FOUR LIMIT CYCLES FROM PERTURBING QUADRATIC INTEGRABLE SYSTEMS BY QUADRATIC POLYNOMIALS*

International Journal of Bifurcation and Chaos, Vol 22, No 22) 25254 28 pages) c World Scientific Publishing Company DOI: 42/S28274252549 FOUR LIMIT CYCLES FROM PERTURBING QUADRATIC INTEGRABLE SYSTEMS BY QUADRATIC POLYNOMIALS* Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom PEI YU,, and MAOAN HAN Department of Mathematics, Shanghai Normal University, Shanghai 2234, P R China Department of Applied Mathematics, The University of Western Ontario, London, Ontario, Canada N6A 5B7 pyu@uwoca Received May 4, 2 In this paper, we present four limit cycles in quadratic near-integrable polynomial systems It is shown that when a quadratic integrable system has two centers and is perturbed by quadratic polynomials, it can generate at least four limit cycles with 3, )-distribution This result provides a positive answer to an open question in this research area Keywords: Hilbert s 6th problem; quadratic near-integrable system; limit cycle; reversible system; Hopf bifurcation; Poincaré bifurcation; Melnikov function Introduction The well-known Hilbert s 6th problem has remained unsolved since Hilbert proposed the 23 mathematical problems at the Second International Congress of Mathematics in 9 [Hilbert, 92] Recently, a modern version of the second part of the 6th problem was formulated by Smale [998], chosen as one of the 8 challenging mathematical problems for the 2st century To be more specific, consider the following planar system: dt = P nx, y), dt = Q nx, y), ) where P n x, y) andq n x, y) representnth degree polynomials of x and y The second part of Hilbert s 6th problem is to find the upper bound Hn) n q on the number of limit cycles that the system can have, where q is a universal constant, and Hn) is called Hilbert number In the early 99 s, Ilyashenko and Yakovenko [99], Écalle [992] proved the finiteness theorem pioneered by Dulac, for given planar polynomial vector fields In general, the finiteness problem has not been solved even for quadratic systems Recent survey articles eg see [Li, 23; Yu, 26] and more references therein) have comprehensively discussed this problem and reported the recent progress If the problem is restricted to the neighborhood of isolated fixed points, then the question on stuing degenerate Hopf bifurcations gives rise to weak fine) focus points In the past six decades, many researchers have considered the local problem and obtained many results eg see [Kukles, 944; Bautin, 952; Malkin, 964; Liu & Li, 989; Li & Liu, 99; Yu & Han, 25a, 25b]) In the last 2 years, much progress on finite cyclicity near a weak focus point or a homoclinic loop has been achieved Roughly speaking, the so-called finite The first draft of this paper has been posted on arxivorg since February 4, 2, No 25v Author for correspondence 25254-

Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom P Yu & M Han cyclicity means that at most a finite number of limit cycles can exist in some neighborhood of focus points or homoclinic loop under small perturbations on the system s parameters In this paper, we particularly consider bifurcation of limit cycles in quadratic systems Early results can be found in a survey article by Ye [982] Some recent progress has been reported in a number of papers eg see [Roussarie, 998; Roussarie & Schlomiuk, 22]) For general quadratic system ) n = 2), in 952, Bautin proved that there exist 3 small limit cycles around a weak focus point or a center [Bautin, 952] After 3 years, until the end of 97 s, concrete examples were given to show that general quadratic systems can have 4 limit cycles [Shi, 979; Chen & Wang, 979], around two foci with 3, )-configuration Since then, many researchers have paid attention to integrable quadratic systems, and a number of results have been obtained A question was naturally raised: Can near-integrable quadratic systems have 4 limit cycles? A quadratic system is called near-integrable if it is a perturbation of a quadratic integrable system by quadratic polynomials On one hand, it is reasonable to believe that the answer should be positive since general quadratic systems have at least 4 limit cycles; while on the other hand, near-integrable quadratic systems have restrictions on their system parameters and thus it is more difficult to find 4 limit cycles in such systems In fact, this is still an open problem after another 3 years since the finding of 4 limit cycles in general quadratic systems, and many researchers are working on this problem It should be mentioned that 4 limit cycles have been discussed by Llibre and Schlomiuk [24], Artés et al [26] using general polynomial perturbations applied to integral quadratic systems, which are not near-integral quadratic systems defined in this paper The stu of bifurcation of limit cycles in near-integrable systems is related to the so-called weak Hilbert s 6th problem [Arnold, 977], which is transformed to finding the maximal number of isolated zeros of the Abelian integral or Melnikov function: Mh, δ) = Q n P n, 2) Hx,y)=h where Hx, y),p n and Q n are all real polynomials of x and y with deg H = n +, and max{deg P n, deg Q n } n The weak Hilbert s 6th problem is a very important problem, closely related to the maximal number of limit cycles of the following near-hamiltonian system [Han, 26]: dt dt = Hx, y) y = Hx, y) x + εp n x, y), + εq n x, y), 3) where Hx, y), p n x, y)andq n x, y)arepolynomials of x and y, and<ε is a small perturbation General quadratic systems with one center have been classified, for example, by Żol adek [994] using a complex analysis on the condition of the center, as four systems: Q LV 3 the Lotka Volterra system; Q H 3 Hamiltonian system; QR 3 reversible system; and Q 4 codimension-4 system In 994, Horozov and Iliev [994] proved that in quadratic perturbation of generic quadratic Hamiltonian vector fields with one center and three saddle points there can appear at most two limit cycles, and this bound is exact Later, Gavrilov [2] extended Horozov and Iliev s method to give a fairly complete analysis on quadratic Hamiltonian systems with quadratic perturbations Quadratic Hamiltonian systems, with at most four singularities, can be classified as three cases [Gavrilov, 2]: i) one center and three saddle points; ii) one center and one saddle point; and iii) two centers and two saddle points Gavrilov [2] showed that like case i), cases ii) and iii) can also have at most two limit cycles Therefore, generic quadratic Hamiltonian systems with quadratic perturbations can have maximal two limit cycles, and this case has been completely solved For the Q R 3 reversible system, there have been many results published For example, Dumortier et al [997] studied a case of Q R 3 system with two centers and two unbounded heteroclinic loops, and presented a complete analysis of quadratic 3- parameter unfolding It was proved that 3 is the maximal number of limit cycles surrounding a single focus, and only the, )-configuration can occur in case of simultaneous nests of limit cycles That is, 3 is the maximal number of limit cycles for the system they studied [Dumortier et al, 997] Later, Peng [22] considered a similar case with a homoclinic loop and showed that 2 is the maximal number of limit cycles which can bifurcate from the system Around the same time, Yu and Li [22] investigated a similar case as Peng considered 25254-2

Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom but with a varied parameter in a certain interval, and obtained the same conclusion as Peng s Later, Iliev et al [25] reinvestigated the same case but for the parameter values varied at a different interval which yields two centers) and got the same conclusion as that of [Dumortier et al, 997], ie 3 is the maximal number of limit cycles which can be obtained from this case Recently, Li and Llibre [2] considered a different case of Q R 3 system which can exhibit the configurations of limit cycles:, ),, ),, ) and, 2) Again, no 4 limit cycles were found In order to explain why the above authors did not find 4 limit cycles from the Q R 3 reversible system, consider the QR 3 system with quadratic perturbations, which can be described by [Dumortier et al, 997] ẋ = y + ax 2 + by 2 + εµ x + µ 2 xy), ẏ = x + cy)+εµ 3 x 2 4), where a, b, c are real parameters, µ i,i=, 2, 3are real perturbation parameters, and <ε When ε =,system4) ε= is a reversible integrable system It has been noted that in all the cases considered in [Dumortier et al, 997; Peng, 22; Yu & Li, 22; Iliev et al, 25], the parameters a and c were chosen a = 3, c= 2, but with b =in [Dumortier et al, 997]; b = in [Peng, 22], b, ), ) in [Yu & Li, 22], and b, 2) in [Iliev et al, 25] In these papers, complete analysis on the perturbation parameters was carried out with the aid of Poincaré transformation and the Picard Fuchs equation, but it needed to fix all or most of) the parameters a, b and c This way it may miss the opportunity to find more limit cycles, such as possible existence of 4 limit cycles As a matter of fact, for the cases considered in [Yu & Li, 22; Iliev et al, 25], a simple scaling on the parameter b b ) can be used to eliminate b So, suppose the nonperturbed system 4) ε= has two free parameters and let us consider the twodimensional parameter plane Then, all the cases studied in the above mentioned articles are special cases, represented by just a point or a line segment in the two-dimensional parameter plane see more details in Sec 2) It has been noted that a different method was used in [Li & Llibre, 2] with Melnikov function up to second order, but no more limit cycles were found It should be mentioned that Zhang [22] has proved that the possible cycle distributions in general quadratic systems with two foci must Four Limit Cycles in Near-Quadratic Integrable Systems be, )-distribution or, i)-distribution, i =,, 2, 3, So far, no results have been obtained for i 4 This result also rules out the possibility of 2, 2)-distribution It is conjectured that at most 3 limit cycles can exist around one focus point The problem of bifurcation of 3 limit cycles near an isolated homoclinic loop is still open In this paper, we turn to a different angle and consider bifurcation of limit cycles in quadratic near-integrable systems with two centers We shall leave more free parameters in the integrable systems, so that we will have chances to find more limit cycles The basic idea is as follows: we first consider bifurcation of multiple limit cycles from Hopf singularity, which does not need to fix any parameters, and use expansion of Melnikov function near centers to get as many as possible such limit cycles This leads to the determination of a maximal number of parameters Then, for the remaining undetermined parameters, we compute the global Melnikov function to look for possible large limit cycles Indeed, although, due to the complex integrating factor in the analysis, we are not able to give a complete analysis for classifying the perturbation unfolding, we do get a positive answer to the open question of existence of 4 limit cycles in quadratic near-integrable systems In particular, we will show that perturbing a reversible, integrable quadratic system with two centers can have at least 4 limit cycles, with 3, )-distribution, bifurcating from the two centers under quadratic perturbations The rest of paper is organized as follows In Sec 2, we give a different classification in real domain for quadratic systems with one center, and compare it with that given by Żol adek [994] Also, we use our classification to present a simple summary on some of the existing results for the reversible near-integrable system Section 3 is devoted to the analysis on bifurcation of small limit cycles from Hopf singularity In Sec 4, we show how to find large limit cycles bifurcating from closed orbits to obtain a total of 4 limit cycles Finally, conclusion is drawn in Sec 5 The main results of this manuscript has been posted on arxivorg since February 2 [Yu & Han, 2] 2 Classification of Generic Quadratic Systems with at Least One Center In this section, we give a different classification in real domain for quadratic systems with a center, 25254-3

P Yu & M Han which is consistent with the Hamiltonian systems considered in [Horozov & Iliev, 994; Gavrilov, 2] We start from the following general quadratic system: with, ) being a { center if a <, saddle point if > Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom dz dt = c + c z + c z 2 + c 2 z 2 + c z z 2 + c 2 z 2 2, dz 2 dt = c 2 + c 2 z + c 2 z 2 + c 22 z 2 + c 2 z z 2 + c 22 z 2 2, 5) where c ijk s are real constant parameters It is easy to show that this system has at most four singularities, or more precisely, it can have, 2 or 4 singularities in real domain In order for system 5) to have limit cycles, the system must have some singularity In this paper, we assume that system 5) has at least two singularities Without loss of generality, we may assume that one singular point is located at the origin, ), which implies c = c 2 =,and the other at p, q) p 2 +q 2 ) Further assume the origin is an elementary center Then introducing a series of linear transformations, parameter rescaling and time rescaling to system 5) yields the following general quadratic system: dt = y + xy + a 2 y 2, dt = x + x2 + a 3 xy + a 4 y 2, 6) which has an elementary center at the origin, ) and another singularity at, ) In order to have the origin of system 6) being a center, we may calculate the focus values of system 6) and find four cases under which, ) is a center, listed in the following theorem here we use Żol adek s notation in our classification) Theorem The origin of 6) is a center if and only if one of the following conditions is satisfied: Q R 3 Reversible system: a 3 = a 2 =, under which system 6) becomes dt = y + xy, dt = x + x2 + a 4 y 2, 7) Q H 3 Hamiltonian system: a 3 = +2a 4 =, under which system 6) is reduced to with, ) being a dt = y + xy + a 2 y 2, dt = x + x2 2 y 2, { center if a <, saddle point if > 8) Q LV 3 Lokta Volterra system: a 2 =+a 4 =, under which system 6) becomes with, ) being a focus node dt = y + xy, dt = x + x2 + a 3 xy y 2, if < + ) 4 a2 3, saddle point if > Q 4 Codimension-4 system: 9) if + ) 4 a2 3 < <, a 3 5a 2 = 5 + 3a 4 ) = a 4 +2+a 2 2 )=, ) under which system 6) can be rewritten as dt = y + 6a2 2 )xy + a 2y 2, dt = x + x2 +5a 2 xy 2 + a 2 2 )y2, with, ) being a node for a 2 ) 25254-4

