turn implies that E x 2 is a center of Ž1.. Proof. By Theorem A, to prove the origin to be a center, one only needs to check whether the equalities in Theorem A hold or not. The four Kukles’ values of system Ž1., i.e., the left-hand sides of the four equalities in Theorem A by substituting a i s bi into them, give a set of polynomials in a i : PS s P1 , P2 , P3 , P4 4 , where P1 , P2 , P3 , and P4 have 8, 26, 10, and 19 terms, respectively. Application of Wu’s method shows that Žk4. implies PS s 0. It follows that for system Ž9., Žk4. holds true when the a i are replaced by the corresponding bi . Hence the origin is a center of Ž9.. THEOREM 1. Kukles’ system Ž1. has two centers if and only if ya12r4 a4 - 0 and one of the following conditions holds: Ži. a2 s a5 s a7 s 0; Žii. a7 s 0, a4 s Ž a1 q a3 . a3 , a5 s yŽ a1 q a3 . a2 , a6 s yŽ a1 q a3 . a32 Ž a1 q 2 a3 .y1 ; Žiii. Žk4.. Remark. Theorem 1 can also be proved by the results of w1x and w7x which state that if a system has a first integral of Darboux type and if the algebraic curves used to construct the first integral do not pass through the second fixed point, then the first integral is analytic at the second fixed point, yielding automatically that the point is a center. However, in this paper, we emphasize the mechanical manipulation of checking the centers without using the first integrals for the second center.
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CHEN, LU, AND WANG
3. TWO FOCI FOR KUKLES SYSTEMS WITH a2 a7 s 0 THEOREM 2. System Ž1. with a2 s 0 has two weak foci if and only if ya12r4 - a4 - 0, a5 s 0, and a7 / 0. Proof. Suppose that both E0 and E x 2 are weak foci. Then Ž3. holds and Ž6. becomes a5 x 2 s 0. So x 2 / 0 leads to a5 s 0. Thus V1 s 3a7 and U1 s 3a1r2 a7 . Clearly, V1 s 0 or U1 s 0 implies a7 s 0. According to Theorem B, both E0 and E x 2 are centers of Ž1.. This shows that if both E0 and E x 2 are weak foci, they have order 1. THEOREM 3. System Ž1. with a7 s 0 has two weak foci if and only if ya12r4 - a4 - 0 and one of the following conditions holds: Ži. a4 s a3 Ž a1 q a3 ., a5 s ya2 Ž a1 q a3 ., a6 / ya23 Ž a1 q a3 .Ž a1 q 2 a3 ; Žii. U0 s 0, U1 s 0, U2 / 0; Žiii. U0 s 0, U1 / 0. .y1
Proof. Ži. First, we show that if system Ž1. with a7 s 0 has two weak foci, then the order of E0 is at most 2. Suppose that E0 is a weak focus of order greater than 2; then V1 s 0 and V2 s 0. Clearly, V1 s 0 is equivalent to a5 s ya2 Ž a1 q a3 . .
Ž 11 .
Substituting Ž11. into U0 s 0 yields a4 s a3 Ž a1 q a3 . .
Ž 12 .
Simplified by Ž11. and Ž12., V2 s 0 leads to a1 a23 q a33 q a1 a6 q 2 a3 a6 s 0.
Ž 13 .
That Ž11., Ž12., and Ž13. are the conditions ŽŽiv. of Theorem C. for E0 to be a center gives a contradiction. Therefore, if E x 2 is a weak focus, it has order at most 1. Namely, U0 s 0, V1 s 0, U1 / 0. Clearly, these are just the conditions in Ži.. Žii. E0 is a focus of order 2, i.e., V1 s 0. If E x is a focus of order 2, 2 then U1 s 0. It is easy to check that U0 s 0, U1 s 0, and V1 s 0 imply, in this case, Živ. of Theorem C. This is a contradiction. Hence, V1 / 0. Žiii. Let E0 be a focus of order 1 and E x have order greater than 2, 2 i.e., U0 s 0, U1 s 0, and U2 s 0. If a1 a2 q 2 a5 s 0, then U0 s 0 leads to
CUBIC SYSTEMS WITH TWO CENTERS
161
a14 q 4 a4 s 0, which contradicts Ž4.. Now eliminating a6 from U1 and U2 and removing the non-zero factors, we obtain a polynomial R1 with 11 terms R1 s y3a1 a42 a3 a4 q 6 a24 a42 q 3a12 a23 a3 a5 q 3a1 a23 a4 a5 y 6 a23 a3 a4 a5 y 8 a12 a22 a52 q 10 a1 a22 a3 a52 q 16 a22 a4 a52 y 28 a1 a2 a53 q 8 a2 a3 a53 y 20 a54 . Factoring the resultant of U0 and R1 with respect to a5 , we get R 2 s a82 a42 Ž ya1 a3 y a32 q a4 .Ž a12 q 4 a4 . .
