ZERO BIFURCATION DIAGRAMS FOR ABELIAN INTEGRALS: A STUDY ON HIGHER-ORDER HYPERELLIPTIC HAMILTONIAN SYSTEMS WITH THREE PERTURBATION PARAMETERS

In this paper, we present the zero bifurcation diagrams for the Abelian integrals of two hyperelliptic Hamiltonian systems with three perturbation parameters using an algebraic-geometric approach. The method can be used to study the bifurcation diagrams for higher-order Hamiltonian systems with polynomial perturbations of any degree.


Introduction
The polynomial vector field y ∂ ∂x + − g(x) + εf (x)y ∂ ∂y = 0 with 0 < ε 1 is a result of perturbation on the Hamiltonian system, x = y,ẏ = −g(x) + εf (x)y, (1.1) The bifurcation diagrams have been obtained for codimensions two, three and four by a sophisticated analysis. In [28], Xiao studied a codimension five bifurcation of cusp case, for which two different truncated systems were obtained by applying the principle re-scaling and central re-scaling. The first truncated system is a Hamiltonian system, and its bifurcation diagram was obtained. The second truncated system is the following 5th-degree perturbed Hamiltonian, (1. 2) The unperturbed system (1.2) can be classified into 11 cases according to the values of α and β when it has at least one period annulus [28]. The Hopf bifurcation surface was obtained for all values of α and β. However, the limit cycle bifurcation surfaces on the whole period annulus are still unknown. The main difficulty of the analysis is due to the higher order of the Hamiltonian. The outer boundaries of the period annulus of the system (1.2) ε=0 have two degenerate cases, one is the nilpotent-saddle loop when α = β = 1, and the other is the cusp-saddle cycle when α = 0 and β = 3 5 , as shown in Figure 1 Perturbation on Hamiltonian systems is also a main topic in the study of the weak Hilbert's 16th problem, proposed by Arnold [1] for studying the maximal number of zeros of the Abelian integral, which is the first order approximation of the return map, q(x, y)dx − p(x, y)dy, h ∈ J, for the perturbed Hamiltonian system, where p(x, y) and q(x, y) are polynomials of degree n ≥ 2, H(x, y) is of degree n + 1, Γ h represents closed algebra curves of H(x, y) which are parameterized by {(x, y)|H(x, y) = h, h ∈ J}, J is an open interval, δ is a set of coefficients of p(x, y) and q(x, y), and ε represents small perturbations satisfying |ε| 1. Some relatively new estimated bounds for general n can be found in [23,24] and references therein. More relative references on study of the number of zeros of Abelian integrals for perturbed integral systems can be found in [17]. A so called "much weaker" case is defined by H = y 2 2 + g(x)dx, p = 0 and q = f (x)y, for which the perturbed Hamiltonian is reduced to system (1.1), and the corresponding Abelian integral has a simple form, System (1.1) is called type (m, n) if g(x) and f (x) are polynomials of degree m and n, respectively. A lot of works on the study of A(h, δ) with various polynomials f (x) and g(x) were reported, for example, see [6] for type (3,2), and [2][3][4]27,32] for type (5,4) with symmetry. For type (3,2), the authors not only deduced the sharp bounds for the number of zeros but also obtained the bifurcation diagrams. The main tools used in these studies are Picárd-Focus equations and Ricatti equations. Because of symmetry in type (5,4), the perturbation terms are reduced to three, and the dimension of Picárd-Focus equations is thus the same as that of type (3,2). The sharp bounds were obtained for most cases, however the bifurcation diagrams have not been obtained and an advanced analysis is needed. We also noticed that the existence and uniqueness of limit cycles in system (1.1) without restriction on the first order bifurcation have been reported in [5,10] and references therein, and the integrability of system (1.1) was summarized in [21]. The lower bounds on the number of system (1.2) with more higher order perturbation terms have been studied in [25,30,31]. When the Hamiltonian is of degree more than four, i.e., the degree of g(x) is more than three without symmetry, it is more difficult to obtain the bifurcation diagrams with three perturbation parameters, since the dimensions of the Picárd-Focus equation and Ricatti equation become higher and some new perturbation terms will appear when analyzing the centroid curve (see the method in [6]). It is almost impossible to analyze the intersection of the lines (planes) and curves (surfaces) by a classical method. In this paper, we will propose a method to analyze the zero bifurcation diagrams of the Abelian integrals for the three parameter perturbation of higher order Hamiltonian systems. We use the method to obtain bifurcation diagrams for two perturbed systems which exhibit truncated codimension-5 bifurcations, and present the results for a higher order perturbation of a Hamiltonian. We will not only show this integral is Chebyshev, but also give the exact zero bifurcation diagram.
The rest of the paper is organized as follows. In section 2, we give the asymptotic expansions of the Abelian integrals A(h) and M(h) for two degenerate cases, which are obtained by using the methods developed in [13,15,16,26]. Then we present the Chebyshev criterion [9,22], which is the generalization of the method developed in [20] to determine the monotonicity of two Abelian integrals. It was efficiently used to bound the number of zeros of some Abelian integrals with more than two generation elements. In our work, we will apply it to "restrict" the perturbation parameters with the help of "combination" skill, which is essential in the bifurcation diagram analysis. In order to achieve this, we need a primary analysis on the unperturbed systems and the Abelian integrals A(h) and M(h). Especially, for the saddle-cusp cycle case, instead of directly studying the original system, we transform the system to an equivalent system with the same Abelian integral of the original one. The main aim of this step is to demonstrate that our method can deal with general polynomial perturbations involving three parameters. In section 3, we study the bifurcation diagram for A(h) using the asymptotic expansions and the combination skill. In section 4, we consider the bifurcation diagram for M(h). Finally, in section 5 we discuss the Hamiltonian systems with higher-degree perturbations and give a complete zero bifurcation diagram for this problem.
satisfying H(x, y) = h for h ∈ (− 108 15625 , 0) and x ∈ (0, 1), as given in Figure 1(b), showing a family of closed orbits Γ h surrounded by a saddle-cusp cycle Γ 0 , with a nilpotent cusp of order 1 at (0, 0) and a hyperbolic saddle at (1, 0). Correspondingly, we have the Abelian integral, The above perturbed Hamiltonian system is a cubic polynomial perturbation to the Hamiltonian, satisfying H * (x, y) = h for h ∈ (− 4 45 , 0) and x ∈ (− 2 3 , 1), as depicted in Figure 2, showing a family of closed orbits Γ * h surrounded by a saddle-cusp cycle Γ * 0 , with a nilpotent cusp of order one at (1, 0) and a hyperbolic saddle at (− 2 3 , 0). The Abelian integral of the perturbed Hamiltonian system (2.6) is given by By [14], we have which will be analyzed in the following sections. The asymptotic expansions of Abelian integrals are proposed to study its zeros near the endpoints of the annuluses, and these zeros correspond to limit cycles near the centers, homoclinic loops and heteroclinic loops, see [11,29], and the book [14]. In this work, we will apply the expansions to study the dynamics of the Abelian integrals on the whole period annulus.
Thus, using the expansions (2.8) and (2.9), we can easily obtain the expansions for I i (h) and its derivative I i (h), which will be used in the following sections.

