Limit cycle bifurcations from a non-degenerate center

Applied Mathematics and Computation 218 (2012) 4703–4709

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Limit cycle bifurcations from a non-degenerate center q Jaume Giné Departament de Matemàtica. Universitat de Lleida. Avda. Jaume II, 69 25001 Lleida, Spain

a r t i c l e

i n f o

a b s t r a c t In this work we discuss the computational problems which appear in the computation of the Poincaré–Liapunov constants and the determination of their functionally independent number. Moreover, we calculate the minimum number of Bautin ideal generators which give the number of small limit cycles under certain hypothesis about the generators. In particular, we consider polynomial systems of the form x_ ¼ y þ P n ðx; yÞ; y_ ¼ x þ Q n ðx; yÞ, where Pn and Qn are a homogeneous polynomial of degree n. We use center bifurcation rather than multiple Hopf bifurcations, used a previous work [19], to estimate the cyclicity of a unique singular point of focus–center type for n = 4, 5, 6, 7 and compare with the results given by the conjecture presented in [18]. Ó 2010 Elsevier Inc. All rights reserved.

Keywords: Poincaré–Liapunov constants Limit cycles Center problem Groebner basis

1. Introduction and statement of the main result Consider the following planar polynomial system of the form

x_ ¼ y þ Pðx; yÞ;

y_ ¼ x þ Q ðx; yÞ;

ð1Þ

where P, Q are polynomials with real coefficients without constants and linear terms. It is well-known that the above system always has either a center or a fine focus at the origin. The center problem consists in distinguishing between a center and a focus at the origin of system (1). In the case of a focus, another related problem is to determine its highest possible order. From Poincaré [29] who defined the notion of center for a real system of differential equations in the plane, the center problem has been historically studied by several authors, see for instance [10–13,25]. In the last decades various kinds of methods and approaches have been attempted, different techniques and algorithms have been developed and an extensive literature has been consequently produced, see for instance [14,15,19,22,23,28,33] and a wide range of references therein. P 2 2 Poincaré’s method consists in finding a formal power series of the form Hðx; yÞ ¼ 1 k¼2 H k ðx; yÞ where H2 = (x + y )/2 and Hk _ _ _ are homogeneous polynomials of degree k, so along the orbits of (1) we have that H ¼ x@H=@x þ y@H=@y ¼ P1 2 2 k k¼2 V 2k ðx þ y Þ , where V2k are called the Poincaré–Liapunov constants. There is an algorithm which involves the solution of a system of linear equations for the coefficients of Hn in terms of the coefficients of P and Q and Hk for k = 2, 3, . . . , n  1, see for instance [26,28]. Another method is to construct a Poincaré’s formal power series in polar coordinates and the Poincaré–Liapunov constants can be computed from recursive linear formulas as definite integrals of trigonometric polynomials, see for instance [4,5]. On the other hand, the bifurcation of limit cycles from critical points (Hopf bifurcation) is now a well-known technique used in the analysis of planar vector fields. But for a reduced classes of systems, the maximum number of bifurcating limit cycles (cyclicity) is known. In particular, quadratic systems can have at most three limit cycles [2] and cubic systems without

q

The author is partially supported by a MCYT/FEDER grant number MTM2008-00694 and by a CIRIT grant number 2009SGR 381. E-mail address: [email protected]