Ž 14 .
Since a12 q 4 a4 / 0, a4 s a2 Ž a1 q a3 . together with R1 and U0 s 0 leads to a contradiction. This gives Žii.. With Ži., Žii., and Žiii., Žiii. is obvious.
4. TWO FOCI FOR KUKLES SYSTEM It is shown in Lemma 4 that if Kukles’ system Ž1. has two weak foci and one of them has order 2, then the other has order at most 1. The remaining question is: if one of the foci has order 1, can the maximal order of the other focus be greater than 6? In fact, we have the following theorem as the main result of this section. THEOREM 4. If Kukles’ system Ž1. has two weak foci, then the sum Nf of the orders of the two foci is less than or equal to 7, i.e., Nf F 7. Proof. By the symmetry of two weak foci, we suppose, without loss of generality, that the origin has order 7 and E x 2 has order 1. In this case, we will get a contradiction from the polynomial system PS s U0 , V1 , . . . , V6 4 s 0. Clearly, that the polynomial set PS has a zero ensures Nf ) 7, provided that the conditions in Theorem 1 fail. Now, we show that this is impossible, i.e., U0 s 0 and Vi s 0, i s 1, . . . , 6, have no common solution. First, solve U0 s 0 and V1 s 0 for a4 and a7 , and substitute the solution into V2 s 0 to obtain an equality with a6 having degree 1. From this equality, a6 can be solved as a function of the other a i ’s for i / 4, 6, 7. Substituting a4 , a6 , and a7 into V3 , . . . , V6 and removing two factors V332 and V333 , we obtain a polynomial set PS1 s V33 , V44 , V55 , V66 4 . Here Vii s Vii Ž a1 , a2 , a3 , a5 ., i s 3, . . . , 6, consist of 13, 49, 122, and 245 terms, respec-
162
CHEN, LU, AND WANG
tively, and V332 s 27a2 a52 a1 q 27a22 a5 a12 q 9 a53 q 9 a13 a32 q 4 a42 a5 q 2 a52 a1 , V333 s a5 q a2 a3 q a2 a1 . V332 s 0 together with U0 s 0, V1 s 0, and V2 s 0 implies Žk4.. V333 s 0 together with U0 s 0, V1 s 0, and V2 s 0 implies Živ. in Theorem C. In both cases, we have two centers. This contradicts Nf ) 7. Now consider the set PS1. By calculating resultants and eliminating a3 from PS1 , we obtain PS2 s V34 , V35 , V36 4 , where V34 s V34Ž a1 , a2 , a5 ., V35 s V35 Ž a1 , a2 , a5 ., and V36 s V36 Ž a1 , a2 , a5 . consist of 30, 72, and 132 terms, respectively. Factoring the resultant of V34 and V35 with respect to a1 leads to V451 ⭈ V452 , where V451 and V452 are polynomials in a2 and a5 , consisting of 13 and 43 terms, respectively. The resultant of V34 and V36 with respect to a1 , denoted by V46 , is a polynomial in a2 and a5 which consists of 82 terms. If we substitute a2 s a2 b 2 and a5 s b into V451 Žor V452 or V46 ., then V451Ž a2 , a5 . s V451Ž a2 , 1. ⭈ b n Žwhere n is a positive integer. with each monomial in V451Ž a2 , 1. having the form a24 k . Therefore, without loss of generality, we substitute a5 s 1 and a2 s c 1r4 into V451 , V452 , and V46 to obtain three polynomials of c; let them be denoted again by V451Ž c ., V452 Ž c ., and V46 Ž c .. The resultants of V451Ž c . and V46 Ž c ., V452 Ž c . and V46 Ž c . give two negative numbers, having 4449 and 18,779 digits, respectively. Namely, V451 ⭈ V452 and V46 have no common solution, which implies that the polynomial set PS has no zero. This completes the proof of the theorem.