Four Limit Cycles in Near-Quadratic Integrable Systems Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom Remark 2 There is one more case found from the above process, defined by the following conditions: a 3 5a 2 = 5 + 3a 4 ) =3a 4 +2)a 4 +) 2 5a 4 +6)a 2 2 = 2) We will show later in this section, when we compare our above real classification with the complex classification given by Żol adek [994], that the case defined by 2) actually belongs to the reversible system Q R 3 Proof Necessity is easy to be verified by computing the focus values of system 6) associated with the origin Some focus values will not equal zero if the condition is not satisfied For sufficiency, we find an integrating factor for each case when the condition holds For the Q H 3 Hamiltonian system 8), we know that the integrating factor is, and the Hamiltonian is given by Hx, y) = 2 x2 + y 2 ) 3 x3 + 2 xy 2 + 3 a 2y 3, 3) which is exactly the same as that given in [Horozov & Iliev, 994; Gavrilov, 2] For the Q R 3 reversible system 7), the integrating factor is γ = + x +2a 4, 4) and the first integral of the system is given by F x, y) = [ 2 sign + x) + x 2a 4 y 2 + + a ] a 4 ) + 2a 4 x) a 4 a 4 ) 2a 4 ) x2 5) a 4 For the Q LV 3 Lokta Volterra system 9), we find the integrating factor as γ = gx, y), where gx, y) =+ x)[x ) 2 + a 3 x )y + )y 2 ], 6) and the first integral of the system is F x, y) = signgx, y)) 2 + ) { 2ln + x + ln + )y 2 a 3 yx ) x ) 2 [ 2 a 3 x ) a + 3 x ) 2 + )y [a 2 3 +4+ )]x ) 2 tanh ]}, [a 2 3 +4+ )]x ) 2 sign + x) 2 + ) when a 2 3 +4+) >, { 2ln + x + ln[ + )y 2 a 3 yx ) x ) 2 ] [ 2 a 3 x ) a 3 x ) 2 + )y [ a 2 3 4 + )]x ) 2 tan ]}, [ a 2 3 4 + )]x ) 2 7) when a 2 3 +4+) < Finally, for the Q 4 codimension-4 system ), we have γ = gx, y) 5/2, where gx, y) = 2 + 2a 2 2 )x 2a 2y ++4a 2 2 )x + a 2y) 2, 8) and the first integral of the system is equal to where F x, y) = signgx, y)) 2a 6 gx, y) 3/2 fx, y), 2 9) fx, y) = + a 2 2)+3x + a 2 y +2a 2 2x) [ + a 2 2 + 3a 2 2)x + a 2 y)] ++3a 2 2 ) + 4a2 2 )x + a 2y) 3 2) The proof is complete 25254-5

Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom P Yu & M Han Note that among the four classifications of the integrable system 6), the first three classified systems 7) 9) have two free parameters, while the last system ) has only one free parameter Remark 22 We now show that our classification in Theorem is equivalent to that given by Żol adek [994] The general quadratic system considered by Żol adek is given in the complex form: dz dt =i + λ)z + Az2 + Bzz + Cz 2, 2) where z = x+iy, anda, B and C are complex coefficients It has been shown by Żol adek [994] that the point z = is a center if and only if one of the following conditions is fulfilled: Q LV 3 : λ = B =, Q H 3 : λ =2A + B =, Q R 3 : λ =ImAB) =ImB3 C) =ImA 3 C)=, Q 4 : λ = A 2B = C B = 22) In the following, we first use real differential equation to give a brief proof different from Żol adek s [994]), and then show that our classification is equivalent to Żol adek s when system 2) is assumed to have a nonzero singularity To prove this, let A = A + ia 2, B = B + ib 2, C = C + ic 2, i 2 = ), and then rewrite the complex equation 2) in the real form: dt = λx + y +A + B + C )x 2 +2A 2 C 2 )xy A B + C )y 2, dt = x + λy A 2 + B 2 + C 2 )x 2 +2A C )xy +A 2 B 2 + C 2 )y 2, 23) where y y has been used Letting λ =yields the focus value v = Then, it is easy to find the first focus value or the first Lyapunov constant) as v = A B 2 B A 2 = ImAB) 24) Letting v =resultsinimab) =, which gives B 2 = B A 2, under the assumption of A A 25) The degenerate case A = can be similarly analyzed and the details are omitted here) Then, we apply our Maple program eg see [Yu, 998]) to system 23), with the conditions λ =and25), to obtain v 2 = fa 2B ) 3A 3, v 3 = ff 3 26A 5, where v 4 = ff 4 972A 7, v 5 = ff 5 46656A 9, f = B 2A + B )C 2 A 3 +3C A 2 A 2 3A 2 2C 2 A C A 3 2), and f 3,f 4,etcarepolynomialsofA,A 2,C,C 2 and B Letting f =, ie B = or 2A + B = or C 2 A 3 +3C A 2 A 2 3A2 2 C 2A C A 3 2 =ImA 3 C)= yields v 2 = v 3 = = Indeed, B = implies B 2 = due to the condition 25), and so B =Thus,weobtainλ = B =, corresponding to the Q LV 3 case For the condition 2A + B =, it follows from 25) that2a 2 B 2 =, ie 2A + B =, which plus the condition λ =givestheq H 3 case The third condition ImA 3 C)=,withλ = and ImAB) =, corresponds to the Q R 3 case Further, it is easy to show that under the condition ImAB) =,ImA 3 C)=andImB 3 C)=are equivalent Thus, the conditions λ = ImAB) = ImB 3 C) = are also applicable for this case So for this case, either ImA 3 C)=orImB 3 C)= is needed, but not both of them In the following, we show one more case to join this case, leading to both the two conditions being needed Note that there is one more condition A =2B which renders v 2 = Letting A =2B,andso A 2 = 2B 2 [see 25)], implying that A 2B = Under the condition A =2B, v = v 2 =,andthe 25254-6

Four Limit Cycles in Near-Quadratic Integrable Systems other focus values become v 3 = 25 8 C2 + C2 2 B2 B2 2 )C 2B 3 3C B 2 B 2 3C 2 B B 2 2 + C B 3 2 ), v 4 = v 3 45 [45B2 + 585B2 2 + 6B C + B 2 C 2 ) 96C 2 + C2 2 )], v 5 = v 3 648 [6487B4 + 24B 2 B 2 2 + 557B 4 2) 396B 2 C 2 768B B 2 C C 2 + 22345B 2 2 C 2 2) + 576B C 6B 2 + 37B 2 2) + 288B 2 C 2 37B 2 + 773B 2 2) 348B 2 C 2 2 + 2685B 2 2 C 2 ) 8688C 2 + C2 2 )B C + B 2 C 2 ) + 8644C 2 + C2 2 ) 2 ], Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom Hence, under the conditions λ = A 2B =,there are two possibilities such that v 3 = v 4 = = The first possibility is C 2 + C2 2 B2 B2 2 =, ie C B =, which is one of the conditions given for the Q 4 case [see 22)] The second possibility is given by the condition: C 2 B 3 3C B 2 B 2 3C 2 B B 2 2 + C B 3 2 =ImB 3 C) = 8 ImA3 C)=, 26) due to A = 2B Since these conditions can be included in the conditions λ = ImAB) = ImB 3 C)=ImA 3 C) =, this possibility belongs to the Q R 3 case The remaining task is to show that the conditions classified in 22) aresufficientthiscanbe done by finding an integrating factor for each case For brevity, we only list these integrating factors below while the lengthy expressions of the first integrals are omitted): +4A 2 x A y)+4a C 2 + A 2 C 2A A 2 )xy +[A + C )A 3C ) +A 2 + C 2 )5A 2 3C 2 )]x 2 +[A 2 + C 2 )A 2 3C 2 ) +A + C )5A 3C )]y 2 +2A 2 + A2 2 C2 C2 2 )[A 2 + C 2 )x 3 A + C )y 3 A 3C )x 2 y +A 2 3C 2 )xy 2 ], for Q LV 3, γ =, for Q H 3, 2A C )y 2A +B A C, for Q R 3, 4B 2 x + B y)+2b 2 + B2 2 )x2 + y 2 ) +2B C + B 2 C 2 )x 2 y 2 )+4B C 2 B 2 C )xy 5/2, for Q 4 27) For the integrating factors of degenerate cases eg A C = ), one can easily find them Next, compare the classification listed in 22) with ours given in Theorem First, consider the Q LV 3 case Letting λ = B = B 2 =in23) yields dt = y +A + C )x 2 +2A 2 C 2 )xy A + C )y 2, dt = x A 2 + C 2 )x 2 +2A C )xy +A 2 + C 2 )y 2 25254-7 28)

Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom P Yu & M Han Then, let sinθ) = k =tanθ), and so k, cosθ) =, 29) +k 2 +k 2 where k is solved from the following cubic polynomial: P k) =A 2 + C 2 )k 3 +A 3C )k 2 +A 2 3C 2 )k + A + C = 3) This cubic polynomial at least has one real solution for k, which gives the slope of the line on which a second fixed point is located k =ifa + C =, otherwise, k Letk be a real root of P k), ie P k) = Further, introducing the linear transformation rotation): x =cosθ)u sinθ)v, 3) y =sinθ)u +cosθ)v, into 28) yields dt = y + m 2x 2 + m xy + m 2 y 2, dt = x + m 22x 2 + m 2 xy + m 22 y 2, where m 2 = m 2 =+k 2 ) 3/2 P k) =, m 22 = m 22 32) =+k 2 ) 3/2 [A + C )k 3 A 2 3C 2 )k 2 +A 3C )k A 2 C 2 ], m = 2 + k 2 ) 3/2 [A C )k 3 A 2 +3C 2 )k 2 +A +3C )k A 2 + C 2 ], m 2 =2+k 2 ) 3/2 [A 2 C 2 )k 3 +A +3C )k 2 +A 2 +3C 2 )k + A C ] Suppose m 22 Then, introducing x = m 22 x, y = m 22 y into 32) resultsin dt = y + m xy, m 22 dt = x + x2 + m 2 m 22 xy y 2, 33) which is identical to 9) as long as letting = m m 22 and a 3 = m 2 m 22 This shows that the four parameters A,A 2,C and C 2 are not independent Thus, alternatively, we may simply take k = which renders the second singularity of 28) onthex-axis), yielding C = A Thus,28) becomes dt = y +2A 2 C 2 )xy, dt = x A 2 + C 2 )x 2 +4A xy +A 2 + C 2 )y 2 Suppose A 2 + C 2 Introducing x = A 2 + C 2 )x, y = A 2 + C 2 )y into the above equations we obtain dt = y 2A 2 C 2 ) xy, A 2 + C 2 34) dt = x + x2 4A xy y 2, A 2 + C 2 which is identical to 9) if letting = 2A 2 C 2 ) A 2 +C 2 and a 3 = 4A A 2 +C 2 In the following, we will use this simple approach for other cases For the Q H 3 case, substituting λ =, B = 2A and B 2 =2A 2 into system 23) resultsin dt = y A C )x 2 +2A 2 C 2 )xy 3A + C )y 2, dt = x 3A 2 + C 2 )x 2 +2A C )xy A 2 C 2 )y 2 Further, taking C = A in the above equations gives another singularity on the x-axis, and introducing x = 3A 2 + C 2 )x, y = 3A 2 + C 2 )y into the resulting equations yields dt = y 2A 2 C 2 ) xy + 4A y 2, 3A 2 + C 2 3A 2 + C 2 dt = x + x2 + A 2 C 2 y 2, 3A 2 + C 2 35) which is identical to 8) ifweset = 2A 2 C 2 ) 3A 2 +C 2 and a 2 = 4A 3A 2 +C 2 25254-8

Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom For the Q R 3 reversible case, it follows from [Żol adek, 994] that all the coefficients A, B and C are real, and thus we obtain the following real form from the complex system 2) where dt = y + ax2 + by 2, dt = x + cxy, a = A + B + C, b = B A C, c =2A 2C 36) Suppose b Then, introducing x = by, y = bx into 36) results in dt = y + c b xy, dt = x + x2 + a b y2, which is identical to 7) if = c b = 2A C ) B A C and 37) a 4 = a b = A + B + C B A C For the last Q 4 case, under the condition λ = A 2B =, by setting C = 3B which renders a nonzero singularity on the x-axis) in 23) weobtain dt = y 22B 2 + C 2 )xy +2B y 2, dt = x +B 2 C 2 )x 2 +B xy 3B 2 C 2 )y 2 Suppose B 2 C 2 Then, introducing x = B 2 C 2 )x, y =B 2 C 2 )y into the above equations yields dt = y 22B 2 + C 2 ) xy + 2B y 2, B 2 C 2 B 2 C 2 dt = x + x2 + B B 2 C 2 xy 3B 2 C 2 B 2 C 2 y 2 38) Comparing the coefficients of the above system 38) with our system 6) resultsin Four Limit Cycles in Near-Quadratic Integrable Systems = 22B 2 + C 2 ) B 2 C 2, a 2 = 2B B 2 C 2, a 3 = B B 2 C 2, a 4 = 3B 2 C 2 B 2 C 2, 39) which in turn implies that a 3 5a 2 = 5 + 3a 4 ) =, and a 4 +2+a 2 2 )=8B2 + C2 2 B2 2 B 2 C 2 ) 2 = C2 + C2 2 B2 B2 2 B 2 C 2 ) 2 =, for C B = The above conditions are the exact conditions given in ) fortheq 4 case Finally, we turn to the conditions given in 2) It follows from 39) that 3a 4 +2)a 4 +) 2 5a 4 +6)a 2 2 4 = B 2 C 2 ) 3 3B3 2 +3B2C 2 2 CB 2 2 BC 2 2 ) 4) On the other hand, under the condition C = 3B, the condition 26) for the second possibility becomes C 2 B 3 3C BB 2 2 3C 2 B B2 2 + C B2 3 = C 2 B 3 + C2 B B 2 3C 2 B B 2 2 3B B 3 2 = B 3B2 3 +3B2 2 C 2 C 2 B 2 B 2 C 2)=, which implies, by Eq 4), 3a 4 +2)a 4 +) 2 5a 4 + 6)a 2 2 =forb Hence, according to Żol adek s classification [see 22)], this case should be included in the Q R 3 case However, one cannot prove this by directly using the conditions in 2) aswellasthat for the Q R 3 case see Theorem ) One must trace back to the original system coefficients In the paper [Żol adek, 994], the author used Bautin s system to verify his classification Bautin s system is described by [Bautin, 952] dt = λ x y + λ 3 x 2 +2λ 2 + λ 5 )xy + λ 6 y 2, dt = x + λ y + λ 2 x 2 +2λ 3 + λ 4 )xy λ 2 y 2 4) It is seen from 23) and4) that Bautin s system has only six parameters, while Żol adek s system has 25254-9