0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.09.025

4704

J. Giné / Applied Mathematics and Computation 218 (2012) 4703–4709

quadratic terms have at most five [3,30,34]. Partial results are also known explicitly for several large classes of systems, see for instance [18,19]. To solve the cyclicity problem from a critical point usually are used the same methods and approaches developed to solve the center problem, see for instance [19,26] and a wide range of references therein. Hence, the objective of all these methods is compute the focal values which are invariant modulo the previous ones. The cyclicity can then be found from examining these values and their common zeros. One way to define focal value is by the Poincaré return map. More exactly, if we let d(r0) = h(r0)  r0, where h is the Poincaré return map in a neighborhood of the origin, and if we denote by vk = d(k)(0)/k!, then the first nonzero focal value v2‘+1 corresponds to an odd number k = 2‘ + 1. This technique is used for instance in [2,10,11,25] and it is based on the computation of the derivatives of the return map from a nonlinear system of recursive differential equations. Another algorithm to compute the focal values is developed in [15] where the method is based in the calculation of the successive derivatives of the first return map associated to the perturbations of some planar Hamiltonian systems. For a polynomial vector field the Hilbert basis theorem ensures the existence of a basis of generators of the ideal generated by the focal values, but it does not provide a constructive method to find it. The existent methods to solve this problem are based in the Buchberger’s algorithm to find a Gröebner basis, see [9]. However, it is only applicable for very trivial cases. Recently, new algebraic manipulators have been developed which can address more complex problems, see for instance [20] and references therein. Therefore it is a computational problem of algebraic nature due to the appearance, already for simple systems, of massive focal values that are polynomials in the parameters of the vector field with rational coefficients. Efficient algorithms do not exist to allow the determination of simple groups of generators. One of the main difficulties ultimately comes from the decomposition in prime numbers of a big integer number. Therefore the resolution of the computational problem goes through efficient algorithms that work with big integers and in decomposition in primes numbers of such big numbers. This is a classical problem in computational mathematics, see [1]. Shi Songling proved in [32] that for polynomials of degree n, under certain hypothesis about the focal values generators of the ideal, the maximum number of limit cycles is given by M(n), being M(n) the minimum number of the ideal generators. An open problem nowadays is the determination of M(n), or in its default an upper bound for it. Hence, from the value of the minimum number of the ideal generators M(n), we can find a sequence of perturbations each of which reverses the stability of the origin M(n) times, provided we have sufficient choice in the parameters that appear in the focal values. Notice that the critical point is a center if, and only if, all the focal values are zero, see [29]. We call the set of parameters for which all the focal values vanish the center variety. By the Hilbert basis theorem, the center variety is an algebraic set. Although the calculation of the focal values is straight forward from the different existent methods, as we have said, the computational complexity of finding the common zeros grows very quickly. The result is that some very simple systems have remained intractable, see [16,18,19,27]. In all these works is used the multiple Hopf bifurcation. For higher degree systems the more realistic approach would be to finding good lower bounds to the cyclicity. In this way, the analysis of global bifurcations of limit cycles from symmetrized systems has given some of the best estimates of the number of limit cycles in such systems, see [8,24]. Other approach is the analysis of global bifurcation of limit cycles from centers to estimate the cyclicity of a system, see [7]. In this paper we focus our attention to finding lower bound to the cyclicity around a unique singular point of focus–center type. Moreover, we will use center bifurcation rather than multiple Hopf bifurcations to estimate the cyclicity of a unique _ singular point of focus–center type. In this direction, Zołdek [35] using abelian integrals has shown that there are cubic _ systems with 11 limit cycles bifurcating from a center of a unique singular point. In the Zołdek’s proof is used second order Poincaré–Pontriagin integral of certain cubic system, showing that the space of this integral has dimension 12. The proof is quite technical and difficultly applicable to systems of higher degree. In [7], Christopher has considered a simple computational approach to estimating the cyclicity of singular point of focus–center type from centers around it. Christopher gives a _ confirmation of Zołdek result, and found a quartic system with 17 limit cycles around a unique singular point. One of the most difficult problem of the list given by Hilbert at the beginning of the last century was the 16th Hilbert problem in its part (b). The present statement of the 16th Hilbert problem part (b), restated by Smale in his proposal of problems for the 21st century [31] is the following: 13th Problem of Smale. 16th Problem of Hilbert part (b). Consider the differential system x_ ¼ Pðx; yÞ and y_ ¼ Pðx; yÞ, in R2 where P and Q are polynomials. Is there a bound K on the number of limit cycles of the form K 6 nq where n is the maximum of the degrees of P and Q, and q is a universal constant? This problem is not solved even for quadratic systems. Because of that Smale has relaxed the problem considering a special class of systems the Liénard systems where the bounds remain open. We propose a more simple problem respect to the preceding ones, the cyclicity around a unique singular point, which is the following. Open problem 1. Consider the differential system (1). System (1) has a singular point at the origin of focus-center type. Is there a bound K on the number of limit cycles around the origin of the form K 6 nq where n is the maximum of the degrees of P and Q, and q is a universal constant? In [18] an upper bound for the number of algebraically independent focal values in a certain basis for planar polynomial differential systems is given. It is also conjectured an upper bound for the number of functionally independent focal values given by:

J. Giné / Applied Mathematics and Computation 218 (2012) 4703–4709

4705

Conjecture 1. The number of functionally independent focal values of system (1) at the origin i.e., the minimum number of ideal generators is M(n) = n2 + 3n  7 where n is the degree of the polynomial differential system. In the case that P and Q are homogeneous polynomials of degree n we have that Mh(n) = 2n  1. The above conjecture means that if we perturb the homogeneous system inside the class of the homogeneous systems of the same degree n we can obtain at most 2n  1 small limit cycles under certain hypothesis about the ideal generators. In the same way, if we perturb inside the class of the general systems of the same degree n we can obtain at most n2 + 3n  7. In the homogeneous case, the conjecture is true for n = 2 and n = 3, see [2,30]. In the general case, the conjecture is true for n = 2 and for n = 3 it is known that M(3) P 11. The objective of this work is to study in the homogeneous case the values of Mh(n) for n = 4, 5, 6, 7 and compare it with the values given by Conjecture 1. Therefore, in this paper we consider homogeneous systems of form (1) where the nonlinear terms are homogeneous polynomials of degree n for n = 4, 5, 6, 7. These systems have been studied in [4,5,17]. However, a complete set of center cases is far from being established. In [19], it was proved that Mh(4) P 6 and Mh(5) P 8. To find the bifurcations of limit cycles from a center we will use the method developed in [7]. Naïvely, we would expect the number of limit cycles to be estimated by one less the maximum codimension of a component of the center variety. We choose a point on the center variety, and we linearize the focal values about this point. We would hope that the point chosen on a component of the center variety of codimension r, then the first r linear terms of the focal values should be independent. If this is the case, in [7] it is proved that the cyclicity is equal r  1. That is, there exist perturbations which can produce r  1 limit cycles, and this number is the maximum possible. In fact, it is proved the following theorem. Theorem 2. Suppose that s 2 K (where K denote the corresponding parameter space of a family of polynomial systems) is a point on the center variety and that the first k of the focal values have independent linear parts (with respect to the expansion of the focal values about s), then s lies on a component of the center variety of codimension at least k and the are bifurcations which produce k  1 limit cycles locally from the center corresponding to the parameter value s. In [7], the computation of the focal values is using the formal construction of a first integral. In our work we used the same method but taking polar coordinates. Using this method and Theorem 2 we can establish the main theorem of this work obtained from the results presented in the following sections. Theorem 3. The minimum number of ideal generators of the homogeneous system (1) where the nonlinear terms are homogeneous polynomials of degree 4, 5, 6, 7 is greater or equal 6, 9, 9, 13, respectively, i.e., Mh(4) P 6, Mh(5) P 9, Mh(6) P 9 and Mh(7) P 13. In the following section from the results presented in this section we describe the algorithm used in this work and we apply it to some centers of systems of the form (1) with a linear center perturbed by homogeneous polynomials in order to find the results that permit to establish Theorem 3.

2. The algorithm In this section we show how is implemented the algorithm to compute the linear part of the focal values in order to obtain the lower bounds given in Theorem 3. Consider a system of the form (1) where P and Q a homogeneous polynomials and with a center at the origin. We are going to perturb these systems inside its class of homogeneous system of the same degree in order to compute the linear part of each focal values. It is known from the time of Liapunov [25] that the computation of focal values of system (1) can be considerably simplified if we introduce a complex structure on the phase plane (x, y) by setting z = x + iy. Then we obtain from system (1) the equation

z_ ¼ iz þ Rn ðz; zÞ:

ð2Þ

where Rn is a homogeneous function in z and z of degree n. Now we perturb Eq. (2) by and arbitrary homogeneous function R of the same degree. In this point we return to the real plane (x, y) and take polar coordinates x = r cosu and y = r sinu. We propose the formal power series as a first integral of the form

Hðr; uÞ ¼

1 X

Hk ðuÞr k ;

ð3Þ

k¼2

P 2k with H2(u) = 1/2 and Hk(u) are homogeneous trigonometric polynomials of degree k, verifying H_ ¼ 1 k¼2 V 2k ðuÞr . In fact, P1 the formal power series (3) for the homogeneous system (1) takes the form Hðr; uÞ ¼ m¼0 Hm ðuÞr mðn1Þþ2 where H0 ðuÞ ¼ 1=2, and Hm ðuÞ are homogeneous trigonometric polynomials of degree m(n  1) + 2, satisfying the differential recurrence equations