5. REMARKS Our work described in this paper uses a computational approach in which computer algebra systems like Maple play a key role. One of the major difficulties encountered in our study is how to handle very large polynomials in equations-solving. Some of the basic tools we have used are the methods of resultants and Ritt᎐Wu characteristic sets. We were able to derive a number of results, but still many occurring polynomial systems are too large to be manageable by using these methods. More advanced elimination techniques are required and being developed to attack other challenging problems on the subject. For instance, we have shown that if Kukles’ system Ž1. has two foci, then their total order is at most 7. It seems that the maximal value 7 is reachable for some concrete example, and
CUBIC SYSTEMS WITH TWO CENTERS
163
6 q 1 small amplitude limit cycles can be constructed. We leave this as a future research problem. We also wish to establish the necessary and sufficient conditions for system Ž1. to have two foci.
ACKNOWLEDGMENTS The authors would like to thank Guoren Dai and Xinan Zhang for helpful discussions. The work of the first author is supported partially by the National Natural Science Foundation of China, and that of the second author by the National Climbing Project of China.
REFERENCES 1. C. Christopher, Invariant algebraic curves and conditions for a centre, Proc. Roy. Soc. Edinburgh Sect. A 124 Ž1994., 1209᎐1229. 2. C. Christopher and N. G. Lloyd, On the paper of Jin and Wang concerning the conditions for a centre in certain cubic systems, Bull. London Math. Soc. 22 Ž1990., 5᎐12. 3. R. Conti, On a class of cubic systems with two centers, Results Math. 14 Ž1988., 30᎐37. 4. R. Conti, On centers of cubic systems, Ann. Polon. Math. 5 Ž1990., 123᎐128. 5. X. F. Jin and D. M. Wang, On the conditions of Kukles for the existence of a centre, Bull. London Math. Soc. 22 Ž1990., 1᎐4. 6. I. S. Kukles, Sur quelques cas de distinction entre un foyer et un centre, Dokl. Akad. Nauk SSSR 42 Ž1944., 208᎐211. 7. C. Rousseau, D. Schlomiuk, and P. Thibaudeau, The centres in the reduced Kukles system, Nonlinearity 8 Ž1995., 541᎐569. 8. A. P. Sadovskii, On the limiting cycles of a cubic system describing nonlinear oscillations, Differentsial’naya Uraneniya 29 Ž1993., 226᎐231. 9. D. M. Wang, Mechanical manipulation for a class of differential systems, J. Symbolic Comput. 12 Ž1991., 233᎐254. 10. W. T. Wu, ‘‘Basic Principles of Mechanical Theorem Proving in Geometries,’’ Science Press, Beijing, 1984. ŽIn Chinese; English translation, Springer-Verlag, ViennarNew York, 1994..
A Class of Cubic Systems with Two Centers or Two Foci Lansun Chen Institute of Mathematics, Academia Sinica, Beijing 100080, China
Zhengyi Lu Department of Mathematics, Sichuan Uni¨ ersity, Chengdu 610064, China
and Dongming Wang Laboratoire LEIBNIZ, Institut IMAG, 46, a¨ enue Felix ´ Viallet, 38031 Grenoble, France Submitted by Raul ´ Manase ´ ¨ ich Received February 11, 1999
In this paper, the necessary and sufficient conditions for a class of cubic differential systems to possess two centers are given. Some conditions for the systems to have two weak foci are also derived by using the computer algebra system Maple. 䊚 2000 Academic Press Key Words: Kukles system; center; focus.
