Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena

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1 Caos, Solitons & Fractals 5 (0) Contents lists available at SciVerse ScienceDirect Caos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Penomena journal omepage: Bifurcation of limit cycles in quadratic Hamiltonian systems wit various degree polynomial perturbations P. Yu a,b,, M. Han a a Department of Matematics, Sangai Normal University, Sangai 003, Cina b Department of Applied Matematics, University of Western Ontario, London, Ontario, Canada N6A 5B7 article info abstract Article istory: Received May 0 Accepted 5 February 0 Available online 6 Marc 0 In tis paper, we consider bifurcation of limit cycles in planar quadratic Hamiltonian systems wit various degree polynomial perturbations. Attention is focused on te limit cycles wic may appear in te vicinity of an isolated center, and up to 0t-degree polynomial perturbations are investigated. Restricted to te first-order Melnikov function, te metod of focus value computation is used to determine te maximal number, H (n), of small-amplitude limit cycles wic may exist in te neigborood of suc a center. Besides te existing results H () = and H (3) = 5, we sall sow tat H ðnþ ¼ ðn þ Þ 3 for n =3,,...,0. Ó 0 Elsevier Ltd. All rigts reserved.. Introduction Te study of te well-known Hilbert s 6t problem [] is still very active toug it as not been completely solved even for quadratic systems, since it as great impact on te development of modern matematics. Consider te following planar system: _x ¼ P n ðx; yþ; _y ¼ Q n ðx; yþ; ðþ were P n (x,y) and Q n (x,y) denote nt-degree polynomials of x and y. Rougly speaking, te second part of Hilbert s 6t problem is to find te upper bound, called Hilbert number H(n), on te number of limit cycles tat system () can ave. A compreensive review on te study of Hilbert s 6t problem can be found in a survey article []. To elp understand and attack te problem te so called weak Hilbert s 6t problem was posed by Arnold [3]. Te problem is to ask for te maximal number of isolated zeros of te Abelian integral or Melnikov function: Z Mð; dþ ¼ Q n dx P n dy; ðþ Hðx;yÞ¼ were H(x,y), P n and Q n are all real polynomials of x and y wit degh = n +, and max{degp n, deg Q n } 6 n. Te weak Hilbert s 6t problem itself is a very important and interesting problem, closely related to te following near-hamiltonian system []: _x ¼ H y ðx; yþþep n ðx; yþ; _y ¼H x ðx; yþþeq n ðx; yþ; ð3þ were H(x,y), p n (x,y) and q n (x,y) are all polynomial functions of x and y, and 0 < e is a small perturbation. Studying te bifurcation of limit cycles for suc a system can be transformed to investigating te zeros of te Melinikov function. Corresponding autor at: Department of Applied Matematics, University of Western Ontario, London, Ontario, Canada N6A 5B7. Fax: address: pyu@pyu.apmats.uwo.ca (P. Yu) /$ - see front matter Ó 0 Elsevier Ltd. All rigts reserved. doi:0.06/j.caos

2 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) On te oter and, if te problem is restricted to a neigborood of isolated fixed points wit Hopf singularity, te problem is ten reduced to studying degenerate Hopf bifurcations, leading to computation of focus values, and many results ave been obtained (e.g., see [5 8]). Alternatively, tis is equivalent to computing te normal form of differential equations associated wit Hopf or degenerate Hopf bifurcations. Suppose te origin of system (), (x, y) = (0, 0), is an element center. Witout loss of generality, we may assume tat te eigenvalues of te Jacobian of system () evaluated at te origin are a purely imaginary pair, ±i. Furter, suppose te normal form associated wit tis Hopf singularity is given in polar coordinates (obtained by using, say, te metod given in [9]): _r ¼ r ðv 0 þ v r þ v r þþv k r k þþ; _ ¼ þ t r þ t r þþt k r k þ; ðþ ð5þ were r and represent, respectively, te amplitude and pase of te limit cycles, and v i, i =0,,,...are called focus values. Te basic idea of finding k small limit cycles around te origin is as follows: First, find te conditions suc tat v 0 = v = v = = v k = 0, but v k 0, and ten perform appropriate small perturbations to prove te existence of k limit cycles. For general quadratic system () (n = ), in 95, Bautin [5] proved tat tere exist 3 small limit cycles around a fine focus point or a center. Until te end of 970 s, concrete examples are given to sow tat quadratic systems can ave limit cycles [0,], wic ave (3, ) configuration: tree limit cycles enclose a fine focus point, wile one limit cycle encloses anoter element focus point. In tis paper, attention is focused on bifurcation of limit cycles in quadratic Hamiltonian system wit various degree polynomial perturbations, and in particular on te limit cycles wic bifurcate from a center. Witout loss of generality, we may assume tat system (3) e=0 as a center at te origin (x,y) = (0,0). For tis problem, we may eiter use te Melnikov metod or te focus value metod to determine te number of limit cycles in te vicinity of te origin. In tis paper, we will apply te metod of focus value computation to study te bifurcation of limit cycles. For e 0, te focus values of system (3) at te origin can be written in te form of v i ¼ v i þ e ~v i þ Oðe Þ; i ¼ 0; ; ;...; ð6þ were v i ¼ 0; i ¼ 0; ; ;... due to te origin being a center. Tus, for sufficiently small e, we may use ~v i to determine te number of small-amplitude limit cycles bifurcating from te origin. For quadratic Hamiltonian system wit nd-degree polynomial perturbation, tere are many results in te literature, mainly publised since te 90 s of last century. Te conclusion is: quadratic Hamiltonian systems wit nd-degree polynomial perturbation can ave maximal two limit cycles, i.e., H () = [,3], were te subscript denotes second-order Hamiltonian systems. Recently, it as been sown tat quadratic Hamiltonian systems wit 3rd-degree polynomial perturbation can ave maximal five limit cycles in te vicinity of a center, i.e., H (3) = 5[]. In[], te Melnikov function metod is used, on te basis of te following Melnikov function: I Z MðÞ ¼ q n dx p n dy ¼ n L n dx were L is a contour around te origin, and D() is te region bounded by te contour. Suppose te general polynomial functions p n (x,y) and q n (x,y) are given by p n ðx; yþ ¼ X a ij x i y j ; q n ðx; yþ ¼ X b ij x i y j : ð8þ 6iþj6n 6iþj6n ð7þ ¼ X 6iþj6n ia ij x i y j þ X 6iþj6n jb ij x i y j ¼ X 06iþj6n ½ði þ Þ a ðiþþj þðj þ Þ b iðjþþ Šx i y j : ð9þ Tus, witout loss of generality, we may assume a ij = 0 (i.e., p n (x,y) 0), and b i0 =0. For a comparison, in tis paper we use te metod of focus value computation to obtain te same result for n = 3. Furter, we will consider bifurcation of limit cycles for quadratic Hamiltonian systems wit 3rd- to 0t-degree polynomial perturbations. More precisely, we ave te following main teorem: Teorem. A quadratic Hamiltonian system wit nt-degree polynomial perturbation can ave maximal 3 ðn þ Þ small limitamplitude cycles bifurcating from a center, i.e., H ðnþ ¼ 3 ðn þ Þ ; 3 6 n 6 0, were [] denotes te integer part. It sould be noted tat te case n = 0 is meaningless since no perturbation is used, wile te cases n = and n = are known: H () = 0 and H () =. Since tese two results do not satisfy te formula H ðnþ ¼ ðn þ Þ 3, and are tus excluded from Teorem. Te rest of te paper is organized as follows. In te next section, we describe general formula for quadratic Hamiltonian systems and apply te metod of focus value computation to re-investigate te cases wit nd- and 3rd-degree polynomial perturbations, confirming te results H () = and H (3) = 5. Te cases of t- to 8t-degree polynomial perturbations are

