EXACT BOUND ON THE NUMBER OF LIMIT CYCLES ARISING FROM A PERIODIC ANNULUS BOUNDED BY A SYMMETRIC

In this paper, the bound on the number of limit cycles by Poincaré bifurcation in a small perturbation of some seventh-degree Hamiltonian system is concerned. The lower and upper bounds on the number of limit cycles have been obtained in two previous works, however, the sharp bound is still unknown. We will employ some new techniques to determine which is the exact bound between 3 and 4. The asymptotic expansions are used to determine the four vertexes of a tetrahedron, and the sharp bound can be reached when the parameters belong to this tetrahedron.


Introduction
The classical Liénard systemẍ + f (x)ẋ + x = 0, (1.1) and its generalized formẍ + f (x)ẋ + g(x) = 0, (1.2) have extensive application in real world. It can describe the oscillation which is evident in many fields of science, not only in electronics, physics and mathematics, but also in civil engineering, chemistry, biology, astronomy, see a very new monograph [2], in which various kinds of nonlinear oscillators and their applications have been studied systematically by methods of numerical analysis and dynamical system. From another point of view, the number of limit cycles of systems (1.1) and (1.2) provides very important information for studying Hilbert's 16th problem [16]. This problem asks for the maximal number of limit cycles for the polynomial differential systemẋ = P (x, y),ẏ = Q(x, y) of a given degree n. It is rather difficult and still open even for n = 2. Several weaker versions had been proposed, for example, S. Smale [21] proposed to study the maximal number of limit cycles of the classical Liénard system (1.1), and VI. Arnold confined the problem to studying limit cycles arising from perturbing a family of ovals of Hamiltonian, see [1]. In detail, H(x, y), p(x, y) and q(x, y) are polynomials, and the Hamiltonian systemẋ = H y (x, y), y = −H x (x, y) has at least one family of closed orbits L h ⊆ {(x, y)|H(x, y) = 0}. After a small polynomial vector perturbation by (p(x, y), q(x, y)), only a finite number of closed orbits persist. These persisting closed orbits are limit cycles of the perturbed Hamiltonian system, x = H y (x, y) + p(x, y),ẏ = −H x (x, y) + q(x, y), where is a sufficiently small parameter. The phenomenon is called Poincaré bifurcation firstly studied by Jules Henri Poincaré . The Poincaré Theorem [9] reveals that the number of limit cycles by Poincaré bifurcation of (1.3) can be evaluated by the number of zeros of the following integral, which is the planar form of system (1.2) with weak damping term − f (x)ẋ. Therefore, studying the maximal number of zeros of the Abelian integral of system (1.5), can be regarded as an intersection version of Smale's problem and Arnold's problem. There have been many excellent works on this rather simple versions, however, it is not easy at all. Note that system (1.5) of a fixed degree n can be classified into different cases according to the topological portraits of system (1.5) =0 . For convenience, the maximal number of zeros of (1.6) for any concrete system (1.5) of degree n is denoted by N (n) when it's need for convenience. Dumortier and Li obtained the sharp bounds of N (3) for 5 different topological portraits of unperturbed system (1.5) in a series paper [4][5][6][7]. For system (1.5) of degree 4, the lower and upper bounds on N (4) for topologically different periodic annulus have been summarized in a new work [22], in which the sharp bounds on N (4) are obtained when the outer boundary of the period annulus is a heteroclinic (homoclinic) loop connecting a nilpotent singularity. The method and analysis are based on the Chebyshev criterion and combination technique. For N (5), the results for system (1.5) with symmetry have been only reported, see [23,31] and the references therein. When system (1.5) is nonsymmetric and has a degree that is larger than 4, the Abelian integrals have more than 4 elements. Then, the classical idea based on Picard-Focus equation fails and applying the Chebyshev criterion leads to rather complicated algebraic computation. The symmetry assumption on (1.5) reduces the generating elements of I(h). For example, I(h) has four generating elements for symmetric system (1.5) of degree 7. The symmetric system (1.5) of degree 7 can be normalized as follows: where ζ = ±1, α and β are real number, f (x) = a 0 + a 1 x 2 + a 2 x 4 + a 3 x 6 . We assume system (1.7) =0 has a heteroclinic loop without other poly-cycle, then we have eight topological