4706

J. Giné / Applied Mathematics and Computation 218 (2012) 4703–4709

dHmþ1 dHm þ ðmðs  1Þ þ 2ÞPs ðuÞHm ðuÞ þ Q s ðuÞ ¼ du du



0;

if ðm þ 1Þðn  1Þ þ 2 is odd;

V mþ1 ; if ðm þ 1Þðn  1Þ þ 2 is even;

ð4Þ

where V(m+1)(n1)+2, for m = 0, 1, . . ., are the Poincaré–Liapunov constants. Taking into account that we only want to compute the linear part of these constant in the parameters of the perturbation we cut, at each step of the process, the higher terms in each Hi ðuÞ for i > 2. In order to do that we made for any parameter ai the substitution sai where s is a scaling parameter. At each step of the computations we take only the constants and linear terms in Hi ðuÞ. Hence we made the mathematica instruction

e i ðuÞ ¼ Coefficient½Hi ðuÞ; s; 0 þ Coefficient½Hi ðuÞ; s; 1: H Without doing these cuts it is impossible to obtain the results presented in this work because the trigonometric polynomials Hi ðuÞ become massive and it is impossible their computation. Doing the same substitution in the parameters ai ? sai, we compute the linear part of each Poincaré–Liapunov constant doing

W i ðuÞ ¼ Coefficient½V i ðuÞ; s; 1: Later, we construct a vector whose each component is the computed Poincaré–Liapunov constant. Let m this vector i.e., m = {V(s1)+2, V2(s1)+2, V3(s1)+2, . . .}. In order to see how many Poincaré–Liapunov constants linear in the parameters are independent we construct the vectors mi = @m/@ai, for any parameter ai. We construct the matrix whose columns are the vectors mi i.e., M = {m1, m2, . . .} and finally we use the mathematica instruction MatrixRang which gives the rank of the matrix M to obtain the number of Poincaré–Liapunov constants linear in the parameters independents. Hence we compute

MatrixRang M ¼ MatrixRang½m1 ; m2 ; . . .: The algorithm can be applied to compute also the Poincaré–Liapunov constants quadratics in the parameters of the perturbation and also try to proof the independence of these Poincaré–Liapunov constants to improve the results presented in this work. In this paper we have applied the algorithm to some known examples of centers found in previous works but obviously it can be applied to any known center. In this way, recently new centers have been found to which we can apply the algorithm to see if the conjecture is reinforced or refuted, see [21]. 2.1. Quartic centers perturbed by quartic homogeneous polynomials We first consider the family of quartic systems given in Theorem 9 case (iii) in [17]. This family in cartesian coordinates takes the form

  x_ ¼ y  k1 x3 y þ k2 y2 2x2  y2 ;   y_ ¼ x þ k2 xy3 þ k1 x2 x2  2y2 ;

ð5Þ

where k1 and k2 are arbitrary constants, with the inverse integrating factor

    2 76 V ¼ 1 þ 2 k1 x3 þ k2 y3 þ k1 x3  k2 y3 : We consider the general perturbation of this system in the class of the quartic homogeneous polynomials. That is we perturb system (5) into the form

  x_ ¼ y  k1 x3 y þ k2 y2 2x2  y2 þ P4 ðx; yÞ;   y_ ¼ x þ k2 xy3 þ k1 x2 x2  2y2 þ Q 4 ðx; yÞ;

ð6Þ

where P4 and Q4 are homogeneous polynomials of degree 4. Automatic computations now show that the linear part of {V8, V14, V20, V26, V32} are independent in the parameters and therefore 5 limit cycles can bifurcate from this center. Moreover, when we vanish these linear parts we obtain that linear part of V38, V44 and V50 also are null. We have studied first this family because in [17] was conjectured from the existence of this family that the number of small amplitude limit cycles was at least seven because the family is defined by seven relations between the original parameters of an arbitrary family of quartic degree. From the results obtained here we can see that the linear parts of the Poincaré–Liapunov constants associated to this center only can give 5 small amplitude limit cycles. Secondly, we consider the family of quartic systems given in Section 3 of [6] with a polynomial inverse integrating factor. In polar coordinates takes the form

x_ ¼ 2x4  y  6x2 y2  12cx2 y2 þ 4cy4  2c2 x4 þ 18c2 x2 y2  4c2 y4  6sx3 y þ 10sxy3 þ 10csx3 y  14csxy3 ; y_ ¼ x  2x3 y þ 6xy3  8cxy3 þ 2c2 x3 y  6c2 xy3  6sx2 y2 þ 2sy4  6csx2 y2 þ 2csy4 ; with the polynomial inverse integrating factor