3 77 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) considered in Sections 3, and te results for te cases of 9t- to 0t-degree polynomial perturbations are summarized in Section. Conclusion is drawn in Section 5.. Quadratic Hamiltonian systems wit nd- and 3rd-degree polynomial perturbations In tis section, we first derive a simple generic quadratic Hamiltonian, and ten re-derive te results of H () = and H (3) = 5... Quadratic Hamiltonian system To obtain a simple generic quadratic Hamiltonian system, we start from te following general quadratic system: dz dt ¼ c 00 þ c 0 z þ c 0 z þ c 0 z þ c z z þ c 0 z ; dz dt ¼ c 00 þ c 0 z þ c 0 z þ c 0 z þ c z z þ c 0 z ; ð0þ were c ijk s are real constant parameters. It is easy to sow tat tis system as at most four singularities, or more precisely, it can ave 0, or singularities in real domain. In order for system (0) to ave limit cycles, te system must ave some singularity. In tis paper, we assume tat system (0) as at least two singularities. Witout loss of generality, we may assume tat one singular point is located at te origin (0,0), wic implies c 00 = c 00 = 0, and te second one at (p,q). Furter assume te origin is an element center (i.e., te linear part of te system as a center). Ten introducing a series of linear transformations, parameter rescaling and time rescaling to system (0) yields te following general quadratic system: dx ¼ y þ a dt xyþa y ; dy ¼xþa ðþ dt 3 x þ a xyþa 5 y ; wic as an element center at te origin (0,0) and anoter singularity at a 3 ; 0. System () becomes a Hamiltonian system wen a = 0 and a 5 ¼ a. Tus, we obtain te following quadratic Hamiltonian system: dx ¼ y þ a dt xyþa y ; dy ¼xþa dt 3 x a y ; ðþ wit center if a < a 3 ; ; 0 being a a 3 saddle point if a > a 3 : Te details for deriving () from (0) can be found in [5]. Since we are interested in te limit cycles bifurcating from te center (0, 0), we will ignore weter te singular point a 3 ; 0 is a center or a saddle point. Te Hamiltonian of system () is given by Hðx; yþ ¼ ðx þ y Þ 3 a 3 x 3 þ a xy þ 3 a y 3 ; In tis paper, we concentrate on te generic case: a a a 3 0. Note tat a 3 = 0 implies tat te second singular point a 3 ; 0 is at te infinity. For a 3 0, we can simply use a scaling to move te singular point a 3 ; 0 to (, 0). Alternatively, we may simply set a 3 =. Tus, in te following analysis, a 3 =, wic does not affect te analysis and results. Te perturbed Hamiltonian system or near-hamiltonian system of () can be generally written as ð3þ dx ¼ y þ a dt xyþa y þ e p n ðx; yþ; dy ¼xþ dt x a y þ e q n ðx; yþ; were, as discussed in te introduction, we ave assumed tat p n (x,y) 0 and q n (x,y) is given in (8) wit b i0 =0. ðþ.. Quadratic Hamiltonian system wit nd-degree polynomial perturbation Wen n =,q (x,y) is given by q ðx; yþ ¼b 0 y þ b xyþ b 0 y : ð5þ For tis case, we ave

4 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) Teorem. Te quadratic near-hamiltonian system () wit nd-degree polynomial perturbation can ave maximal two smallamplitude limit cycles bifurcating from te origin, i.e., H () =. Proof. First of all, in order to satisfy te condition suc tat te origin (0,0) is an element center, it requires ~v 0 ¼ b 0 ¼ 0 ) b 0 ¼ 0: ð6þ Ten, applying te Maple program [9] to system () wit q (x,y) given in (5), we obtain te focus value ~v, given by ~v ¼ 6 ½ða Þ b þ a b 0 Š: ð7þ We may solve b 0 from te equation ~v ¼ 0, as b 0 ¼ ða Þ a b : ð8þ and ten we ave te following simplified focus values: i ~v ¼ 5 b 9 ða þ Þ ða Þ a ; i ~v 3 ¼ 5 b 059 ða þ Þða Þ a 53a 8a þ 388 þ 868a ; i ~v ¼ b ða þ Þða Þ a 65969a 79000a3 þ 76a 338a þ þa 75900a 8758a þ þ 99568a ;.. It is seen tat letting ~v ¼ 0 yields ~v i ¼ 0; i ¼ 3; ;..., leading to a center. In fact, in general, if b = 0 and so b 0 = 0, ten system () is reduced to unperturbed Hamiltonian system. Hence, for te quadratic perturbation, one may set ~v 0 ¼ ~v ¼ 0 by coosing b 0 = 0 and b 0 given in (8), but ~v 0. Tis indicates tat we can ave at most two small-amplitude limit cycles around te center (0,0). Furter, it is easy to sow tat by proper perturbations to b 0 and b 0, one can obtain two small-amplitude limit cycles, and ence H () =. ð9þ.3. Quadratic Hamiltonian system wit 3rd-degree polynomial perturbation Now we turn to consider te case n = 3, for wic q 3 (x,y) is given by q 3 ðx; yþ ¼b 0 y þ b xyþ b 0 y þ b x y þ b xy þ b 03 y 3 : For tis case, we ave te following teorem. ð0þ Teorem 3. Te quadratic near-hamiltonian system ()wit 3rd-degree polynomial perturbation can ave maximal five smallamplitude limit cycles bifurcating from te origin, i.e., H (3) = 5. Proof. Again set b 0 = 0 in order for te origin (0,0) of te perturbed Hamiltonian system () to be an element center, under wic ~v 0 ¼ 0. Ten, applying te Maple program to system () wit q 3 (x,y) given in (0), we obtain te focus value ~v as ~v ¼ 6 ½ða Þ b þ a b 0 b 6 b 03 Š: ðþ Solving () for b 03 results in b 03 ¼ 6 ½ða Þ b þ a b 0 b Š: ðþ Ten, we obtain ~v as follows: ~v ¼ 8 a ð5a Þ ða b 0 b Þ 9 0a þð3a þ 0Þða Þ ðb þ b Þ: ð3þ First, suppose a 5. We solve ~v ¼ 0 to obtain b ¼ a b 0 þ 0 a þð3 a þ 0Þða Þ a ð5 a Þ under wic ~v 3 ; ~v, etc. become ðb þ b Þ; ðþ