1. INTRODUCTION We consider the cubic system of center and focus type
˙x s y,
˙y s yx q Q Ž x, y . ,
Ž 1.
together with its associated system
˙x s y,
˙y s yx q y q Q Ž x, y . ,
where Q Ž x, y . s a1 x 2 q a2 xy q a3 y 2 q a4 x 3 q a5 x 2 y q a6 xy 2 q a7 y 3 . 154 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
Ž 2.
CUBIC SYSTEMS WITH TWO CENTERS
155
The conditions for E0 s Ž0, 0. to be a center of Ž1. were given by Kukles w6x in 1944. Two subclass systems of Ž1., i.e., the cases in which a2 a7 s 0, were considered recently in w2, 5, 8x. For a2 s 0, an example given in w5x suggests that Kukles’ conditions are incomplete; it is then proved in w1x that the conditions are indeed incomplete by showing that the origin is a center in this case. Although Kukles’ conditions are not necessary, w1, 6x proved that they are sufficient. Especially, we have THEOREM A. Conditions Žk4. are sufficient for E0 to be a center of system Ž1., a22 a4 q a5 k s 0,
Ž 3a7 k q k 2 q a6 a22 . a5 y 3a7 k 2 y a6 a22 k s 0, k q a1 a2 q a5 s 0,
Ž k4.
9 a6 a22 q 2 a42 q 9k 2 q 27a7 k s 0, where k s a2 a3 q 3a7 . In w8x, Sadovskii considered Ž1. with a2 s 0 and proved the following THEOREM B. For system Ž1. with a2 s 0, the origin is a center if and only if one of the following conditions holds: Ži. Žii. 3'2 a1.
a5 s a7 s 0; a3 s y2 a1 , a6 s 0, a4 s y1r3a12 , a5 s .1r3'2 a12 , a7 s "1r
It is also shown in w8x that if E0 is a focus of Ž1. with a2 s 0, then the maximal order of E0 is 6 and six small amplitude limit cycles can be constructed for its perturbation system Ž2. with a2 s 0. The following result is established in w2x. THEOREM C. For system Ž1. with a7 s 0, the origin is a center if and only if one of the following conditions holds: Ži. a2 s a5 s 0; Žii. a1 s a3 s a5 s 0; Žiii. a4 s a5 s a6 s 0, a1 q a3 s 0; Živ. a4 s Ž a1 q a3 . a3 , a5 s yŽ a1 q a3 . a2 , a6 s yŽ a1 q a3 . a23 Ž a1 q y1 2 a3 . . In this case, the maximal order for E0 to be a focus of Ž1. with a7 s 0 is 5 and five small amplitude limit cycles can be constructed for system Ž2. with a7 s 0.
156
CHEN, LU, AND WANG
As pointed out in w1x, determining the necessary and sufficient conditions for E0 to be a center of Kukles’ system Ž1. is still a problem that remains open. On the other hand, the problem of finding conditions for cubic systems to have two or more centers has also been considered in the literature w3, 4, 7x. In the present paper, we are concerned with this problem for the Kukles system and derive necessary and sufficient conditions for the system to have two centers or two foci. The two-center conditions obtained are given in Section 2. The sufficient and necessary conditions for Kukles’ system to have two weak foci are established in Section 3 for the case a2 a7 s 0. In Section 4, the two weak foci problem is considered with the help of Maple in dealing with large polynomials. The paper is concluded with some remarks.
2. TWO CENTERS Clearly, system Ž1. has at most two fixed points E x 1 s Ž x 1 , 0. and E x 2 s Ž x 2 , 0. other than the origin. All the fixed points are located on x-axis, which is an isoline. By an index theorem, if x 1 s x 2 , then E x 1 cannot be of center-focus type; if x 1 / x 2 , then at most one of them, say, E x 2 has a center-focus type. Note that the fixed points along the x-axis are alternatively saddles and anti-saddles, therefore if x 1 - 0 - x 2 , then system Ž1. has two saddles. In the sequel, we consider two cases 0 - x 1 - x 2 and 0 ) x 1 ) x 2 . Clearly, x 1 and x 2 are distinct solutions of the quadratic equation y1 q a1 x 2 q a4 x 22 s 0,
Ž 3.
and x 1 x 2 ) 0 if and only if ya12r4 - a4 - 0.