ð7Þ

4707

J. Giné / Applied Mathematics and Computation 218 (2012) 4703–4709

  Vðx; yÞ ¼ 1  2y þ 2y2  2cy2  2sxy 1 þ 6x2 y þ 2y3  4cy3  6c2 x2 y þ 2c2 y3  2sx3  6sxy2  2csx3 þ 6csxy2 Þð1 þ 2y þ 2y2 þ 4y3 þ 4y4 þ 2cy2  4cy3  8cy4 þ 4c2 y4  þ 2sxy  4sxy2  8sxy3 þ 8csxy3 þ 4s2 x2 y2 ;

ð8Þ

where c = cosw and s = sinw and with arbitrary w 2 [0, 2p]. We consider the general perturbation of this system in the class of the quartic homogeneous polynomials. That is we perturb system (7) into the form

x_ ¼ 2x4  y  6x2 y2  12cx2 y2 þ 4cy4  2c2 x4 þ 18c2 x2 y2  4c2 y4  6sx3 y þ 10sxy3 þ 10csx3 y  14csxy3 þ P4 ðx; yÞ;

ð9Þ

y_ ¼ x  2x3 y þ 6xy3  8cxy3 þ 2c2 x3 y  6c2 xy3  6sx2 y2 þ 2sy4  6csx2 y2 þ 2csy4 þ Q 4 ðx; yÞ;

where P4 and Q4 are homogeneous polynomials of degree 4. Automatic computations now show that the linear part of {V8, V14, V20, V26, V32, V38} are independent in the parameters and therefore 6 limit cycles can bifurcate from this center. Moreover, when we vanish these linear parts we obtain that linear part of V44 and V50 also are null. In conclusion, in this case system (9) has a fine focus of order 6. Moreover, the number of functionally independent Poincaré–Liapunov constants is also 6 and consequently the minimum number of ideal generators is at least 6, i.e., Mh(4) P 6. 2.2. Quintic centers perturbed by quintic homogeneous polynomials In this section, we consider a quintic system with center perturbed by quintic homogeneous polynomials. In particular, we consider the family of quintic systems given in Theorem 10 case (iii) in [17]. This family in cartesian coordinates takes the form 2 2 x_ ¼ y þ 2k1 x5 þ 2k1 k2 x5 þ 6x4 y  10k1 x4 y  6k1 k2 x4 y  8x3 y2 2

2

 4k1 x3 y2  20k1 k2 x3 y2  8x2 y3 þ 20k1 x2 y3  4k1 k2 x2 y3 2

2

þ 8xy4  6k1 xy4 þ 10k1 k2 xy4 þ 2y5  2k1 y5 þ 2k1 k2 y5 ; y_ ¼

ð10Þ

2 2 x  2x5 þ 2k1 x5 þ 2k1 k2 x5  8x4 y þ 6k1 x4 y þ 10k1 k2 x4 y 2 2 þ 8x3 y2  20k1 x3 y2  4k1 k2 x3 y2 þ 8x2 y3 þ 4k1 x2 y3 2 2  20k1 k2 x2 y3  6xy4 þ 10k1 xy4  6k1 k2 xy4  2k1 y5 þ 2k1 k2 y5 ;

where k1 = cosw and k2 = sinw and with arbitrary w 2 [0, 2p], with the inverse integrating factor

 14  2 2 2 2 2 4 2 V ¼ 1  4k2 x4 þ 16k1 k2 x3 y  16k1 x2 y2 þ 8k2 x2 y2  16k1 k2 xy3  4k2 y4 1  4x4 þ 4k1 x4 þ 8k1 k2 x4 þ 4k1 k2 x8 3 3