Asymptotic Expansion of M(h)
By [13,15,16,26], we obtain the expansions of M(h) as follows: and B 00 are constants with B 00 > 0, and The coefficients e i (δ) can be computed by applying the methods developed in [15,16,26], given by and d j (δ) can be obtained by employing the program in [13], (25 a 0 + 15 a 1 + 9 a 2 ) , (1025 a 0 + 435 a 1 + 369 a 2 ) , Similarly, we can easily obtain the expansions of J i (h) and its derivative J i (h) by using the expansions (2.10) and (2.11).

Chebyshev criterion
For convenience of analysis in the next section, we introduce some preliminary results related to Chebyshev criterion.
. . , f n−1 (x) are analytic functions on an real open interval J.
(i) The family of sets {f 0 (x), f 1 (x), . . . , f n−1 (x)} is called a Chebyshev system ( T-system for short) provided that any nontrivial linear combination, has at most n − 1 isolated zeros on J.
. . , f n−1 (x)} is called complete Chebyshev system ( CT-system for short) provided that any nontrivial linear combination, has at most i − 1 zeros for all i = 1, 2, · · · , n. Moreover it is called extended complete Chebyshev system ( ECT-system for short) if the multiplicities of zeros are taken into account.
is an ECT-system on J, so it is a CT-system on J, and thus a T-system on J. However the inverse is not necessary true.
Let res(f 1 , f 2 ) denote the resultant of f 1 (x) and f 2 (x), where f 1 (x) and f 2 (x) are two univariate polynomials of x on rational number field Q. As it is known, res(f 1 (x), f 2 (x)) = 0 if and only if f 1 (x) and f 2 (x) have at least one common root.
Let res(f, g, x) and res(f, g, z) denote respectively the resultants between f (x, z) and g(x, z) with respect to x and z, where f (x, z) and g(x, z) are two polynomials in {x, z} on rational number field. res(f, g, x) is a polynomial in z and res(f, g, z) is a polynomial in x. Regarding the relation between the common roots of two polynomials and their resultants, the following result can be found in many works on polynomial algebra, such as [8]. For completeness we give a short proof.
(ii) Let res(f, g, z) have a unique real root on some open interval (α, β), and res(f, g, x) have a unique real root on some open interval (γ, θ). Then, there exists at most one common real root of f (x, z) and g(x, z) on (α, β) × (γ, θ).
Proof. (ii) is obvious if (i) is true. So we only prove (i). A two-variable polynomial can be treated as one univariate polynomial of one variable with the other treated as a parameter. Taking f (x, z) = f z (x) and g(x, z) = g z (x) which are polynomials of x with parameter z. Let x 0 be the common root of f z0 (x) and g z0 (x), where z 0 is the common root of f x0 (z) and g x0 (z). Then, res(f z0 (x), g z0 (x)) = res(f, g, z 0 ) = 0, and therefore, res(f x0 (z), g x0 (z)) = res(f, g, x 0 ) = 0.
Let H(x, y) = U(x) + y 2 2 be an analytic function. Assume there exists a punctured neighborhood P of the origin foliated by ovals The projection of P on the x-axis is an interval (x l , x r ) with x l < 0 < x r . Under these assumptions, it is easy to verify that xU (x) > 0 for all x ∈ (x l , x r )\{0}, and U (x) has a zero of even multiplicity at x = 0 and there exists an analytic involution z(x), defined by U(x) = U(z(x)), for all x ∈ (x l , x r ). Let where f i (x), i = 0, 1, . . . , n−1, are analytic functions on (x l , x r ) and s ∈ N . Further, define .
Then, we have ). Under the above assumption,