P Yu & M Han seven in real domain) parameters This indicates that Żol adek s system has one redundant parameter In fact, putting Bautin s system in Żol adek s complex form gives the following expressions: λ = λ, A = 4 λ 3 + λ 4 λ 6 iλ 5 ), B = 2 λ 3 λ 6 ), C = 4 [ 3λ 3 + λ 4 + λ 6 )+i4λ 2 + λ 5 )] Then, applying the formulas given in 23) will immediately generate the center conditions obtained by Bautin [952] The above expressions clearly show that B 2 = As a matter of factor, the integrating factor for the system, corresponding to the second possibility, ie when λ = A 2B =ImB 3 C) =, is given by Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom [ +2 C B 2 + B2 2 ) ] B B 2 3B2 2 ) 2 B 2 x + B y) For B 2 =, the above expression is reduced to 5B 5B B 2 3B2 2 ) C B 2 +B2 2 ) 2B B2 3B2 2 ) C 22B C )y 2B 2 = 2A C )y 2A +B A C due to A =2B ), which is the integrating factor for the Q R 3 system, asshownin27) Now we return to system 6) Among the four classifications, the Hamiltonian system Q H 3 ) has been completely studied in [Horozov & Iliev, 994; Gavrilov, 2]: the system can have maximal two limit cycles In this paper, we will concentrate on the Q R 3 reversible case Special cases for the reversible system have been investigated by a number of authors eg see [Dumortier et al, 997; Peng, 22; Yu & Li, 22; Iliev et al, 25; Li & Llibre, 2]) It is easy to see that system 7) is invariant under the mapping t, y) t, y), where and a 4 can be considered as perturbation parameters The singular point, ) of 7) is a center when < ; but a saddle point when > = gives a degenerate singular point at, ) Further, it is easy to verify that when +)a 4 >, there are no more singularity; while when +)a 4 <, there exist additional two saddle points, given by x,y )= ) a4 +), ± a 4 a 4 = is a critical value, yielding the two additional saddle points at infinity: x,y ) =, ± ) In summary, the distribution of singularity of the reversible system 7) has the following possibility see Fig, wherec +S stands for one center and one saddle point, similar meaning applies to 2C, 2C +2S and C +3S): Two centers when < anda 4 < ; Two centers and two saddle points when < anda 4 > ; One center and one saddle point when > anda 4 > ; One center and three saddle points when > anda 4 < 42) In this paper, we pay particular attention to <,a 4 <, for which system 7) has only two singularities at, ) and, ), both of them are centers By adding quadratic perturbations to system 7) we obtain the following perturbed quadratic system: dt = y + x)+εp x, y) = y + x)+ε x + y + a 2 x 2 + xy + a 2 y 2 ), dt = x + x2 + a 4 y 2 + εqx, y) = x + x 2 + a 4 y 2 + εb x + b y + b 2 x 2 + b xy + b 2 y 2 ), 43) where <ε, a ij s and b ij s are perturbation parameters 25254-

Four Limit Cycles in Near-Quadratic Integrable Systems Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom 5 4 3 4 3 2 2 2 3 4 5 2 3 4 Fig Case studies for the Q R 3 reversible system Remark 23 The special system considered by Dumortier et al [997] is system 4) with a = 3, c = 2, b = This is equivalent to our system when = 2 and a 4 = 3 for which the system has only two elementary centers at, ) and, ) Consider the a 4 parameter plane, as shown in Fig It can be seen that this case is just a point,,a 4 )= 2, 3), in the parameter plane, marked by a blank circle in the third quadrant on the line a 4 = 3 2 see Fig ) The special system studied by Peng [22] is system 4) with a = 3, c = 2, b = This is equivalent to our system when =2and a 4 = 3, for which the system has one center at, ) and one saddle point at, ) Thus, this case is again a point,,a 4 )=2, 3), in the a 4 parameter plane, marked by another blank circle in the first quadrant on the line a 4 = 3 2 see Fig ) The cases considered in [Yu & Li, 22; Iliev et al, 25] correspond to system 4) witha = 3, c = 2, and b, ), ) in [Yu & Li, 22], and b, 2) in [Iliev et al, 25] When ε = in system 4), one can use the following transformation: x = ỹ b, y = x b, to transform system 4) ε= to d x dt =ỹ + c ), b x dỹ dt = x + x2 + a b ỹ2, 44) which is our system 7) with Equation 45) yields = c b, a 4 = a b 45) a 4 = a c b ), 46) which represents a line in the a 4 parameter plane, passing through the origin with the slope a c In particular, the parameter values: a = 3,c = 2, b, ), ), 2), yielding = 2 b and a 4 = 3 b, correspond to a part of the line, described by a 4 = 3 2, ), ), 47) as shown in Fig, where the dotted line for [, ] is excluded from the stu in [Yu & Li, 22; Iliev et al, 25] It should be noted that when a = 3,c = 2, the point, b ) is a saddle point if and only if + c b = 2 > b, ) 2, + ) b Thus, the case considered in [Yu & Li, 22] has one center and one saddle point; while the case studied in [Iliev et al, 25] has two elementary centers But even these two studies together do not cover the whole line a 4 = 3 2 the missing part is denoted by a dotted line segment in Fig ) Another alternative form for a special case of our system 7) considered by Han [997] is described by [ dt = y +2 e) x + )], d 48) dt = x + 2 + ey 2, where e and d ) are parameters This system has a saddle point at the origin and a center at x, y) = d, ) Based on the two parameters, seven cases are classified [Han, 997] We can apply the following transformation: to system 48), yielding x = d x ), y = d y, dt = y [ + ] 2 e) x, d 25254-

Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom P Yu & M Han dt = x + x2 + e d y2, 49) which has a center at the origin and a saddle point at, ) Then, setting 2 e) =, a 4 = e d d, 5) in system 49) leads to our system 7) Equation 5) denotes a line, given by e a 4 = 2 e), 5) in the a 4 parameter plane, passing through the e origin with the slope 2 e) However, it is easy to see that using our system 7) inanalysisissimpler to using system 48) In fact, all the seven cases classified in [Han, 997] together denote a region in Fig, see the shaded area in this figure This area covers most of the region, defined by > But the stu given in [Han, 997] for the seven cases is restricted to local analysis on the bifurcation of limit cycles near a homoclinic loop, except the two lines see Fig ): a 4 =, ), ), 52) which corresponds to the parameter value e = 2 3, and a 4 = 2, ), 53) which corresponds to e ± It has been shown in [Han, 997] that except the above two lines, for the parameter values in the shaded area, system 48) can have at most 2 limit cycles near a homoclinic loop under quadratic perturbation Figure shows the a 4 parameter plane associated with the reversible system 7), where the above mentioned case studies are indicated on the line a 4 = 3 2 as well as in the shaded area More precisely, a complete global analysis given in [Yu & Li, 22], which includes the result in [Dumortier et al, 997] as a special case, shows that corresponding to each point on the line segment a 4 = 3 2 > ), the system has one center and one saddle point, and has maximal 2 limit cycles In [Han, 997] it is shown for each point in the shaded area [except the two line segments a 4 = > ) and a 4 = 2 > )], which contains the above line segment, the system has one center and one or three) saddles), and has maximal 2 limit cycles, but restricted to local analysis near one homoclinic loop Similarly, a global analysis given in [Iliev et al, 25], which contains the result in [Dumortier et al, 997] as a special case, proves that corresponding to each point on the line segment a 4 = 3 2 < ), the system has two centers, and exhibits maximal 3 limit cycles around one center The technique of Poincaré transformation and Picar Puchs equation, used for the above mentioned global analysis on parameter unfolding, seems not possible to be generalized to consider the general situation for arbitrary points in the a 4 parameter plane The two particular dash-dotted lines: a 4 = 3 5), ), ), and a 4 = 3 6 +5), ),aswellas the five dark circles correspond to our results, presented in the next two sections In particular, we will show that there exist 3 small limit cycles on the two dash-dotted lines, and at least 4 limit cycles for the parameter values marked by the five dark circles In the following, we will use the perturbed quadratic system 43) for our stu on bifurcation of limit cycles First, we need the following lemma, which will greatly simplify the analysis Lemm The perturbed quadratic reversible system 43) can have three independent perturbation parameters Proof First, note that the integrating factor for the unperturbed reversible system 7) isγ = + x +2a 4 Thus,lett = γτ Thensystem7) can be transformed to dτ = +x +2a 4 y + xy), dτ = +x +2a 4 x + x 2 + a 4 y 2 ) which has the Hamiltonian function: Hx, y) = 2 sign + x) + x 2a 4 54) [ y 2 + + a 4 ) + 2a 4 x) a 4 a 4 ) 2a 4 ) ] x2 55) a 4 25254-2