2 4

5

6

2

5

4 2

 16k1 k2 x8 þ 24k1 k2 x8  16k1 k2 x8 þ 4k2 x8  8x3 y þ 16k1 x3 y þ 16k1 k2 x3 y  16k1 k2 x7 y þ 48k1 k2 x7 y   þ

3 3 2 4 5 6 2 6 4 2 64k1 k2 x7 y þ 64k1 k2 x7 y  48k1 k2 x7 y þ 16k2 x7 y þ 8x2 y2  24k1 x2 y2 þ 16k1 x6 y2  16k1 k2 x6 y2 3 3 2 4 5 6 2 6 32k1 k2 x6 y2 þ 48k1 k2 x6 y2  32k1 k2 x6 y2 þ 16k2 x6 y2  8xy3 þ 16k1 xy3  16k1 k2 xy3  64k1 x5 y3 5 4 2 3 3 5 6 2 48k1 k2 x5 y3  48k1 k2 x5 y3 þ 64k1 k2 x5 y3 þ 16k1 k2 x5 y3  16k2 x5 y3  4y4 þ 4k1 y4  8k1 k2 y4 6

4 2

2 4

6

6

6

6

4 2

4 2

3 3

5

4 2

þ 96k1 x4 y4 þ 24k1 k2 x4 y4  16k1 k2 x4 y4  40k2 x4 y4  64k1 x3 y5  48k1 k2 x3 y5  48k1 k2 x3 y5 3 3

5

3 3

2 4

 64k1 k2 x3 y5  16k1 k2 x3 y5  16k2 x3 y5 þ 16k1 x2 y6  16k1 k2 x2 y6 þ 32k1 k2 x2 y6 þ 48k1 k2 x2 y6 5

6

5

2 4

5

6

þ 32k1 k2 x2 y6 þ 16k2 x2 y6 þ 16k1 k2 xy7 þ 48k1 k2 xy7 þ 64k1 k2 xy7 þ 64k1 k2 xy7 þ 48k1 k2 xy7 þ 16k2 xy7  4 2 3 3 2 4 5 6 þ 4k1 k2 y8 þ 16k1 k2 y8 þ 24k1 k2 y8 þ 16k1 k2 y8 þ 4k2 y8 : We consider the general perturbation of system (10) in the class of the quintic homogeneous polynomials. Automatic computations show that the linear part of {V6, V10, V14, V18, V22, V26, V30, V30, V34, V38} are independent in the parameters and therefore 9 limit cycles can bifurcate from this center. Moreover, when we vanish these linear parts we obtain that linear part of V42 also is null. In conclusion, in this case the perturbation of system (10) has a fine focus of order 9. This fact means that there are linear centers perturbed by quintic homogeneous polynomials having a fine focus of order 9. Moreover, the number of functionally independent Poincaré–Liapunov constants is also 9 and consequently the minimum number of ideal generators is at least 9, i.e., Mh(5) P 9. 2.3. Sextic centers perturbed by sextic homogeneous polynomials We consider the family of sextic systems given in Theorem 11 case (iii) in [17]. This family in cartesian coordinates takes the form

4708

J. Giné / Applied Mathematics and Computation 218 (2012) 4703–4709

  x_ ¼ y  k1 x5 y þ k2 y4 2x2  y2 ;   y_ ¼ x þ k2 xy5 þ k1 x4 x2  2y2 ;

ð11Þ

where k1 and k2 are arbitrary constants, with the inverse integrating factor

    2 109 V ¼ 1 þ 2 k1 x5 þ k2 y5 þ k1 x5  k2 y5 : We consider the general perturbation of this system in the class of the sextic homogeneous polynomials. That is we perturb system (11) into the form

  x_ ¼ y  k1 x5 y þ k2 y4 2x2  y2 þ P6 ðx; yÞ;   y_ ¼ x þ k2 xy5 þ k1 x4 x2  2y2 þ Q 6 ðx; yÞ;