5 776 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) i 307 ð5 a ða Þ þ Þða Þ a ðb þ b Þ 3a þ a a ; i 368ð5a ða Þ þ Þða Þ a ðb þ b Þ 673a þ 770a3 93a þ 859a 70 þ a 80a þ 99a a ; i ð5 a ða Þ þ Þ ða Þ a ðb þ b Þ a 6 þ 88868a a þ a3 8058a þ a þ a a þ035779a a þ 078a 606 a a 38768a þ þ 57350a ~v 3 ¼ 35 ~v ¼ 35 ~v 5 ¼ 7.. Tere is a common factor ða þ Þ ða Þ a to a center. Tus, in order for it to be non zero, let a ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ða þ Þ 6 for a ; p ffiffiffi 3![ 3 p ffiffiffi! 3 ; ð; :3090Þ[ð0:3090; Þ: 3 Under te condition (6), ~v and ~v 5 are reduced to ð5þ i ðb þ b Þ in te expressions of ~v i. Setting tis factor to equal zero leads 7 ~v ¼096 ð5 a ða Þ þ Þ a 8 a þ ðb þ b Þ a 3 þ 6 a 8 a þ ; 7 ~v 5 ¼95 ð5 a ða Þ þ Þ a 8 a þ ðb þ b Þ ð8þ 638 a 5 þ 3873 a þ 8 a a þ 568 a 70 : Terefore, te only possibility for ~v ¼ 0 but ~v 5 0 is te roots of te polynomial ð6þ ð7þ F ða Þ¼ a 3 þ 6 a 8 a þ : ð9þ It is easy to sow tat te discriminant of te equation F (a )=0is D ¼ < 0; indicating tat F (a ) = 0 as tree real roots, given by a ¼5: ; 0: ; : ; wic are all located in te interval given in (7). Furter, wen a satisfies (9), ~v 5 is reduced to ~v 5 ¼ 9 73ð5a Þ ða þ Þ a 8a þ 3073a 57a þ 500 ðb þ b Þ 0 for b + b 0 and a taking one of te real roots of F (a )=0. Te above results sow tat one may coose te perturbation parameters b 0, b 03, b, a and a suc tat ~v i ¼ 0; i ¼ 0; ; ; 3;, but ~v 5 0. Combining wit (6), we obtain a total of six sets of solutions for (a,a ). Tus, for tis case, at most 5 small limit cycles can bifurcate from te origin. Furter, by proper perturbations on te parameters a, a, b, b 03 and b 0, we can obtain five small-amplitude limit cycles. Alternatively, we can sow te existence of five limit cycles by verifying te @a D 3 ¼ 5 ¼ 5 a a 8a þ ða þþ ðb þb Þ 65a ð5a Þ 3 38a5 568a þ896a3 þ688a 696a þ38 5 if b + b 0, were ~v 3 and ~v are given in (5), evaluated at te critical point determined by (6) and (9), since te 6tdegree polynomial in te above expression as te real roots: a ¼:75669 ; 0:60575 ; 0:889 ; 6:06837 ; none of tem satisfies (9). Tis non-zero determinant implies tat proper perturbations on a and a can be found suc tat j~v jj~v 5 j < and ~v ~v 5 < 0. Oter tree perturbations on b, b 03, b 0 can be easily obtained one by one, satisfying j~v j jj~v jþ j and ~v j ~v jþ < 0 for j =0,,,3.

6 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) Now, consider te case wen a ¼ 5. For tis special case, b 03 and b become b 03 ¼ ðb þ 8 a b 0 b Þ; b ¼ 5 b 0 þ 5 68 a 6 a þ 7 ðb þ b Þ; under wic ~v ¼ ~v ¼ 0, and ~v 3 ¼ 5 ð3 756 a 7Þ 6 a ~v ¼ 3a ~v 5 ¼.. Hence, 79 ðb þ b Þ; 006a 88796a 978 ðb þ b Þ; 3 a a a a ðb þ b Þ; a ¼ pffiffiffiffiffiffiffiffi 79; ð3þ and ten ~v ¼ ðb þ b Þ; ð33þ sowing tat wen a ¼ 5, te system can ave at most four small-amplitude limit cycles around te origin. In conclusion, we ave sown tat H (3) = 5. ð30þ ð3þ Remark (i) Te metod and formulas presented in tis section for proving Teorem 3 are different from tat given in [], but lead to te same conclusion. (ii) Te coefficients a and a appeared in te Hamiltonian function do not play any role in determining H () =, wile tey are used to obtain two additional limit cycles in obtaining H (3) = 5. (iii) In te case of cubic perturbation, te number of b ij coefficients is more tan needed. For example, we may set b = b 0 = 0. Tus, b 03 ¼ 3 b ; b ¼ 0 a þð3 a þ 0Þða Þ a ð5 a Þ wic will greatly simplify te computation in analysis. b ; ð3þ 3. Quadratic Hamiltonian systems wit t- to 8t-degree polynomial perturbations In tis section, we consider te quadratic near-hamiltonian system () wit iger-degree polynomial perturbations. In particular, we sall study te cases n =,5,...,8, and leave cases n = 9,0,...,0 to be discussed in te next section. 3.. t-degree polynomial perturbation H () = 6 Wen n =, we ave q ðx; yþ ¼b xyþ b 0 y þ b x y þ b xy þ b 03 y 3 : þ b 3 x 3 y þ b x y þ b 3 xy 3 þ b 0 y ; were b 0 as been set zero in order for te origin (0,0) to be an element center under perturbation. ð35þ Teorem. Te quadratic near-hamiltonian system () wit t-degree polynomial perturbation can ave maximal six smallamplitude limit cycles bifurcating from te origin, i.e., H () = 6. Proof. Te first focus value, ~v, is te same as tat given in (), and tus te solution given in () also applies to tis case. Ten, we obtain ~v as follows: 9 a ð5a Þða b 0 b Þþ 0a þð3a þ 0Þða Þ ðb þ b Þ þ ð3 a 0Þ b 3 þ 8 a b þ 6 ð3 a Þ b 3 þ 80 a b 0 g: ~v ¼ To ave maximal number of limit cycles, suppose a 5. Ten solving b from te equation ~v ¼ 0 results in ð36þ

7 778 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) b ¼ a b 0 þ a ð5a Þ Next, solving ~v 3 ¼ 0 for b yields 0 a þð3 a þ 0Þða Þ ðb þ b Þþ ð3a 0Þb 3 þ 8a b þ 6 ð3a Þb 3 þ 80a b 0 : b ¼b 7 a 3 3a þ 57a a 3a þ a a 300a3 þ 68a þ 336a þ a 0a þ 0 a b3 a 36a þ 0 8a a a b 3a b 3 þ 8a b 0 Ten we obtain ð37þ ð38þ ~v ¼ 88 Q ða ; a Þ F ða ; a Þ; ~v 5 ¼ 368 Q ða ; a Þ F ða ; a Þ; ~v 6 ¼ Q ða ; a Þ F 3 ða ; a Þ; ð39þ were Q ¼ a3 a b3 þ a a b 6a b 3 þ 6a b 0 3a þ a ; a F ¼880a 863a 0a a þ 8a 68a3 68a þ 63a 880; F ¼00 a a 986 a 088 a 6 03 a 0968 a3 þ a 580 a þ 5936 a þ 67a a5 þ 96a þ 958a3 38a þ 98a 3500; F 3 ¼ a a a 9308 a a a3 þ a 3600 a þ78850 Þ a a6 þ a5 þ a a 3 þ a 663 a þ 3600 a þ 6339a a7 þ 5888a6 þ a a þ 77886a a þ a : Eliminating a from te equations F (a,a )=F (a,a ) = 0 yields te solution for a : a ¼ G ða Þ¼ 079a6 886a5 797a þ 099a3 396a 03a þ ð509 a þ 076 a a 6766 a ; ð0þ þ 89Þ and a resultant equation: F ða Þ¼ða þ Þ F ða Þ¼0, were F ¼ 7757a9 560a8 þ 9070a a a5 þ a 66a3 þ a 0880a þ 960: ðþ Since G () = <0,a = is not a solution. Te equation F ¼ 0 as 7 real solutions, among wic 6 solutions satisfy G > 0, given by a ¼:39995 ; 0:8877 ; 0: ; 0: ; : ; 0: : Tus, tere are in total solutions suc tat ~v i ; i ¼ 0; ; ; 5. To sow tat under tese solutions, ~v 6 0, we simplify F 3 under te condition (0) to obtain 08 ða þ Þ 3 F 5 ða Þ¼ 509a þ 076a a þ 6766a F 5 þ 89 ða Þ;