Four Limit Cycles in Near-Quadratic Integrable Systems Then, the Melnikov function of system 43) along a loop defined by : Hx, y) =h, can be expressed as Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom Note that Mh,,a 4,a ij,b ij ) = γqx, y, b ij ) γpx, y, a ij ) = + x +2a 4 b + b x)y + x +2a 4 x + a 2 x 2 + a 2 y 2 ) = + x +2a 4 [b + b x)+ +2a 2 x a ] +2a 4 + x) x + a 2 x 2 ) y a 2 + x +2a 4 y 2 = + x +2a 4 [ + b )+b +2a 2 )x a ] +2a 4 + x) x + a 2 x 2 ) y 3 +2a 4 )a 2 + x a 4 + x Further, it follows from Eq 7) that +2a 4 + x + x +2a 4 y 3 56) y 3 = 3a 4 + x +2a 4 y 2 57) +2a 4 x + x 2 + a 4 y 2 ) =+ x)y, 58) which is multiplied by +x +2a 4 + x y on both sides and then the resulting equation is integrated along to yield + x +2a 4 x + x 2 + a 4 y 2 + x )y = + x + x Combining the above equation with 57) weobtain + x +2a 4 y 2 = a +2a 4 a 4 +2a 4 + x + x +2a 4 + x)y 2 x + x 2 )y 59) Substituting the above result into 56) yields Mh,,a 4,,b,b,a 2,a 2 ) = + x +2a 4 [ + b )+b +2a 2 )x a ] +2a 4 + x) x + a 2 x 2 ) y a +2a 4 a 2 a 4 + x + x +2a 4 x + x 2 )y 25254-3

Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom P Yu & M Han = + x +2a 4 { + b )+b +2a 2 )x } +2a 4 a 4 ) + x) [ a 4 ) x + a 2 x 2 )+a 2 x + x 2 )] y Next, rewriting the term in the square bracket of 6) gives Mh,,a 4,,a 2,a 2,b,b ) = + x +2a 4 a +2a 4 { + b )+b +2a 2 )x a 4 ) + x) [ [ a 4 )a 2 + a 2 ] + x)x + a 2 [ a 4 ) a 2 ) + )a 2 ] + x) ]} a 2 [ a 4 ) a 2 ) + )a 2 ] where and { = + x +2a 4 a +2a a 4 a 2 a 4 ) [ a 4 ) a 2 ) + )a 2 ] + x + b + +2a 4 a 2 a 4 ) [ a 4 ) a 2 ) + )a 2 ] [ b +2a 2 a ]) +2a 4 a 4 ) [ a 4 )a 2 + a 2 ] + b +2a 2 +2a 4 a 4 ) [ a 4 )a 2 + a 2 ] = c I + c I + c 2 I 2, I = sign + x) + x) I = sign + x) + x) I 2 = sign + x) + x) a 2 ) 2 +a 4 ) y, } + x) 6) 6) 2 +a 4 ) y + x), 62) 2 +a 4 ) y + x) 2, c = +2a 4 +2a 4 a 2 + ) +2a 4 ) a 2 a a 2, 63) a 4 ) c = b 2a 4 b + 4a 4 a 2 + 2 + ) +2a 4 ) a 2 a a 2, 64) a 4 ) a 2 c 2 = b + 2a 4 a 2 a 2 + +2a 4 a 2 a 4 ) a 2 65) 25254-4

Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom It is obvious that the expression in 6) contains only three independent perturbation parameters, though the parameters and a 4 are involved in I i,i =,, 2 Thus, we may let two of them equal to zero For example, letting a 2 = a 2 =yields c = + 2a 4 ), c = b 2a 4 + b, 66) c 2 = b, which indeed shows that,b and b can be used as the three independent perturbation parameters Thus, without loss of generality, we may assume that = a 2 = = a 2 = b = b 2 = b 2 =, under which system 43) is reduced to dt = y + x)+ε x, dt = x + x2 + a 4 y 2 + εb y + b xy), where < and<ε 67) 3 Hopf Bifurcation Associated with the Two Centers In this section, we stu Hopf bifurcation of system 67) from two centers, ) and, ), leading to the bifurcation of multiple limit cycles The result is summarized in the following theorem Theorem 2 When <, the quadratic near-integrable system 67) can have small limit cycles bifurcating from the two centers, ) and, ) with distributions: 3, ),, 3), 2, ),, 2) and, ) 2, )- or, 2)-distribution does not exist Proof Consider system 67) for < The system 67) ε= is a reversible integrable system In order to compute the Melnikov function near the two centers, ) and, ), we multiply 67) by the integrating factor γ [given in 4)] to obtain the following perturbed Hamiltonian system: dτ = γy + xy)+εγ x, Four Limit Cycles in Near-Quadratic Integrable Systems dτ = γ x + x2 + a 4 y 2 )+εγb y + b xy), 68) with the Hamiltonian of the unperturbed system 7) ie 68) ε= ), given by 55), with a 4, a 4, 2a 4 Thecasesa 4 =, = a 4 or =2a 4 will not be considered in this paper Note that h = H, ) = + a 4 2a 4 a 4 ) 2a 4 ), h = H, ) for + x>, = +)a 4 +) 2a 4 a 4 ) 2a 4 ) ) 2a 4, for + x< 69) Since in this paper, we concentrate on the case that system 67) ε= has only two centers, we assume <,a 4 < Thus, lim Hx, y) =+ and x lim Hx, y) = x + It is easy to see from system 67) thatthetrajectories of 67) ε= rotate around the center, ) in the clockwise direction, while rotating around the center, ) in the counter clockwise direction, as shown in Fig 2Thus,thevaluesofhin Hx, y) =h y 5 5-5 - -5-25 -2-5 - -5 5 5 2 25 3 35 x Fig 2 A phase portrait of the reversible system 7) with two centers for = 3,a 4 = 8 3 25254-5