ð12Þ

where P6 and Q6 are homogeneous polynomials of degree 6. Automatic computations now show that the linear part of {V12, V22, V32, V42, V52, V62, V72, V82, V92} are independent in the parameters and therefore 9 limit cycles can bifurcate from this center. Moreover, when we vanish these linear part we obtain that linear parts of V38, V44 and V50 also are null. In conclusion, in this case system (12) has a fine focus of order 9. This fact means that there are linear centers perturbed by sextic homogeneous polynomials having a fine focus of order 9. Moreover, the number of functionally independent Poincaré–Liapunov constants is also 9 and consequently the minimum number of ideal generators is at least 9, i.e., Mh(6) P 9. 2.4. Septic centers perturbed by Septic homogeneous polynomials In this section, we consider system septic centers perturbed by quintic homogeneous polynomials. In particular, we consider the family of septic systems given in Theorem 12 case (iii.2) in [17]. This family in cartesian coordinates takes the form

 2 2 x_ ¼ y þ 4k1 x7 þ 4k1 k2 x7 þ 14x6 y  22k1 x6 y  14k1 k2 x6 y  8x5 y2 2

2

 8k1 x5 y2  44k1 k2 x5 y2  6x4 y3 þ 30k1 x4 y3  22k1 k2 x4 y3  16x3 y4 2

2

 28k1 x3 y4  20k1 k2 x3 y4  14x2 y5 þ 46k1 x2 y5  2k1 k2 x2 y5 þ 24xy6  2 2 16k1 xy6 þ 28k1 k2 xy6 þ 6y7  6k1 y7 þ 6k1 k2 y7 =3;  2 2 y_ ¼ x þ 6x7 þ 6k1 x7 þ 6k1 k2 x7  24x6 y þ 16k1 x6 y þ 28k1 k2 x6 y 2

ð13Þ

2

þ 14x5 y2  46k1 x5 y2  2k1 k2 x5 y2 þ 16x4 y3 þ 28k1 x4 y3  20k1 k2 x4 y3 2

2

þ 6x3 y4  30k1 x3 y4  22k1 k2 x3 y4 þ 8x2 y5 þ 8k1 x2 y5  44k1 k2 x2 y5  2 2 14xy6 þ 22k1 xy6  14k1 k2 xy6  4k1 y7 þ 4k1 k2 y7 =3; where k1 = cosw and k2 = sinw and with arbitrary w 2 [0, 2p], and which has an inverse integrating factor with similar form of quintic system (10). We consider the general perturbation of system (13) in the class of the septic homogeneous polynomials. Automatic computations show that the linear part of {V8, V14, V20, V26, V32, V38, V44, V50, V56, V62, V68, V74, V80} are independent in the parameters and therefore 13 limit cycles can bifurcate from this center. Moreover, when we vanish these linear parts we obtain that linear part of V86 also is null. In conclusion, in this case a perturbation of system (13) has a fine focus of order 13. This fact means that there are linear centers perturbed by septic homogeneous polynomials having a fine focus of order 13. However, the number of functionally independent Poincaré–Liapunov constants is 13 and consequently the minimum number of ideal generators is at least 13, i.e., Mh(7) P 13. Acknowledgments The author is grateful to the referees for their valuable remarks which helped to improve the manuscript. References [1] M. Agrawal, N. Kayal, N. Saxena, Primes is in P, Ann. Math. 160 (2) (2004) 781–793. [2] N.N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sb. 30 (72) (1952) 181–196; N.N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Amer. Math. Soc. Transl. 100 (1954) 397–413. [3] T.R. Blows, N.G. Lloyd, The number of limit cycles of certain polynomial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 98 (3–4) (1984) 215– 239. [4] J. Chavarriga, J. Giné, Integrability of a linear center perturbed by fourth degree homogeneous polynomial, Publ. Mat. 40 (1) (1996) 21–39. [5] J. Chavarriga, J. Giné, Integrability of a linear center perturbed by a fifth degree homogeneous polynomial, Publ. Mat. 41 (2) (1997) 335–356. [6] J. Chavarriga, J. Giné, M. Grau, Integrable systems via polynomial inverse integrating factors, Bull. Sci. Math. 126 (4) (2002) 315–331.