8 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) were F 5 ¼ a a0 þ a a a 7 þ a6 þ a a a 3 þ a a a0 þ a a a 7 þ a a 5 þ a a 3 þ a a þ : ðþ It can be sown tat for te six roots of F ða Þ¼0, F 5 ða Þ 0. For example, taking te second root and truncated up to 00 decimal points yields te following critical parameter values: a ¼ 0:88 ; a ¼ 0: ; b ¼0:9805 b 3 ; b ¼ :6593 b 3 ; b 03 ¼ 0: b 3 ; b 0 ¼ 0; were we ave set b = b 0 = b = b 3 = b 0 = 0, under wic te computed focus values, up to 00 decimal points, are ~v 0 ¼ 0; ~v ¼ 0; ~v ¼ 0: b 3 ; ~v 3 ¼0: 0 99 b 3 ; ~v ¼ 0: b 3 ; ~v 5 ¼ 0: b 3 ; ~v 6 ¼ 0:00056 b 3 : Witout loss of generality, we may take b 3 =. It sould be noted tat for te exact solutions (in oter words, if we obtain te solutions up to infinite decimal points), te focus values ~v i ; i ¼ ; 3; ; 5 sould be exactly equal to zero. Furter, using (39) and (0) it can be sow D ¼ ðq ¼ F ðq F ðq F ðq F ~v @a ¼ F þ F @Q F @a F @F ðdue to F ¼ F ¼ 0 at te critical 99 a ða þ Þ ða þ Þ Q ¼ a þ 076 a a 6766 a 3 þ a a5 þ a þ a a þ a þ a a a 8 þ a a a5 þ a a3 þ a a þ since Q 0, te denominator (wic is te denominator of te solution for a ) is non zero, and te 6t-degree polynomial factor in te above expression as eigt real solutions for a, but none of tem satisfies F ða Þ¼0. In fact, for te above cosen parameter values, we obtain D ¼ 0: b 3 0 ðb 3 0Þ. Summarizing te above results sows tat one can coose b 0, b 03, b, b, a and a suc tat ~v i ¼ 0; i ¼ 0; ; ; 5, but ~v 6 0. Furter, we can perturb tese coefficients in backwards to generate j~v j jj~v jþ j and ~v j ~v jþ < 0 for j ¼ 0; ; ; 5: Tis finises te proof. Remark. Again it is noted tat te number of b ij coefficients is more tan needed. For example, we may set b = b 0 = b = b 3 = b 0 = 0, wic will greatly simplify te analysis.

9 780 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) t-degree polynomial perturbation H (5) = 8 For n = 5, we ave te following result. Teorem 5. Te quadratic near-hamiltonian system ()wit 5t-degree polynomial perturbation can ave maximal eigt small limit cycles bifurcating from te origin, i.e., H (5) = 8. Proof. For tis case, q 5 (x,y) is given by q 5 ðx; yþ ¼b xy þ b 0 y þ b x y þ b xy þ b 03 y 3 þ b 3 x 3 y þ b x y þ b 3 xy 3 þ b 0 y þ b x y þ b 3 x 3 y þ b 3 x y 3 þ b xy þ b 05 y 5 ; ð3þ were b 0 as again been set zero in order for te origin (0,0) to be an element center under perturbation. Now, based on te focus value computation, solving ~v ¼ 0 for b 03 gives te same solution as given in (). Solving ~v ¼ 0 for b, ~v 3 ¼ 0 for b, ~v ¼ 0 for b 05, and ~v 5 ¼ 0 for b, yields b ¼ a b 0 þ 3a a ð5a Þ þ a 0 þ 0a ðb þ b Þþð3a 0Þ b 3 þ 8 a b þ 6 ð3a Þ b 3 þ80 a b 0 ðb þ b 3 Þ60 b 05 g; b ¼b 7 a 3 3a þ 57a a 3a þ a a 300a3 þ 68a þ 336a þ a 0a þ 0 a b3 a 36a þ 0 8a a a b 3 a b 3 þ 8a b 0 36a 3 3a 8a þ 8 þ ð6a Þa b þ 8a ð5a Þð3a Þ b 3 6a 9a þ a þ 8a þ 6 a ð7a Þð5a Þ b 0 3 7a 3 9a þ 8a 8ða Þ a b05 and b 05 ¼ bn 05, b 0 b D ¼ bn, were 05 6 a b D b N 05 ¼ 8a 68a3 68a þ 63a a 0a þ0a a a 3 a b3 6a b 3 þa a b þ 6a b 0 Šþ a 387a þ 0a3 þ 600a 66a þ 50 a 363a þ 78a3 576a þ a þ 30 þ a 87a a 5 þ 3a b a 7a 5 þ 0a 8a3 768a þ 360a 35 8a 3 þ 93a a þ 8ða Þa a b 3 þ 6 a 7a 5 þ a 600a3 0a 7a þ 58 8a 3 þ 30a 8a þ56 ð33a þ 9Þa a b 3 6 a 5a a3 68a þ 68a 7 þa 77a3 þ 9a þ 3a a b ; b3 b D 05 ¼ 3 36a5 09a 6a3 þ 50a 38a þ 76 6 a a3 87a 0a þ 8 þða 9Þa ; n b N ¼ 3a ða Þ 7a 3 þ 378a þ 68a 0 8a 80a 6 þ 975a5 658a þ 658a3 þ88a 070a þ 675 þ a 7a5 86a 6a3 þ 358a þ 8a 396 3a 3 þ 56a þ 3a 30 30a a a 3 a b3 þ a a b 6a b 3 þ 6a b 0 n þ 3a 3 ða þ Þða Þ 7a 3 þ 378a þ 68a 0 a ða Þ 0a 7 þ 508a6 þ6a 5 396a þ 98a3 þ 35a 976a þ 96 þ 6a 9a7 þ 98a6 77a5 þ 586a 7a3 008a þ 6768a 760 a a 30a 939a3 þ 670a 37a þ a 3a þ 0 39a a b n þ a ða Þ 3a ða Þ 7a 3 þ 378a þ 68a 0 8a ða Þ 80a þ 695a3 98a 988a þ 688 þ a 7a5 86a 6a3 þ 358a þ8a 396 a 3a3 þ 56a þ 3a n 30 30a b3 6a 3a ða Þ 7a 3 þ 378a þ 68a 0 8a ða Þ 80a þ 695a3 98a 988a þ 688 þ a 7a5 86a 6a3 þ 358a þ8a 396 a 3a3 þ 56a þ 3a 30 30a b3 ;

10 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) b D ¼ 3 7a7 þ 6a6 908a5 þ 360a þ 00a3 a þ 086a 66 þ a 68a8 380a7 8056a6 þ 0955a 5 þ 8a 776a3 þ 58a 83a 33 a 5a6 þ 39a þ 663a5 659a3 þ0a þ 0a 396 a 3a 3 þ 56a þ 9a 350 5a : Ten, te 6t focus value becomes ~v 6 ¼ 3 99 Q 75 5ða ; a Þ F 5 ða ; a Þ and ~v 7 ¼ 3608 Q 5ða ; a Þ F 53 ða ; a Þ; were Q 5 ða ; a Þ¼ F 5ða ;a Þ F 50 ða ;a Þ, and F 50 ¼ 3ða Þ 7a 3 þ 378a þ 68a 0 þ a ða Þ 68a 7 a6 5a 5 33a þ 658a3 þ 30a þ a þ 66 a 5a6 þ 663a 5 þ 39a 659a3 þ 0a þ 0a 396 a 3a3 þ56a þ 9a 350 5a ; F 5 ¼ a a 3 3a þ 3 a a 3 a b3 þ a b a 6a b 3 þ 6a b 0 þ a 3 þ a a b þ a ða Þb 3 6a b 3 ; F 5 ¼ 05 a þ 66 a3 þ 66 a 566 a þ a 99 a þ 708 a þ 5 70 a ; F 53 ¼ 899a 6 þ 968a a 39778a3 þ 0980a 06387a þ 773 þ a 553a þ 6906a3 þ 559a 3803a 8938 a 8307a þ 669a þ 33 08a : Next, eliminating a from te equations F 5 (a,a )=F 53 (a,a ) = 0 yields a ¼ G 5ða Þ¼ 0565a6 þ 955a5 þ 06996a þ 60a3 6057a þ 958a a þ 78836a3 þ 666a þ 66a ; 776 and a resultant equation: F 5 ða Þ¼ða þ Þ F 5 ða Þ¼0, were F 5 ða Þ¼8a 9 þ 7507a8 þ 5630a a a5 þ 5033a a3 þ a a þ 8658: a = is not a solution since G 5 () = < 0. Te polynomial F 5 ða Þ as tree real roots, given by a ¼: ; 0: ; 0: ; all of tem satisfy G 5 (a ) > 0. Tus, tere are in total six solutions. It can be sown tat for tese six solutions, ~v i ¼ 0; i ¼ 0; ; ; 7, but ~v 8 0. For example, taking te second root of F 53 ða Þ¼0 yields te critical parameter values, up to 00 decimal points: a ¼ 0:0876 ; a ¼ 0:57080 ; b ¼ 0:06 b ; b 05 ¼0: b ; b ¼0:99805 b ; b ¼:67999 b ; b 03 ¼ 0:3368 b ; b 0 ¼ 0; were we ave set b = b 0 = b 3 = b = b 3 = b 0 = b 3 = b 3 = 0, and te corresponding focus values are ~v 0 ¼ 0; ~v ¼ 0; ~v ¼ 0: 0 00 b ; ~v 3 ¼ 0: b ; ~v ¼0: b ; ~v 5 ¼ 0: b ; ~v 6 ¼0: b ; ~v 7 ¼ 0: b ; ~v 8 ¼ 0: b : Tis sows tat tere exist at most eigt small-amplitude limit cycles around te origin. Moreover, by a similar argument, one can sow tat