P Yu & M Han Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom are taken from the two intervals: h h, ) for + x>, and h,h )for+ x< It should be noted that h is not necessarily larger than h The analyses on the two half-plane in the x y plane see Fig 2), divided by the singular line + x =, are independent Next, introduce : Hx, y) { h h, ), for + x>, = h h,h ), for + x<, 7) and define the Melnikov function: Mh, a ij,b ij ) = qx, y, b ij ) px, y, a ij ), 7) µ =2π + b ), where px, y, a ij )=γ x and qx, y, b ij )=γb + b x)y Using the results in [Han, 2, 26; Han & Chen, 2], we can expand M near h = h and h = h as M h, a ij,b ij ) = µ h h )+µ h h ) 2 + µ 2 h h ) 3 + µ 3 h h ) 4 + Oh h ) 5 ), for <h h, M h, a ij,b ij ) = µ h h)+µ h h) 2 + µ 2 h h) 3 + µ 3 h h) 4 + Oh h) 5 ), for <h h, 72) where the coefficients µ ij,i=, ; j =,, 2,can be obtained by using the Maple programs developed in [Han et al, 29] as follows: µ = π 2 [ 3 4a 4 +3a 2 +7a 4 2a 2 4 ) + +a 4 + a 2 5 a 4 +4a 2 4 )b + 2 + a 4 )b ], µ 2 = π 864 [54 98 28a 4 + 86a 2 52a 4 3948a 2 4 626a3 + 566a2 a 4 + 62 a 2 4 µ 3 = 4432a 3 4 + 33a4 8a3 a 4 279a 2 a2 4 + 38a 3 4 296a4 4 ) + 54 + 7 + 38a 4 +2a 2 + 68 a 4 + 2772a 2 4 2a 3 + 26a 2 a 4 828 a 2 4 + 424a 3 4 + a 4 58a 3 a 4 + 369a 2 a 2 4 72 a 3 4 + 4a4 4 )b +24b + a 4 )7 + 35 +7a 4 + a 2 7a 4 +52a 2 4 )b ], π 6228 [3434 33 + 33a 4 + 6969a 2 498498 a 4 732732a 2 4 5885a 3 + 33485a 2 a 4 44358 a 2 4 7864a 3 4 + 3232a 4 2444439a 3 a 4 + 4228a 2 a2 4 + 5794692a 3 4 833936a4 4 6863a5 + 425a4 a 4 325355a 3 a2 4 296282a 2 a 3 4 + 2794 a 4 4 462368a 5 4 + 562a 6 523a 5 a 4 + 47287a 4 a 2 4 4753a 3 a 3 4 4589772a 2 a 4 4 + 749264 a 5 4 35966a 6 4) + 3434 + 3333 + 22a 4 + 6969a 2 + 582582a 4 + 429428a 2 4 + 935a3 + 4425a2 a 4 + 2728 a 2 4 + 223684a 3 4 699a 4 379a 3 a 4 + 262458a 2 a 2 4 8938 a 3 4 + 699474a 4 4 + 47a5 + 38a4 a 4 499a 3 a2 4 + 2838a2 a3 4 2964576a 4 4 + 267432a 5 4 39a6 29a5 a 4 + 46227a 4 a2 4 36693a3 a3 4 + 76988a2 a4 4 33526 a 5 4 + 578624a 6 4)b + 3636 + 3636 + 88a 4 + 792a 2 25254-6

Four Limit Cycles in Near-Quadratic Integrable Systems + 659736 a 4 + 52424a 2 4 + 26a 3 + 486a 2 a 4 + 376488 a 2 4 + 25432a 3 4 + 8a 4 8a3 a 4 + 34a 2 a2 4 6848a 3 4 + 65664a4 4 + 8a4 a 4 234a 3 a2 4 + 323a 2 a 3 4 3896 a 4 4 + 8692a 5 4)b ], and µ =2π ) 3/2 [ 2a 4 ) ++ )b + b )], Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom µ = π 2 ) 2 a 4 ) [ + 33 6a 4 +36a 2 2a 4 24a 2 a 4 +3 a 2 4 8a3 4 ) ++ ) + 2 a 4 +2a 2 5a 4 +4a 2 4 )b + ) + a 4 )2 + 3 4a 4 )b ], µ 2 = π 864 ) 5 8a 4 ) 2 [54 + 74 28a 4 + 34a 2 592a 4 + 222a 2 4 + 448a3 µ 3 = 872a 2 a 4 + 8628 a 2 4 2a 3 4 + 752a 4 4 224a 3 a 4 + 223a 2 a 2 4 5232 a 3 4 + 432a 4 728a4 a 4 + 692a 3 a2 4 7938a2 a3 4 + 4272a 4 4 8a5 4 ) ++ )54 + 546 38a 4 + 76a 2 972a 4 + 2772a 2 4 + 44a3 93a 2 a 4 + 6372 a 2 4 424a 3 4 + 864a 4 396a 3 a 4 + 3969a 2 a 2 4 236 a 3 4 + 4a 4 4)b + ) + a 4 )4 + 42 42a 4 + 423a 2 996a 4 + 576a 2 4 + 44a 3 633a 2 a 4 + 888 a 2 4 4a 3 4)b ], π 2446 ) 2 3a 4 ) [3434 + 2727 9898a 4 + 532439a 2 542348 a 4 + 3938a 2 4 + 74334645a 3 237355a 2 a 4 + 655766 a 2 4 77692a 3 4 + 62396a 4 47959a 3 a 4 + 3934978a 2 a2 4 49682268a 3 4 + 6439744a4 4 + 2726a5 95462a 4 a 4 + 3238865a 3 a 2 4 89486a 2 a 3 4 + 292788 a 4 4 35274a 5 4 + 54432a 6 289466a 5 a 4 + 63998532a 4 a 2 4 7487963a 3 a 3 4 + 48498336a 2 a 4 4 6296336 a 5 4 + 28248a6 4 5552a6 a 4 + 96336a 5 a2 4 246752a4 a3 4 + 3232358a 3 a 4 4 2272536a 2 a 5 4 + 8296 a 6 4 57248a 7 4) ++ )3434 + 77 22a 4 + 352259a 2 452258 a 4 + 429428a 2 4 + 37785825a3 792297a2 a 4 + 5445594 a 2 4 223684a3 4 + 2225636a 4 683729a 3 a 4 + 77864598a 2 a 2 4 386828 a 3 4 + 699474a 4 4 + 66294a 5 2898468a4 a 4 + 49687587a 3 a2 4 464974a2 a3 4 + 6953984a 4 4 267432a 5 4 + 7776a6 4868a5 a 4 + 23876a 4 a2 4 56679a3 a3 4 + 36268a 2 a 4 4 45648 a 5 4 + 578624a 6 4)b 25254-7