Ž 4.
If a1 - 0, then x2 s
1 2 a4
q 4 a4 .
1r2
/ - 0;
y Ž a12 q 4 a4 .
1r2
/ ) 0.
ž ya q Ž a 1
2 1
if a1 ) 0, then x2 s
1 2 a4
ž ya
1
157
CUBIC SYSTEMS WITH TWO CENTERS
By using the transformation x s x y x2 ,
y s y,
system Ž1. leads to
˙x s y, ˙y s yax q ⌬ y q a1 x 2 q a2 xy q a3 y 2 q a4 x 3 q a5 x 2 y
Ž 5.
q a6 xy 2 q a7 y 3 , where x and y are used instead of x and y, and a s a1 x 2 y 2, ⌬ s Ž a 2 q a5 x 2 . x 2 , a1 s a1 q 3a4 x 2 , a 2 s a 2 q 2 a5 x 2 , a 3 s a 3 q a6 x 2 . Since E x 2 is supposed to be of center-focus type, ⌬ ' 0, that is Ž x 2 / 0. a1 q a5 x 2 s 0.
Ž 6.
It is easy to know from Ž3. that
¡1 2 a4 a s a1 x 2 y 2 s
~
¢
a1 y Ž a12 q 4 a4 .
1r2
Ž a12 q 4 a4 .
1r2
) 0,
a1 - 0, 1 2 a4
ya1 y Ž a12 q 4 a4 .
1r2
Ž a12 q 4 a4 .
1r2
) 0,
a1 ) 0.
From the above discussions, we obtain LEMMA 1.
E x 2 is of center-focus type if and only if Ž3., Ž4., and Ž6. hold.
Clearly, Ž3., Ž4., and Ž6. imply in the case of a5 / 0 that U0 s a22 a4 y a1 a2 a5 y a52 s 0.
Ž 7.
LEMMA 2. System Ž1. with a2 s 0 has two centers if and only if Ž4. holds and a5 s a7 s 0.
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Proof. Suppose that both the origin and E x 2 are centers of Ž1.. Then by Theorem B and Ž6., we have a5 s 0,
a7 s 0.
Ž 8.
Note that x 2 / 0. Observe that a ) 0, and under Ž8., Ž5.< a 2s0 is invariant under the transformation Ž x, y, t . ª Ž x, yy, yt .. It follows that the origin is a center of Ž5.< a 2s0 by the symmetry principle. This proves the lemma. Now consider system Ž1. with a2 a5 / 0. By the transformation x s a1r2 x,
y s y,
s a1r2 t
with a s a1 x 2 y 2 s y2 y a1 a2ra5 , system Ž5. is changed to
˙x s y, ˙y s yx q b1 x 2 q b 2 xy q b 3 y 2 q b4 x 3 q b5 x 2 y q b6 xy 2 q b 7 y 3 ,
Ž 9.
where x and y are used instead of x and y, and b1 s ay3r2 a1 y
ž
b 3 s ay1r2 a3 y
ž
b6 s ay1 a6 ,
3a2 a4 a5 a 2 a6 a5
/
/ ,
b 2 s yay1a2 ,
,
b4 s ay2 a4 ,
b5 s ay3r2 a5 ,
b 7 s ay1r2 a7 .
LEMMA 3. Let Ž4. hold true and let a7 s 0. Then system Ž1. has two centers if and only if Živ. in Theorem C holds. Proof. Clearly, we only need to check the sufficiency. The corresponding values in Živ. of Theorem C for system Ž5. at the origin are c1 s b4 y Ž b1 q b 3 . b 3 , c 2 s b5 y Ž b1 q b 3 . b 2 , c 3 s b6 q Ž b1 q b 3 . b 32 Ž b1 q 2 b 3 .
Ž 10 . y1
.
Substituting each concrete bi Ž i s 1, . . . , 6. which is a function of a i Ž i s 1, . . . , 6. into c j Ž j s 1, 2, 3. and removing the nonzero factors, we obtain a set of three polynomials, denoted by c1 , c 2 , c 3 , in a i Ž i s 1, . . . , 6..