11 78 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) ~v ~v D 5 ¼ 5 05 ~v ~v 05 Q 5 ¼ ðq 5 F ðq 5 F 53 ðq 5 F @ ðq 5 F a ða þ Þ ða þ Þ Q 5 ¼ a þ a3 þ 666 a þ 66 a a a a a a a þ a0 þ a a a7 þ a 6 þ a a þ a a þ a Þ 0 at te critical point. Indeed, for te above cosen parameter values, D 5 ¼ 0: b 0(b 0). Tis implies tat appropriate perturbations can be made to a, a, b, b 05, b, b, b 03, b 0 to obtain eigt limit cycles t-degree polynomial perturbation H (6) = 9 In tis section, we consider te case n = 6, for wic we ave te following teorem. Teorem 6. Te quadratic near-hamiltonian system () wit 6t-degree polynomial perturbation can ave maximal nine small-amplitude limit cycles bifurcating from te origin, i.e., H (6) = 9. Proof. For tis case, q 6 (x,y) can take te following form: q 6 ðx; yþ ¼ X b ij x i y j ; wit b i0 ¼ 0; i ¼ ; 3; ; 6: ðþ 6iþj66 First note tat te focus values ~v and ~v are identical to tat of te case n = 5. Tus, te solutions of b 03 solved from ~v ¼ 0 and b from ~v ¼ 0 are identical to tat for case n = 5. We ten solve ~v 3 ¼ 0 for b, ~v ¼ 0 for b 05, ~v 5 ¼ 0 for b, and ~v 6 ¼ 0 for b 3 to obtain b :¼ð35 ða^3 3 a^ þ a^þð3a^ þ a a^þb þ 5 ð57 a^ 300 a^3 þ 68 a^ þ 336 a þ a^ ð a^ 0 a þ 0ÞÞ b3 þð a^ 36 a þ 0 8 a^þð0 a a b þ 30 a^ b3 80 a b0þ0 ð36 a^3 3 a^ 8 a þ 8 þ 7 a a^ a^þb 30 a ð9a^ þ a þ 8 a^þb3 00 ð a^3 7 a^ þ a 8 a a^ þ 8 a^þb05 þ ð5a Þð0 a ð3a Þb3 þ 0 a ð7a Þb 5 ð3a Þb5 a b 3 ð9a 0Þb33 0 a b 5 ð5a Þb5 0 a b06þþ =35=ð3 a^5 þ 3 a^ 0 a^3 þ a^ þ 8 a 6 a^3 a^ 8 a a^ þ 6 a^þ : b05 :¼ð5ð8 a^ 68 a^3 68 a^ þ 63 a a^ ð63 a^ 0 a þ 0 a^þþ ðða^3 a^þb3 þ a a b 6 a^ b3 þ 6 a b0þ5ða^ ð387 a^ þ 0 a^3 þ 600 a^ 66 a þ 50Þa^ ð363 a^ 576 a^ þ 78 a^3 þ a þ 30 þ a^ ð87 a^ a 5 þ 3 a^þþþ b 0 a ð7 a^5 þ 0 a^ 8 a^3 768 a^ þ 360 a 35 8 a^ ðð a^3 þ 93 a^ a þ Þ6ð a Þa^ÞÞ b3 þ 30 a ð7 a^5 þ a^ 600 a^3 0 a^ 7 a þ 58 8 a^ ð a^3 þ 30 a^ 8 a þ a a^ 38 a^þþ b3 30 a ð5 a^ a^3 68 a^ þ 68 a 7 þ a^ ð77 a^3 þ 9 a^ þ 3 a a^þþ b 80 ða ð36 a^ 99 a^3 750 a^ þ 60 a þ 88Þa^ ð0 6 a^ þ 8 a 7 a^ þ 08 a a^þþ b5 þ 6 a ð7 a^ þ 375 a^3 9 a^ þ a þ 5 a^ ð05 a^ þ 9 a a^þþ b 6 ð59 a^5 þ 990 a^ þ 86 a^ 6 a^3 69 a þ a^ ð05 a^3 þ a a^ a^ þ 98 a 67 a^þþ b33 þ 80 a ð35 a^ þ 7 a^3 88 a^ 58 a þ 88 8 a^ ð8 a^ þ 3 a a^þþ b 0 ð3ð35 a^5 þ 9 a^ þ 80 a^3 þ a^ 58 a þ 88Þa^ ð630 a^3 þ 5 a^ a 76 0 a a^ 76 a^þþ b5 þ 680 a ð5 a^3 70 a^ 56 a þ 8 þ a^ a^ þ 36 a a^ þ 0 a^þb06þ=00=ð3 ð36 a^5 09 a^ 6 a^3 þ 50 a^ 38 a þ 76Þ 6 a^ ð a^3 87 a^ 0 a þ 8 þ a a^ 9 a^þþ : b :¼... b3 :¼...

12 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) Te lengty expressions for b and b 3 are omitted ere for brevity. Under te above conditions, computing te 7t and 8t focus values yields 3 ~v 7 ¼ Q 3 6ða ; a Þ F 6 ða ; a Þ and ~v 8 ¼ Q 6ða ; a Þ F 63 ða ; a Þ were Q 6 ða ; a Þ¼ F 6ða ;a Þ, and F 60 ða ;a Þ F 60 ¼ 05a þ 66a3 þ 66a 566a þ 360 8a 99a þ 708a þ 5 70a ; F 6 ¼ 5a a3 a b5 þ a 3 þ 8a ða b 3a b 33 Þþ80a a b 0 3a 3 þ a b5 þ 0a a b 06 F 6 ¼ 3779a a7 778a6 366a5 þ 77370a þ 309a a þ 69579a a 393 a a a a3 06 a þ 38 a þ 70 a 3 75a 3596a3 7896a 38688a þ 97 þ8088 a 3 a þ a a ; F 63 ¼ 50566a a a a7 þ a 6 þ a a þ a a þ a a a a a6 þ 9387a5 þ a a 3 þ a a þ a a6 6585a5 þ 39000a þ a3 þ a 79639a þ þ a 68887a 868a a a þ þ5a 6957a þ 35808a a : Now, eliminating a from te two equations F 6 (a,a )=F 63 (a,a ) = 0 yields a ¼ G 6ða ÞA, were A :¼ ð a^ a^ a^ þ a^0 þ a^9 þ a^ a^ a^6 þ a^5 þ a^ a^ a^ þ a^ þ a^ a^ a^8 þ a^ a^ a^5 þ a^ a^ a^ þ a Þ =ð a^ a^ a^9 þ a^8 þ a^7 þ a^ a^ a^ þ a^ a^ a^ þ a^ a^ a^8 þ a^7 þ a^ a^5 þ a^ a^ a^ þ a Þ : and a resultant equation: F 6 ða Þ¼ða þ Þ F 6 ða Þ¼0, were