P Yu & M Han + ) + a 4 )22 + 99 88a 4 + 683a 2 33236 a 4 + 5376a 2 4 + 397655a3 459648a2 a 4 + 585 a 2 4 8824a3 4 + 594864a4 3266a 3 a 4 + 559422a 2 a 2 4 45682 a 3 4 + 379936a 4 4 + 972a 5 736776a 4 a 4 + 26279a 3 a 2 4 38268a 2 a 3 4 + 236528 a 4 4 578624a 5 4)b ], Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom Remark 3 The coefficients µ j listed above are applicable as long as, ) is a center, and the coefficients µ j are applicable as long as, ) is a center, regardless of the number and distribution of the system s singularities Therefore, for each point on the whole line a 4 = 3 5) see Fig ), there always exist 3 small limit cycles bifurcating from the center, ), no matter whether the system has two centers, or one center and three saddle points, or one center and one saddle point For each point on the line segment a 4 = 3 6 +5) < ), the system can have 3 limit cycles bifurcating from the center, ) This indicates that the results given in [Dumortier et al, 997; Peng, 22; Han, 997] showing that the reversible near-integrable systems with one center and one saddle point can have maximal 2 limit cycles is conservative, since on the part of the line a 4 = 3 5) in the first quadrant > 5) such a system can have at least 3 limit cycles First, we consider the maximal number of limit cycles which can bifurcate from the center, ) Setting µ =yields andthenwehave b =, 73) µ = π[ a 4 ) +2a 4 ) ++a 4 )b ] 74) In order to have µ =, we suppose a 4 and choose b = a 4 ) +2a 4 ) 75) +a 4 Then, µ 2 and µ 3 are simplified to µ 2 = π 3 a 4 ) +2a 4 ) 3a 4 5), µ 3 = π 44 a 4 ) +2a 4 )77 + 5 + 4a 4 +42a 2 434a 4 + 274a 2 4 3a 3 + 28a2 a 4 45 a 2 4 + 444a3 4 ) 76) There are five choices for µ 2 = Except the choice 3a 4 5 =, all other choices lead to µ i =,i =3, 4, Thus, letting a 4 = 3 5), 77) which implies 2 when a 4 Since we assume <, for this case ie when the condition 77) holds),a 4 is guaranteed Then, we have µ 3 = 25π 62 +) 2) 2 2 +5), µ 4 = 5π 8748 +) 2) 2 2 +5) + 4)7 + 58) µ = π 3 ) 3/2 2 +5), µ = 25π 324 ) 22 +5) 3 2) 2 2 +5), implying that in addition we need 2 +5) 78) Under the above conditions 73), 75), 77) and 78), we obtain µ = µ = µ 2 =, but µ 3, µ Hence, at most 3 small limit cycles can bifurcate from the center, ) with no limit cycles bifurcating from the center, ) Further, giving proper perturbations to the parameters a 4 or ), 25254-8

Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom b and b, we can obtain 3 small limit cycles bifurcating from the origin This shows that the conclusion is true for the case of 3, )-distribution Next, consider the, 3)-distribution Similarly, letting µ =yields b = b + 2a 4 79) + Then, µ becomes Hence, we set µ = π ) 2 a 4 ) [ +2a 4 )2 a 4 +) + ) 2 a 4 +)b ] 8) b = +2a 4 )2 a 4 +) + ) 2 a 4 +), to yield µ =,and a 4 + ), 8) µ 2 = π 3 ) 5 8a 4 2 a 4 ) +2a 4 )6 3a 4 +5), µ 3 = π 288 ) 2 3a 4 ) a 4 ) +2a 4 )77 + 225 4a 4 + 242a 2 3234a 4 + 274a 2 4 72a3 962a 2 a 4 + 689 a 2 4 444a3 4 ) 82) The only choice for µ 2 =is6 3a 4 +5=, from which we have a 4 = 3 6 +5) 83) This implies that a 4 += + 2 3 ) > for < Further, we obtain µ 3 = 25π 324 ) +a 3 +2) 2 3 +5), µ 4 = 5π 7496 ) 8+87a 6 3 +2) 2 3 + 5)3 + 4)5 + 58) Four Limit Cycles in Near-Quadratic Integrable Systems µ = π 3 + ) 2 3 +5), 25π µ = 324 + ) 2 3 + 5)3 +2) 2, implying that in addition we require 3 +5) 84) Under the above conditions 79), 8), 83) and 84), we have µ = µ = µ 2 =, but µ 3,µ Further, by properly perturbing the parameters a 4 or ), b and b,wecan obtain 3 small limit cycles bifurcating from the center, ), but no limit cycles from the origin This provesthecaseof, 3)-distribution For the case of 2, )-distribution, it follows from the conditions 73) and75), and a 4 that µ = µ =,and µ 2 = π 3 a 4 ) +2a 4 ) 3a 4 5), 2π µ = + a 4 ) ) 3/2 a 4 ) +2a 4 ) Thus, µ 2 implies µ, indicating that the conclusion holds for the case of 2, )-distribution, if a 4, When a 4 =, 74) becomes µ = π 2) for < and Under the conditions b = and a 4 =, µ and µ becomes µ = 2π ) 3/2 [ 2) + )b ], µ = π ) 2+ 2), 85) which shows that µ for < and But we can choose b = 2 + to obtain µ =Thus,forthiscasewehavea, )-distribution 25254-9

Int J Bifurcation Chaos 2222 Downloaded from wwwworldscientificcom P Yu & M Han Similarly, for the, 2)-distribution, we use the conditions 79) and8) toobtain µ 2 = π 3 ) 5 8a 4 2 a 4 ) +2a 4 ) µ = 6 3a 4 +5), 2π + ) 2 a 4 +) a 4 ) +2a 4 ) This indicates that µ 2 implies µ,andso the conclusion for the case of, 2)-distribution is also true if a 4 + When a 4 + =, ie a 4 = + <, 8) is reduced to µ = π ) 2 a 4 ) 3 +2) and µ and µ become for < and, µ = 2π + [ +2) + )b ], µ = π + 3 +2), 86) which clearly shows that µ for < and However, we may choose b = +2 + to obtain µ =Thus,for a 4 + =, we have a, )-distribution Finally, suppose the condition given in 73) is satisfied, ie b =, then substitute this into µ to solve b to obtain b = +2a 4 87) + Then, under the conditions 73) and87), we obtain µ = π a 4 ) +2a 4 ), + µ = π ) 2 a 4 ) a 4 ) +2a 4 ), 88) which shows that µ implies µ, and thus in general the conclusion is true for the case of, )-distribution As we have seen in the above analysis, if the condition 77), a 4 = 3 5), is not used, then we can only have 2 limit cycles bifurcating from the origin, but no limit cycles can occur from the center, ) In other words, we can obtain one more limit cycle, by using the condition a 4 = 3 5), only bifurcating from the center, ) Similarly, if the condition 83), a 4 = 3 6 + 5), is not used, then we can have only 2 limit cycles bifurcating from the center, ), but no limit cycles can bifurcate from the origin Then, condition a 4 = 3 6 +5) can be only used to get one more limit cycle around the center, ), rather than the origin Therefore, 2, )- or, 2)-distribution is not possible This completes the proof of Theorem 2 4 Limit Cycles Bifurcating from Closed Orbits In this section, based on the results of the small limit cycles obtained in the previous section, we wish to investigate the possibility of existence of large limit cycles by applying the Melnikov function, defined in 7) We have the following result Theorem 3 For the case of bifurcation of small limit cycles from the two centers, ) and, ) with 3, )-distribution resp,, 3)-distribution) there exists at least one large limit cycle near for some h,h ) resp for some h h, )) For the case of limit cycles with 2, )- distribution resp,, 2)-distribution) there exist at least two large limit cycles, one near for some h,h ) and one near 2 for some h 2 h, ) The corresponding values of the parameters and a 4 for the existence of 4 limit cycles can appear at least in some regions in the a 4 parameter plane Remark 4 Theorem 3 gives a positive answer to the open question of existence of limit cycles in near-integrable quadratic systems: at least 4 limit cycles can exist For the case of, )-distribution, so far no more large limit cycles have been found Proof We use the formulas given in 6) and66) in the following calculations Since one cannot find the closed form of the integrals I i h,,a 4 ), i =,, 2, for general and a 4,northetechniqueof Picard Fuchs equation can be applied here, we shall choose some values for and a 4 and then find numerical values of the integral We first use the 25254-2