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By using Wu’s method w10x, it can be proved that Živ. of Theorem C implies that c1 s c 2 s c 3 s 0. This shows that the origin is a center of Ž5.. The proof is complete. Denote the ith focal value of E0 by Vi and of E x 2 by Ui ; V1 , V2 , V3 , V4 , V5 , V6 , and U1 consist of 4, 13, 49, 131, 292, 577, and 11 terms, respectively. How to compute focal values of a system of differential equations of center-focus type is described in w9x. LEMMA 4. V1 s V2 s U0 s U1 s 0 and a2 a7 / 0 if and only if the four equalities Žk4. in Theorem A hold. Lemma 4 means that, when E0 is a center, Žk4. are necessary conditions for E x 2 to be a center. Lemma 4 also implies that if both E0 and E x 2 are weak foci and one of them has order 2, then the other has order at most 1. Now we show that Žk4. are also sufficient to ensure E x 2 to be a center. LEMMA 5. If the equalities in Theorem A hold, then the origin is a center of Ž9., which in turn implies that E x 2 is a center of Ž1.. Proof. By Theorem A, to prove the origin to be a center, one only needs to check whether the equalities in Theorem A hold or not. The four Kukles’ values of system Ž1., i.e., the left-hand sides of the four equalities in Theorem A by substituting a i s bi into them, give a set of polynomials in a i : PS s P1 , P2 , P3 , P4 4 , where P1 , P2 , P3 , and P4 have 8, 26, 10, and 19 terms, respectively. Application of Wu’s method shows that Žk4. implies PS s 0. It follows that for system Ž9., Žk4. holds true when the a i are replaced by the corresponding bi . Hence the origin is a center of Ž9.. THEOREM 1. Kukles’ system Ž1. has two centers if and only if ya12r4 a4 - 0 and one of the following conditions holds: Ži. a2 s a5 s a7 s 0; Žii. a7 s 0, a4 s Ž a1 q a3 . a3 , a5 s yŽ a1 q a3 . a2 , a6 s yŽ a1 q a3 . a32 Ž a1 q 2 a3 .y1 ; Žiii. Žk4.. Remark. Theorem 1 can also be proved by the results of w1x and w7x which state that if a system has a first integral of Darboux type and if the algebraic curves used to construct the first integral do not pass through the second fixed point, then the first integral is analytic at the second fixed point, yielding automatically that the point is a center. However, in this paper, we emphasize the mechanical manipulation of checking the centers without using the first integrals for the second center.
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3. TWO FOCI FOR KUKLES SYSTEMS WITH a2 a7 s 0 THEOREM 2. System Ž1. with a2 s 0 has two weak foci if and only if ya12r4 - a4 - 0, a5 s 0, and a7 / 0. Proof. Suppose that both E0 and E x 2 are weak foci. Then Ž3. holds and Ž6. becomes a5 x 2 s 0. So x 2 / 0 leads to a5 s 0. Thus V1 s 3a7 and U1 s 3a1r2 a7 . Clearly, V1 s 0 or U1 s 0 implies a7 s 0. According to Theorem B, both E0 and E x 2 are centers of Ž1.. This shows that if both E0 and E x 2 are weak foci, they have order 1. THEOREM 3. System Ž1. with a7 s 0 has two weak foci if and only if ya12r4 - a4 - 0 and one of the following conditions holds: Ži. a4 s a3 Ž a1 q a3 ., a5 s ya2 Ž a1 q a3 ., a6 / ya23 Ž a1 q a3 .Ž a1 q 2 a3 ; Žii. U0 s 0, U1 s 0, U2 / 0; Žiii. U0 s 0, U1 / 0. .y1
Proof. Ži. First, we show that if system Ž1. with a7 s 0 has two weak foci, then the order of E0 is at most 2. Suppose that E0 is a weak focus of order greater than 2; then V1 s 0 and V2 s 0. Clearly, V1 s 0 is equivalent to a5 s ya2 Ž a1 q a3 . .
Ž 11 .
Substituting Ž11. into U0 s 0 yields a4 s a3 Ž a1 q a3 . .