portraits, see Figure 1. For cases a and b, it was proved that 4 is the sharp bound of N (7) in [24] and [30] by different researchers, while in the latter work the same result was obtained for case c only on the inner part of the periodic annulus. Kazemi and Zangeneh [17] studied the perturbation of period annulus belonging to case e, and proved 3 and 4 are the lower and upper bounds on the corresponding N (7). An equivalent system was studied in [25] and almost same results were obtained with different methods on asymptotic expansion of I(h). Sun and Zhao [26] obtained that 3 and 4 are the lower and upper bounds on the corresponding N (7) by perturbing the period annulus of case f . There are no results reported on cases d, g and h up to now, and it is still unknown that which is the exact bound on the corresponding N (7) between 3 and 4 for cases e and f . The sharp bounds are only obtained for system (1.7) belonging to cases a, b, it is interesting that there exist a heteroclinic loop connecting two hyperbolic saddles and an elementary center. In a very recent work [32], an equivalent system that belongs to case a was investigated by choosing different α and β from those in [24]. They obtained that 3 and 4 are the lower bound and upper bound on the corresponding N (7), respectively. The authors also claimed that the non-existence of nilpotent singularity on the boundary of period annulus may not be the key condition for obtaining the sharp bound 4. An additional technique based on combination of two elements in I(h) has been used in [24,26]. It plays an key role in obtaining the exact bound and the upper bound. The sharp bound in [24] could not be obtained without the combination technique and only a larger upper bound 5 can be derived by directly applying the Chebyshev criterion, see the final remark in [24]. Chebyshev criteria [8,19] failed to bound N (7) on case f . However, the combination produces a parameter in the integral systems and the computational analysis becomes more difficult.
In this paper, we concern about part of the remaindering problems on system (1.7) with a unique heteroclinic loop bounding a periodic annulus. We employ the combination technique to determine which is the exact bound between 3 and 4 for N (7) on case e. The main idea to prove our result is as follows: we bound the parameter space in R 3 to obtain a cubic set, which might yield maximal 4 zeros of the corresponding Abelian integrals. Furthermore, we exclude existence of 4 zeros on this cubic set, and thus the sharp bound is 3. We adopt the system given below rather than the system (1.7) α=β=1 investigated in [17], and part of the analysis is based on [24],ẋ where is sufficiently small and a i s (i = 0, 1, 2, 3) are bounded parameters. System (1.8) can be transformed into system (1.7) α=β=1 by scaling x = x 2 , y = y 16 , t = 8 t and a i = 2 2i+1 a i .
The unperturbed system ( = 0) is an integrable system with the Hamiltonian, The bounded level sets are sketched in Figure 2. H(x, y) = h defines a family of closed curves γ h (orbits of system (1.8) = 0) forms a periodic annulus PA for 0 < h < 1 2048 . The inner boundary is an elementary center at the origin (0, 0) and the outer one is a heteroclinic loop γ l connecting two nilpotent saddles (± 1 2 , 0) with H(γ l ) = 1 2048 . Corresponding to the discussion above, the first order approximation of the return map constructed on PA is given as below: a 1 , a 2 , a 3 ). The following results have been obtained. 25]). For all possible (a 0 , a 1 , a 2 , a 3 ) ∈ R 4 , I(h, δ) has at most 4 zeros, and there exist some (a 0 , a 1 , a 2 , a 3 ) ∈ R 4 such that I(h, δ) has at least 3 zeros.
As we mentioned before the same result was given in [17] for the related Abelian integral. However, the exact bound had not been obtained. Our new result is given as belows: Theorem 1.2. The exact bound on the number of zeros of I(h, δ) is 3.
As we have shown that (1.7) α=β=1 is equivalent to system (1.8). Therefore, Theorem 1.2 holds for system (1.7) α=β=1 . The rest of the paper is organized as follows: in section 2, we give some primary results of the elements I i (h) near the endpoints of (0, 1 2048 ) and give a short description of the main tool that is used in this paper. In section 3, we prove our main result and give four vertexes of a tetrahedron T 4 . I(h) has exact 3 zeros when the parameter belongs to T 4 . Finally, we end this paper with a discussion and a summary on the bound of N (7) associated with the eight periodic annuli bounded by a heteroclinic loop.