13 78 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) F 6 ¼ 5 a3 þ 3 a 08a þ a a6 þ a 5 þ a a a þ a þ a a a8 þ a 7 þ a a 5 þ a a 3 þ a a a0 þ a a8 þ a a6 þ a 5 þ a a 3 þ a þ a Þ: Again a = is not a solution since G 6 () = < 0. Te polynomial F 6 ða Þ as ten real roots, all of tem satisfying G 6 (a ) > 0, but only seven of tem leading to ~v i ¼ 0; i ¼ 0; ; ; 8, but ~v 9 0. Tese solutions are a ¼:5853 ; :797 ; 0:876 ; 0:090 ; 0:87 ; :509 ; 5:568 : Tus, tere are in total twelve solutions. One example for taking te parameter values is given as follows (up to 00 decimal points): a ¼0:876 ; a ¼ :595 ; b 3 ¼6:69700 b 5 ; b ¼ :7006 b 5 ; b 05 ¼:73869 b 5 ; b ¼0: b 5 ; b ¼:3539 b 5 ; b 03 ¼ 0:38893 b 5 ; b 0 ¼ 0; were b, b 0, b 3, b, b 3, b 0, b, b 3, b, b 33, b, b 5, b 06 ave been set zero, associated wit te following focus values: ~v 0 ¼ ~v ¼ 0; ~v ¼ 0: b 5 ; ~v 3 ¼ 0: b 5 ; ~v ¼ 0: b 5 ; ~v 5 ¼0: b 5 ; ~v 6 ¼ 0: b 5 ; ~v 7 ¼ 0: b 5 ; ~v 8 ¼0: b 5 ; ~v 9 ¼0: b 5 : Furter, we can similarly sow tat for all te twelve solutions (critical points), ~v ~v D 6 ¼ 5 ~v For example, for te above cosen parameter values, D 6 ¼0: b 5 0 ðb 5 0Þ. Tus, we can perturb te parameters a, a, b 3, b, b 05, b, b, b 03, b 0 to obtain nine limit cycles t-degree polynomial perturbation H (7) = 0 For n = 7, we ave te following result. Teorem 7. Te quadratic near-hamiltonian system () wit 7t-degree polynomial perturbation can ave maximal eleven small-amplitude limit cycles bifurcating from te origin, i.e., H (7) = 0. Proof. For tis case, q 7 (x,y) takes te following form q 7 ðx; yþ ¼ X b ij x i y j ; wit b i0 ¼ 0; i ¼ ; 3; ; 7: ð5þ 6iþj67 Te focus values ~v and ~v for tis case are identical to tat of cases n = 5, 6, and tus te solutions for b 03 and b are te same as tat given in te previous two cases. Ten, similarly solving ~v 3 ¼ 0 for b, ~v ¼ 0 for b 05, ~v 5 ¼ 0 for b, ~v 6 ¼ 0 for b 3, ~v 7 ¼ 0 for b 07, and ~v 8 ¼ 0 for b 6, yields a family of solutions, for wic computing te 9t and 0t focus values yields ~v 8 ¼ Q 7ða ; a Þ F 7 ða ; a Þ ~v 9 ¼ were Q 7 ða ; a Þ¼ F 7ða ;a Þ F 70 ða ;a Þ, and Q 7ða ; a Þ F 73 ða ; a Þ;

14 F 70 ¼ a 8337a a8 þ a7 þ a a a þ a a þ a a 609a8 9960a7 þ 8795a6 þ a5 5586a 80336a 3 þ 8888a 77370a þ 9736 a 3 975a6 0550a5 þ 5956a 5379a a 5578a þ 73788Þþ36a 587a þ 3930a3 þ 03a þ 83960a þ 66 a 79a þ 358a þ 75 ; F 7 ¼ 3a 3 þ 8a 5a a3 a b5 þ a a 3 þ 8a b 6a a 3 þ 8a b33 þ80a a b 0 3a 3 þ a b5 þ 0a a b a 3 þ a b5 þ 5a a 7 a6 8a a 8a3 a þ 6a b6 a 3 þ 8a 0a 3a a b5 3a 3 5a a a b3 þ 0a a a 3 a þ a b3 0a 3a 3 8a a 8a b6 ; F 7 ¼ a þ a a0 þ a9 þ a a7 059a6 þ a5 þ a a3 þ a a þ a a 0 þ 6308a9 83a8 þ a 7 þ a6 þ 300a a þ889388a 3 þ a a þ a a8 þ a7 6708a a5 þ a þ a a þ a þ þþ6a a6 þ a5 0338a a 3 þ 55780a þ 39800a a 3 75a 76a3 55a 9966a 63 8a 5a 7988a þ 860a ; F 73 ¼ a þ a a þ a þ a a9 þ a8 þ a a6 þ a5 þ a a3 þ a a þ a P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) a þ a a0 þ a a a 7 þ a6 þ a a þ a a a þ a 095a0 þ 68930a a8 þ a a a5 þ a þ a a 55568a þ þ a a8 þ a a 6 þ a5 þ a a a þ a þ a a6 þ a a 99660a 3 þ a a a 5655a a a 7306a a 8955a 0830a þ 50500a : Ten, eliminating a from te two equations F 7 (a,a )=F 73 (a,a ) = 0 yields a ¼ G 7ða Þ, and a resultant equation: F 7 ða Þ¼ða þ Þ F 7 ða Þ¼0, were

15 786 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) F7 :¼ ð8 a^9 þ 7507 a^8 þ 5630 a^ a^ a^5 þ 5033 a^ a^3 þ a^ a þ 8658Þ ð a^5 þ a^53 þ a^ a^ a^50 þ a^9 þ a^ a^7 þ a^6 þ a^ a^ þ a^ a ^ þ a^ þ a^ a^39 þ a^38 þ a^ a^36 þ a^35 þ a^ a^33 þ a^3 þ a^ a^30 þ a^ a^ a^7 þ a^ a^5 þ a^ a^ a^ þ a^ a^0 þ a^ a^8 þ a ^ a^6 þ a^ a^ þ a^ þ a^ a^ þ a^ a^9 þ a^ a^ a^6 þ a^ a^ þ a^ a^ þ a : Note tat a = is not a solution since G 7 () = < 0. Te polynomial F 7 ða Þ as eleven real roots, nine of tem satisfying G 6 (a ) > 0. But only six of tem satisfy ~v i ¼ 0; i ¼ 0; ; ; 9, but ~v 0 0. Tese solutions are a ¼: ; : ; 0: ; 0: ; 0: ; 0:90977 : Tus, tere are in total twelve solutions. One example for taking te parameter values is given as follows (up to 00 decimal points): a ¼0:3898 ; a ¼ :7876 ; b 07 ¼0:8880 b 6 ; b 3 ¼ 0:5636 b 6 ; b ¼ 0:69003 b 6 ; b 05 ¼ 0:3879 b 6 ; b ¼0:309 b 6 ; b ¼ :63703 b 6 ; b 03 ¼ 0:03697 b 6 ; b 0 ¼ 0; were we ave set b = b 0 = b 3 = b = b 3 = b 0 = b = b 3 = b 5 = b = b 33 = b = b 5 = b 06 = b 5 = b 3 = b 3 = b 5 = b 6 =0. Te corresponding focus values are:

16 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) ~v 0 ¼ 0; ~v ¼ 0: 0 00 b 6 ; ~v ¼ 0: b 6 ; ~v 3 ¼0: 0 99 b 6 ; ~v ¼ 0: 0 99 b 6 ; ~v 5 ¼0: b 6 ; ~v 6 ¼0: b 6 ; ~v 7 ¼ 0: b 6 ; ~v 8 ¼ 0: b 6 ; ~v 9 ¼0: b 6 ; ~v 0 ¼0: b 6 : Furter, we can similarly sow tat for all te twelve solutions, ~v ~v D 7 ¼ 5 ~v In particular, for te above cosen parameter values, D 7 ¼0:0663 b 6 0 ðb 6 0Þ. Tus, we can perturb te parameters a, a, b 07, b 3, b, b 05, b, b, b 03, b 0 to obtain ten limit cycles. Remark 3. Following te pattern seen from solving te cases n = 3,, 5, 6, it seems tat b 6 can be used to solve ~v 8 ¼ 0, and tus te case n = 7 may ave eleven limit cycles. If tis is true, ten te number of small limit cycles around te origin would obey te rule H ðnþ ¼ ð3n þ Þ, rater tan H ðnþ ¼ ðn þ Þ 3. However, it as been observed from te expressions ~v8 and ~v 9 tat at tis step all te remaining b ij coefficients, b 5, b, b 33, b, b 5, b 06, b 6, b 5, b 3, b 3, b 5 and b 6 appear in a common factor F 7 (tis factor also appears in ~v 0 ; ~v ; ), and so using any of tese coefficients to solve ~v 8 ¼ 0 would result in ~v 9 ¼ ~v 0 ¼¼0. Tis unusual pattern will also appear in te cases n = 3, 9, to be discussed in te next section t-degree polynomial perturbation H (8) = To end tis section, we consider te case n = 8, for wic we ave te following teorem. Teorem 8. Te quadratic near-hamiltonian system () wit 8t-degree polynomial perturbation can ave maximal twelve small-amplitude limit cycles bifurcating from te origin, i.e., H (8) =. Proof. For tis case, q 8 (x,y) can be written as q 8 ðx; yþ ¼ X b ij x i y j ; wit b i0 ¼ 0; i ¼ ; 3; ; 8: ð6þ 6iþj68 First it is noted tat te focus values ~v ; ~v and ~v 3 are identical to tat of case n = 7, and tus te solutions b 03, b and b solved from ~v ¼ ~v ¼ ~v 3 ¼ 0 are also te same as tat obtained in case n = 7. Ten, solving ~v ¼ 0 for b 05, ~v 5 ¼ 0 for b, ~v 6 ¼ 0 for b 3, ~v 7 ¼ 0 for b 07, ~v 8 ¼ 0 for b 6, and ~v 9 ¼ 0 for b 5 we obtain a family of solutions under wic computing te 0t and t focus values yields ~v 0 ¼ Q 8ða ; a ÞF 8 ða ; a Þ; ~v ¼ Q 8ða ; a ÞF 83 ða ; a Þ; were Q 8 ða ; a Þ¼ F 8ða ;a Þ F 80 ða ;a Þ, and F 8i, i =,, 3 are polynomials of a and a. Teir lengty expressions are omitted ere for brevity. Now eliminating a from te two equations F 8 (a,a )=F 83 (a,a ) = 0 yields a ¼ G 8ða Þ and a resultant equation: F 8 ða Þ¼ða þ Þ F 8 ða Þ¼0, were F8 :¼ ð075 a^9 þ a^8 þ 5905 a^ a^6 988 a^5 þ a^ a^3 þ 5055 a^ þ 50 a þ 5Þ ð a þ a^ a^3 þ a^ a^5 þ a^ a^7 þ a^ a^ a^0 þ a^ a^ þ a^ a^

17 788 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) a^5 þ a^ a^7 þ a^ a^9 þ a^ a^ þ a^ þ a^ a^ þ a^ a^ a^7 þ a^ a^ a^30 þ a^ a^ a^33 þ a^ a^ a^36 þ a^37 þ a^ a^39 þ a^ 0 þ a^ a^ a^ a^ þ a^5 þ a^6 þ a ^ a^ a^ a^50 þ a^5 þ a^ a^53 þ a^5þ : a = is not a solution since G 8 () = < 0. Te polynomial F 8 ða Þ as tirteen real roots, eleven of tem satisfying G 8 (a ) > 0. By verifying te original focus values, only eigt of tem satisfy ~v i ¼ 0; i ¼ 0; ; ;, but ~v 0. Tese solutions are (up to 00 decimal points): a ¼:66530 ; : ; 0:76099 ; 0:8085 ; 0: ; 0: ; 0: ; : : Tus, tere are in total sixteen solutions. One example for taking te parameter values is given as follows (up to 00 decimal points): a ¼:50858 ; a ¼ 0:80 ; b 08 ¼0:730 b 7 ; b 6 ¼:7050 b 7 ; b 07 ¼ :0705 b 7 ; b 3 ¼0:5763 b 7 ; b ¼ 5:5653 b 7 ; b 05 ¼0:30 b 7 ; b ¼0:98573 b 7 ; b ¼:08588 b 7 ; b 03 ¼ 0:3857 b 7 ; b 0 ¼ 0; were we ave set b = b 0 = b 3 = b = b 3 = b 0 = b = b 3 = b 5 = b = b 33 = b = b 5 = b 06 = b 6 = b 5 = b 3 = b 3 = b 5 = b 6 = b 53 = b = b 35 = b 6 = b 7 = 0. Te corresponding focus values are: ~v 0 ¼ ~v ¼ ~v ¼ 0; ~v 3 ¼0: 0 99 b 7 ; ~v ¼ 0: 0 98 b 7 ; ~v 5 ¼0: 0 97 b 7 ; ~v 6 ¼ 0: b 7 ; ~v 7 ¼0: b 7 ; ~v 8 ¼0: b 6 ; ~v 9 ¼0: b 7 ; ~v 0 ¼ 0: b 7 ; ~v ¼ 0: b 7 ; ~v ¼0: b 7 : Furter, we can sow tat for all te sixteen solutions, ~v ~v D 8 ¼ 5

18 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) In particular, for te above cosen parameter values, D 8 ¼ 0:00089 b 7 0 ðb 7 0Þ. Tus, we can perturb te parameters a, a, b 08, b 6, b 07, b 3, b, b 05, b, b, b 03, b 0 to obtain twelve limit cycles. Remark. Following te pattern in te previous cases, b 5 migt be used to solve ~v 9 ¼ 0. However, it as been found tat after solving ~v 8 ¼ 0 for b 6, te expression of ~v 9 actually does not contain te coefficient b 5. More precisely, it does not contain any remaining of te 7t-order coefficients: b 6, b 5, b 3, b 3 and b 5. So one must use one 8t-order coefficient to solve ~v 9 ¼ 0. Here, we ave used b 08. Tis pattern will also appear in te cases n =, 0, to be seen in te next section.. Quadratic Hamiltonian systems wit 9t-to 0t-degree polynomial perturbations In tis section, we sall consider te cases n = 9, 0,, 0. In order to simplify te presentation, first we establis a rule, based on te cases studied in te previous section, and tis rule as been verified for 9 6 n 6 0 by using general computation wit all b ij coefficients retained in te system, like wat we ave done for te cases 3 6 n 6 8. Ten we present tree representative cases n =, 3,, sowing te transfer from te regular pattern to te unusual pattern, as we ave seen in te previous section. Te results for oter cases ave been obtained, but computations are more involved for larger n... Te rule of using b ij coefficients in solving focus values We ave te following result. Teorem 9. Te b ij coefficients used in solving ~v i ¼ 0 follows te rule sown in Table. Te pattern can be clearly seen from te table... t-degree polynomial perturbation H () = 7 In tis section, we consider te case n =, for wic we ave te following teorem. Teorem 0. Te quadratic near-hamiltonian system() wit t-degree polynomial perturbation can ave maximal seventeen small-amplitude limit cycles bifurcating from te origin, i.e., H () = 7. Table Te rule of using coefficients b ij in solving v i =0. n v i due to b ij v i due to (a,a ) Nonzero v i due to Nonzero b ij LC Note b 0? v b 0 v 0 b 03? v 3 b? v (v 3,v ) b 0? v b? v (v,v 5 ) b 3 0? v b 05? v 5 b? v 5 (v 6,v 7 ) b 0? v b 3? v 6 (v 7,v 8 ) b 5 0? v b 07? v 7 (v 8,v 9 ) b 6 0? v a b 6 b 6? v 8 8 (b 08? v 9 ) (v 0,v ) b 7 0? v 0 b b 5 b 09? v 9 9 b 8? v 0 (v,v ) b 8 0? v b 7? v (v,v 3 ) b 9 0? v 0 b 0? v b 0? v 3 (v,v 5 ) b 0 0? v b 9? v (v 5,v 6 ) b, 0? v b 03? v 5 (v 6,v 7 ) b 0? v a b, b,? v 6 (b 0? v 7 ) (v 8,v 9 ) b 3 0? v b b b 05? v 7 5 b,? v 8 (v 9,v 0 ) b 0? v 0 6 b 3? v 9 (v 0,v ) b 5 0? v 0 b 07? v 0 7 b,6? v (v,v ) b 6 0? v b 5? v (v 3,v ) b 7 0? v b 09? v 3 (v,v 5 ) b 8 0? v a b,8 b,8? v 0 (b 00? v 5 ) (v 6,v 7 ) b 9 0? v b b 7 a b Indicates tat te coefficient can not be used. Denotes tat te coefficient does not appear in te process of computation.