Ž 12 .
Simplified by Ž11. and Ž12., V2 s 0 leads to a1 a23 q a33 q a1 a6 q 2 a3 a6 s 0.
Ž 13 .
That Ž11., Ž12., and Ž13. are the conditions ŽŽiv. of Theorem C. for E0 to be a center gives a contradiction. Therefore, if E x 2 is a weak focus, it has order at most 1. Namely, U0 s 0, V1 s 0, U1 / 0. Clearly, these are just the conditions in Ži.. Žii. E0 is a focus of order 2, i.e., V1 s 0. If E x is a focus of order 2, 2 then U1 s 0. It is easy to check that U0 s 0, U1 s 0, and V1 s 0 imply, in this case, Živ. of Theorem C. This is a contradiction. Hence, V1 / 0. Žiii. Let E0 be a focus of order 1 and E x have order greater than 2, 2 i.e., U0 s 0, U1 s 0, and U2 s 0. If a1 a2 q 2 a5 s 0, then U0 s 0 leads to
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a14 q 4 a4 s 0, which contradicts Ž4.. Now eliminating a6 from U1 and U2 and removing the non-zero factors, we obtain a polynomial R1 with 11 terms R1 s y3a1 a42 a3 a4 q 6 a24 a42 q 3a12 a23 a3 a5 q 3a1 a23 a4 a5 y 6 a23 a3 a4 a5 y 8 a12 a22 a52 q 10 a1 a22 a3 a52 q 16 a22 a4 a52 y 28 a1 a2 a53 q 8 a2 a3 a53 y 20 a54 . Factoring the resultant of U0 and R1 with respect to a5 , we get R 2 s a82 a42 Ž ya1 a3 y a32 q a4 .Ž a12 q 4 a4 . .
Ž 14 .
Since a12 q 4 a4 / 0, a4 s a2 Ž a1 q a3 . together with R1 and U0 s 0 leads to a contradiction. This gives Žii.. With Ži., Žii., and Žiii., Žiii. is obvious.
4. TWO FOCI FOR KUKLES SYSTEM It is shown in Lemma 4 that if Kukles’ system Ž1. has two weak foci and one of them has order 2, then the other has order at most 1. The remaining question is: if one of the foci has order 1, can the maximal order of the other focus be greater than 6? In fact, we have the following theorem as the main result of this section. THEOREM 4. If Kukles’ system Ž1. has two weak foci, then the sum Nf of the orders of the two foci is less than or equal to 7, i.e., Nf F 7. Proof. By the symmetry of two weak foci, we suppose, without loss of generality, that the origin has order 7 and E x 2 has order 1. In this case, we will get a contradiction from the polynomial system PS s U0 , V1 , . . . , V6 4 s 0. Clearly, that the polynomial set PS has a zero ensures Nf ) 7, provided that the conditions in Theorem 1 fail. Now, we show that this is impossible, i.e., U0 s 0 and Vi s 0, i s 1, . . . , 6, have no common solution. First, solve U0 s 0 and V1 s 0 for a4 and a7 , and substitute the solution into V2 s 0 to obtain an equality with a6 having degree 1. From this equality, a6 can be solved as a function of the other a i ’s for i / 4, 6, 7. Substituting a4 , a6 , and a7 into V3 , . . . , V6 and removing two factors V332 and V333 , we obtain a polynomial set PS1 s V33 , V44 , V55 , V66 4 . Here Vii s Vii Ž a1 , a2 , a3 , a5 ., i s 3, . . . , 6, consist of 13, 49, 122, and 245 terms, respec-
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tively, and V332 s 27a2 a52 a1 q 27a22 a5 a12 q 9 a53 q 9 a13 a32 q 4 a42 a5 q 2 a52 a1 , V333 s a5 q a2 a3 q a2 a1 . V332 s 0 together with U0 s 0, V1 s 0, and V2 s 0 implies Žk4.. V333 s 0 together with U0 s 0, V1 s 0, and V2 s 0 implies Živ. in Theorem C. In both cases, we have two centers. This contradicts Nf ) 7. Now consider the set PS1. By calculating resultants and eliminating a3 from PS1 , we obtain PS2 s V34 , V35 , V36 4 , where V34 s V34Ž a1 , a2 , a5 ., V35 s V35 Ž a1 , a2 , a5 ., and V36 s V36 Ž a1 , a2 , a5 . consist of 30, 72, and 132 terms, respectively. Factoring the resultant of V34 and V35 with respect to a1 leads to V451 ⭈ V452 , where V451 and V452 