J. Giné / Applied Mathematics and Computation 218 (2012) 4703–4709

4709

[7] C.J. Christopher, Estimating limit cycle bifurcations from centers, Differential Equations with Symbolic Computation, Trends Math., Birkhuser, Basel, 2005. pp. 23–35. [8] C.J. Christopher, N.G. Lloyd, Polynomial systems: a lower bound for the Hilbert numbers, Proc. Roy. Soc. London Ser. A 45 (1938) (1995) 219–224. [9] D. Cox, J. Little, D. O’Shea, Ideals, Varieties and Algorithms, second ed., Undergraduate Texts Math., Springer-Verlag, 1992. [10] H. Dulac, Détermination et intégration d’une certain classe d’équations différentielle ayant pour point singulier un center, Bull. Sci. Math. Sér. 32 (2) (1908) 230–252. [11] M. Frommer, Über das Auftreten von Wirbeln und Strudeln (geschlossener und spiraliger Integralkurven) in der Umgebung rationaler Unbestimmheitsstellen, Math. Ann. 109 (1934) 395–424. [12] W. Kapteyn, On the midpoints of integral curves of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland 19 (1911) 1446–1457. Dutch. [13] W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. 20 (1912) 1354–1365; W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. 21 (1912) 27–33. Dutch. [14] W.W. Farr, C. Li, I.S. Labouriau, W.F. Langford, Degenerate Hopf-bifurcation formulas and Hilbert’s 16th problem, SIAM J. Math. Anal. 20 (1989) 13–29. [15] J.P. Françoise, Successive derivatives of a first return map, application to the study of quadratic vector fields, Ergodic Theor. Dyn. Syst. 16 (1996) 87–96. [16] J. Giné, Polynomial first integrals via the Poincaré series, J. Comput. Appl. Math. 184 (2) (2005) 428–441. [17] J. Giné, The center problem for a linear center perturbed by homogeneous polynomials, Acta Math. Sin. 22 (6) (2006) 1613–1620. [18] J. Giné, On the number of algebraically independent Poincaré–Liapunov constants, Appl. Math. Comput. 188 (2) (2007) 1870–1877. [19] J. Giné, J. Mallol, Minimum number of ideal generators for a linear center perturbed by homogeneous polynomials, Nonlinear Anal. 71 (12) (2009) e132–e137. [20] J. Giné, V.G. Romanovski, Linearizability conditions for Lotka–Volterra planar complex cubic systems, J. Phys. A 42 (22) (2009) 225206. [21] J. Giné, V.G. Romanovski, Integrability conditions for Lotka–Volterra planar complex quintic systems, Nonlinear Anal. Real World Appl. 11 (3) (2010) 2100–2105. [22] J. Giné, X. Santallusia, On the Poincaré–Liapunov constants and the Poincaré series, Appl. Math. (Warsaw) 28 (1) (2001) 17–30. [23] J. Giné, X. Santallusia, Implementation of a new algorithm of computation of the Poincaré–Liapunov constants, J. Comput. Appl. Math. 166 (2) (2004) 465–476. [24] Ji Bin Li, Qi Ming Huang, Bifurcation of limit cycles forming compound eyes in the cubic systems, Chinese Ann. Math. Ser. B 8 (4) (1987) 391–403. [25] M.A. Liapunov, Problème général de la stabilité du mouvement, Ann. Math. Stud., 17, Pricenton University Press, 1947. [26] N.G. Lloyd, J.M. Pearson, REDUCE and the bifurcation of limit cycles, J. Symbolic Comput. 9 (1990) 215–224. [27] N.G. Lloyd, J.M. Pearson, Computing center conditions for certain cubic systems, J. Comput. Appl. Math. 40 (1992) 323–336. [28] J.M. Pearson, N.G. Lloyd, C.J. Christopher, Algorithmic derivation of centre conditions, SIAM Rev. 38 (1996) 619–636. [29] H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. Math. 37 (1881) 375–422; H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. Math. 8 (1882) 251–296; H. Poincaré, Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951. pp. 3–84. [30] K.S. Sibirskii, On the number of limit cycles in the neighborhood of a singular point, Differ. Eqn. 1 (1965) 36–47. [31] S. Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998) 7–15. [32] Shi Songling, A method of constructing cycles without contact around a weak focus, J. Differ. Eqn. 41 (1981) 301–312. [33] Shi Songling, On the structure of Poincaré–Lyapunov constants for the weak focus of polynomial vector fields, J. Differ. Eqn. 52 (1984) 52–57. _ [34] H. ZoŁdek, On certain generalization of Bautin’s theorem, Nonlinearity 7 (1994) 273–279. _ [35] H. ZoŁdek, Eleven small limit cycles in a cubic vector field, Nonlinearity 8 (1995) 843–860.