Bifurcation Diagram of A(h)
In this section, we study A(h). We write H(x, y) = y which defines the involution z(x) on the period annulus. We have the following result.
Computing the resultant of Q 1 and q with respect to z, and applying Sturm's Theorem, we can show that Q 1 and q have no common zeros for x ∈ (0, 1). Therefore, W [L 0 ] does not vanish for x ∈ (0, 1). Similarly, we can show that P (x, z) and W [L 0 , L 1 ] do not vanish for x ∈ (0, 1).
Computing the resultant of γ 12 and q with respect to z, and applying Sturm's Theorem, we can prove that γ 12 and q have no common zeros for x ∈ (0, 1). Therefore, γ 12 does not vanish for x ∈ (0, 1). Solving γ 12 (x, z)λ − γ 11 (x, z) = 0 gives for which we have the following result.
Therefore, the double limit cycle curve C is tangent to the line L 1 at DH, and tangent to the line L 2 at DL. Table 1 shows the distribution of the number of zeros of A(h) and the corresponding phase portraits of system (1.2) with α = β = 1 in different regions V i , i = 1, 2, . . . , 5. It should be noted that the singular point (1, 0) is a nilpotent saddle when η +λ+1 = 0 and a degenerated saddle when η +λ+1 = 0. , showing the Hopf bifurcation line L1, the homoclinic bifurcation line L2, the degenerate bifurcation points DL, DH and HL, and the curve C characterized by one zero of multiplicity two.
, in which γ * 11 (x, z) and γ * 12 (x, z) are two-variate polynomials of degree 27 and 26, respectively, and Q * 3 (x, z) is of degree 27. Computing the resultant of P * and q * with respect to z, and applying Sturm's Theorem, we can show that P * and q * have no common zeros for x ∈ (0, 1). Therefore, P * does not vanish, and so W [L * 0 , L * 1 ] and W [L * 0 , L * 1 ] are well defined. Similarly, Q * 1 (x, z) does not vanish by Sturm's theorem, therefore W [L * 0 ] does not vanish for x ∈ (0, 1).

Proof. A direct computation shows that
.
The asymptotic expansions (2.10) and (2.11) yield the values of ϑ(h) at the end of annulus,  ). Proof. We first give a short proof for the first assertion by using an argument of contradiction. Let h * be a zero of ϑ (h) with l multiplicities, l ≥ 2. Then there must exist an η such that η + ϑ(h) has a zero at h = h * with l + 1 (≥ 3) multiplicities. Because J 0 (h) > 0, the relationship between M(h) and η + ϑ(h) implies that M(h) has a zero at h = h * with l + 1 (≥ 3) multiplicities. This contradicts Lemma 4.4.
• DC • HC • DH  Similarly, we can prove that the double limit cycle curve C is tangent to L 1 at DH, and tangent to L 2 at DC. Therefore, system (1.2) with α = 0 and β = 3 5 has one limit cycle in the regions V 1 and V 3 , one limit cycle of multiplicity two on the curve C, two limit cycles in V 5 , and no limit cycles in regions V 2 and V 4 .

Conclusion and Discussion
In this paper, we have successfully analyzed the hyperelliptic Hamiltonian systems (1.2) with three perturbation parameters by using the algebraic criterion and asymptotic property with the help of combination of Abelian integrals. We obtained complete bifurcation diagrams for two cases: α = β = 1 and α = 0, β = 3 5 . The method we developed provides an efficient tool to study zero bifurcation diagrams of Abelian integrals for a wide class of perturbed Hamiltonian systems. The method can indeed deal with general polynomial perturbations, in particular for the systems with the degree of perturbations higher than that of the unperturbed Hamiltonian systems. As an example, consider the following perturbed Hamiltonian system,ẋ = y,ẏ = x(x − 1) 3 + ε(a 0 + a 5 x 5 + a 7 x 7 )y.
We can follow the procedure developed in this paper to obtain the bifurcation diagram, as shown in Figure 5. ) and DH = (0, 0). Zero bifurcation in the regions is as follows: one zero in V1 and V3, one zero of multiplicity two on C, two zeros in V5, and no zeros in V2 and V4.