are polynomials in a2 and a5 , consisting of 13 and 43 terms, respectively. The resultant of V34 and V36 with respect to a1 , denoted by V46 , is a polynomial in a2 and a5 which consists of 82 terms. If we substitute a2 s a2 b 2 and a5 s b into V451 Žor V452 or V46 ., then V451Ž a2 , a5 . s V451Ž a2 , 1. ⭈ b n Žwhere n is a positive integer. with each monomial in V451Ž a2 , 1. having the form a24 k . Therefore, without loss of generality, we substitute a5 s 1 and a2 s c 1r4 into V451 , V452 , and V46 to obtain three polynomials of c; let them be denoted again by V451Ž c ., V452 Ž c ., and V46 Ž c .. The resultants of V451Ž c . and V46 Ž c ., V452 Ž c . and V46 Ž c . give two negative numbers, having 4449 and 18,779 digits, respectively. Namely, V451 ⭈ V452 and V46 have no common solution, which implies that the polynomial set PS has no zero. This completes the proof of the theorem.
5. REMARKS Our work described in this paper uses a computational approach in which computer algebra systems like Maple play a key role. One of the major difficulties encountered in our study is how to handle very large polynomials in equations-solving. Some of the basic tools we have used are the methods of resultants and Ritt᎐Wu characteristic sets. We were able to derive a number of results, but still many occurring polynomial systems are too large to be manageable by using these methods. More advanced elimination techniques are required and being developed to attack other challenging problems on the subject. For instance, we have shown that if Kukles’ system Ž1. has two foci, then their total order is at most 7. It seems that the maximal value 7 is reachable for some concrete example, and
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6 q 1 small amplitude limit cycles can be constructed. We leave this as a future research problem. We also wish to establish the necessary and sufficient conditions for system Ž1. to have two foci.
ACKNOWLEDGMENTS The authors would like to thank Guoren Dai and Xinan Zhang for helpful discussions. The work of the first author is supported partially by the National Natural Science Foundation of China, and that of the second author by the National Climbing Project of China.
REFERENCES 1. C. Christopher, Invariant algebraic curves and conditions for a centre, Proc. Roy. Soc. Edinburgh Sect. A 124 Ž1994., 1209᎐1229. 2. C. Christopher and N. G. Lloyd, On the paper of Jin and Wang concerning the conditions for a centre in certain cubic systems, Bull. London Math. Soc. 22 Ž1990., 5᎐12. 3. R. Conti, On a class of cubic systems with two centers, Results Math. 14 Ž1988., 30᎐37. 4. R. Conti, On centers of cubic systems, Ann. Polon. Math. 5 Ž1990., 123᎐128. 5. X. F. Jin and D. M. Wang, On the conditions of Kukles for the existence of a centre, Bull. London Math. Soc. 22 Ž1990., 1᎐4. 6. I. S. Kukles, Sur quelques cas de distinction entre un foyer et un centre, Dokl. Akad. Nauk SSSR 42 Ž1944., 208᎐211. 7. C. Rousseau, D. Schlomiuk, and P. Thibaudeau, The centres in the reduced Kukles system, Nonlinearity 8 Ž1995., 541᎐569. 8. A. P. Sadovskii, On the limiting cycles of a cubic system describing nonlinear oscillations, Differentsial’naya Uraneniya 29 Ž1993., 226᎐231. 9. D. M. Wang, Mechanical manipulation for a class of differential systems, J. Symbolic Comput. 12 Ž1991., 233᎐254. 10. W. T. Wu, ‘‘Basic Principles of Mechanical Theorem Proving in Geometries,’’ Science Press, Beijing, 1984. ŽIn Chinese; English translation, Springer-Verlag, ViennarNew York, 1994..