19 790 P. Yu, M. Han / Caos, Solitons & Fractals 5 (0) Proof. For tis case, according to Table, q (x,y) takes te form: q ðx; yþ ¼b x y þ b xy þ b 03 y 3 þ b 3 x y 3 þ b xy þ b 05 y 5 þ b 6 xy 6 þ b 07 y 7 þ b 7 x y 7 þ b 8 xy 8 þ b 09 y 9 þ b 9 x y 9 þ b 0 xy 0 þ b 0 y þ b ; x y: ð7þ We use te coefficients listed in Table for n = to solve te focus value equations ~v i ¼ 0 to obtain b03 :¼ =3 b : b :¼ = ða b þ 3 a^ b þ 0 a^ b 0 b b3 60 b05þ=a=ð þ 5 aþ : b :¼ ð3 b a þ 0 b05 a 80 a b05 a^ 70 b05 a^ þ a^ b3 36 a b3 0 b07 a þ 0 b05 a^3 þ 80 b05 a^ þ 7 a^3 b3 0 b05 þ 68 b07 þ 9 a b a þ 8 a a^ b3 80 a^ b aþ =ð8 a þ a^ 8 a a^ þ 3 a^ þ 6 a^ 0 a^3 6 þ 3 a^5 a^3 a^þ=7 : b05 :¼ ð6 a a^ b3 þ 30 b a þ 58 a b3 86 a^ b3 3 b6 a þ 9 a^ a^ b3 þ 06 b07 a 360 a^3 b3 þ 80 b07 a^ þ 8 a^ b a 556 b07 þ 67 b7 þ 608 b09 þ 77 a^3 b a a b a^3 þ 696 b07 a^ þ 63 a^5 b3 06 a b7 þ 8 a^6 b3 þ 58 a^ a^ b3 þ 08 b a^3 390 a b07 a^ 976 a b a 536 a^3 b6 a a^ b07 a^ 50 a^ a^ b3 8 b09 a 800 a^ b a^ b a^3 6 a^3 b a^3 70 a^3 a^ b3 0 a^ b a þ 0080 a b6 a þ 788 b07 a^3 þ 9 a a^ b3 þ 68 b07 a^ 50 a^ b7 þ 9856 b07 a^ þ 08 b a^5 þ 67 a^ b7 þ 608 a b6 a^3 736 a^ b6 a 3 b6 a^3 þ 608 b09 a^ 536 b09 a^þ =ð39 a^ a^ 08 a^ þ 08 a^5 67 a^ 8 a^3 þ 5 a^ 67 a^3 a^ 5 a þ 760 a a^ 35 a^ a þ 58 þ 30 a^þ=0 : b :¼ ð5690 a^5 b09 a^ a^6 b07 a^ þ 59 a^5 a^6 b3 þ a^6 a^ b3 þ 397 a^ b6 a^ a^3 a^ b9 307 a^ b8 a^5 707 a^5 a^ b7 937 a^3 a^ b a^3 b09 a^ 535 a^5 b07 a^ þ 800 b07 a^ a^5 b07 a^ a^ b07 a^ þ 86 a^6 b6 a^3 þ a^ b a^ a^ b3 þ a^3 b07 a^ þ a^ a^ b b07 a^ a^ b07 a^ þ a b07 a^ 506 a a^ b9 þ a^ b6 a þ a^3 a^ b3 þ 3096 a b6 a^ a a^5 b8 þ 55 a^3 b07 a^ a^ a^ b3 590 a^ a^6 b3 þ 0778 a^3 b8 a 766 a a^ b7 308 a^ a^ b a b8 a^3 398 a b6 a^5 þ a^5 b6 a 3680 a^ b6 a^ a^3 b6 a 7873 a^3 b6 a^3 006 b07 a^ a b09 a^ a^ b8 a a^5 a^ b3 þ a^ b6 a þ 3538 a^3 b07 a^ b07 a a^ a^ b7 þ a^ b09 a^ þ a^ b07 a^ þ 65 a^6 a^ b a^ b07 a^ a^5 b3 þ 7006 a^5 a^ b3 þ 3368 a b8 a^ a a^ b8 þ 786 a^ b6 a^5 þ 6589 a^ a^6 b3 þ 6678 a^ b0 a^ þ a b09 a^ 9688 a a^7 b6 590 a^ a^8 b3 þ 6989 a^ b07 a^ a^7 a^ b a b09 a^ a^ b6 a^ a^3 b0 a^ a b6 a a b07 a^ 5779 a^ b8 a^ a b6 a^7 þ 5776 a b07 a^6 þ 603 a^ a^ b9 þ a^3 b6 a^ a^ b09 a^ a^3 b09 a^ 6736 a^3 b3 þ a^ b8 a^3 þ a a^ b7 09 a^3 a^ b7 þ a^ a^ b a^5 b6 a^3 67 a^ b6 a^3 355 a^7 a^ b a a^6 b6 þ 03 a^ b09 a^ þ a^3 a^6 b3 þ 8336 a^ a^ b7 þ a a^ b9 þ a^3 b8 a^ a b0 a^ 750 a^8 a^ b3 þ a b0 a^ 0 a^3 a^ b7 þ 6888 a a^6 b8 860 a a^6 b7 þ 5738 b6 a þ b a^ b7 þ b07 a^ þ 7099 a^ b7 þ 360 a^7 b b0 a þ 978 b09 a^3 þ 675 a b8 a b6 a^3 þ b09 a^ þ b b09 a^ þ 9839 b07 a^ b07 a^ b09 a^ b8 a 30 b07 a^6 060 a^ b7 þ a^6 b b09 a^5 þ 5576 b07 a^7 þ 6589 b6 a^ b0 a^3 þ b09 a þ 5978 b09 a^ b07 a^6 þ 7850 b0 a^ þ a b7 þ a^ b b0 a^ 8889 a^8 b3 þ 536 b0 a^5 þ 608 b0 a^ b0 a^ þ b09 a^ a b9 þ b8 a^3 þ 3970 b09 a^6 þ 6538 a^ b b7 856 b09 þ 8096 b b07 a^ b09 a^7 þ 3398 a^3 b b6 a^ a^ b9 556 a^6 b7 þ 338 a^ b b8 a^5 976 a^6 b a^ b9