The ratios of some I i (h)
I 0 (h) at the endpoints and Chebyshev criterion In this section, we analyze the ratios of Ii  Proof. From section 3 of [25], I(h, δ) has the following asymptotic expansion for h psotive and near h = 0, The following result is taken from [25] which was obtained based on Lemma 4.1 of [8].
The technique to bound the number of zeros of the Abelian integral used in [17,25] is the Chebyshev criterion. Generally speaking, we call a set of integrals, taking {I 0 (h), I 1 (h), I 2 (h), · · · , I n−1 (h)} for example, forms a Chebyshev system if any linear combination α 0 I 0 (h) + α 1 I 1 (h) + α 2 I 2 (h) + · · · + α n−1 I n−1 (h) has at most n − 1 zeros for n ∈ N + . A clever criterion to determine the Chebyshev system for a special integral set was proposed in [8,19], which reduces the problem into determining a set of analytic functions forming a Chebyshev system. The reduced problem is algebraic and one only check if the related Wronskians vanishes or not. For completeness and readability, we give the exact criterion of the main criterion [19] based on the integrals { γ h f i (x)y 7 dx} for system (1.8) which is defined on the periodic annulus PA, see [8,17,19,25] for more details.

Proof of Main result
In this section, we prove the main result. The main idea is to intercept the parameter space to obtain a cubic set S, only on which I(h, δ) may have maximal 4 zeros. Then, the maximum of I(h, δ) on (0, 1 2048 ) for (a 0 , a 1 , a 2 ) ∈ S contradicts existence of zeros of I(h, δ). Hence, the sharp bound is 3.
The following lemma is obtained by direct computation, and part of them had been proved in [25]. The proof is omitted.
We have the following two propositions by applying Lemma 2.3 and Lemma 3.1.  It is without loss of generality to assume a 3 = 1 and denote δ = (a 0 , a 1 , a 2 ) if a 3 = 0. Introduce The following equations inherit from Lemma 2.2.
Then, we have a 2

I2(h)
I0(h) ≤ 0. Then Hence,  [20]. Then, there exists a tetrahedron T 4 bounding the parameter space R 3 such that I(h, δ) has exact 3 zeros when δ ∈ T 4 . The four vertexes of T 4 can be obtained by the asymptotic of expansions of I(h, δ) near the boundary of PA. It is usually very difficult to obtain the expansions near a homoclinic loop or heteroclinic loop connecting singularities, see a series of papers [10][11][12][13]15,27,28,33] for the expansions of the Abelian integrals near various kinds of singular loops. The method has been extended to non-smooth systems [14] and higher dimensional systems [29] The asymptotic expansion of I(h, δ) near the heteroclinic loop γ l was given in [25] based on the relatively new results in [12] at that time,

Conclusion
In this paper, we provide a rigorous proof that the Ablian integral of system (1.8) has at most 3 zeros, and then the sharp bound of the number of zeros is 3. The main tools are Chebyshev criterion and combination technique. The problem on the exact bound of zeros of the Abelian integral for system (1.7) of case e is completely solved. However, there still exist some open problems on the bound of N (7) for system (1.7) of other cases: (I) it is still unknown that the exact bounds on N (7) for system (1.7) of cases a, b and c when the parameters α and β in unperturbed system are not fixed; (II) what are the exact bounds on N (7) for system (1.7) of cases d and g with fixed and unfixed parameter α; (III) what are the exact bound for cases f and h. Those problems are difficult by applying recent methods and techniques. We need improve the algebraic criterion and develop more efficient symbolic computations.