Centers of reversible cubic perturbations of the symmetric 8-loop Hamiltonian
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Archiv der Mathematik
Centers of reversible cubic perturbations of the symmetric 8-loop Hamiltonian Fatma Sassi, Ameni Gargouri, Lubomir Gavrilov, and Bassem Ben Hamed
Abstract. We show that the center set of reversible cubic systems, close to the symmetric Hamiltonian system x = y, y = x−x3 , has two irreducible components of co-dimension two in the parameter space. One of them corresponds to the Hamiltonian stratum, the other to systems which are a polynomial pull back of an appropriate linear system. Mathematics Subject Classification. Primary 37F75; Secondary 34C14. Keywords. Center-focus problem, Cubic vector field, Limit cycles.
1. Introduction. The paper is a contribution to the study of the center set of plane polynomial cubic vector fields, in a neighbourhood of the symmetric Hamiltonian cubic vector field X0 x˙ = y, X0 : (1.1) y˙ = x − x3 . X0 is Hamiltonian, and has a first integral 1 1 1 H(x, y) = y 2 − x2 + x4 . 2 2 4 Under a small analytic perturbation, the centers of X0 near (1, 0) and (−1, 0) are either simultaneously destroyed, or simultaneously persistent. The set of cubic vector fields close to X0 and having a center near (±1, 0) is the center set C0 . It is known that the center set is an algebraic set in the space of parameters, but even the number of its irreducible components is unknown. Recently, Iliev, Li, and Yu [4] studied special one-parameter families of perturbations of the form x˙ = y + εP (x, y), (1.2) Xε : y˙ = x − x3 + εQ(x, y),
Arch. Math.
F. Sassi et al.
where P, Q are arbitrary fixed real cubic polynomials. The displacement function near the singular points (±1, 0) has an analytic expansion d(h, ε) = εM1 (h) + ε2 M2 (h) + ε3 M3 (h) + · · ·
(1.3)
where as usual h is the restriction of the Hamiltonian H on a cross-section of the vector field. The so called Melnikov functions Mk vanish if and only if the displacement map is the zero map, that is to say, the centers (1, 0) and (−1, 0) are simultaneously persistent. It was shown then in [4, Theorem 1] that if M1 = M2 = M3 = M4 = 0, then the displacement map is identically zero and therefore Xε has a center near (1, 0) and (−1, 0). The set of such a system is an algebraic set C l contained in C. It turns out that C l is a union of vector spaces, and a vector field Xε which belongs to an irreducible component of C l is either Hamiltonian, or y-reversible, or x-reversible. It is clear that when a vector field Xε is Hamiltonian, or y-reversible, then it has a center near (±1, 0). If, however, the vector field is x-reversible, it does not follow that it has a center near (±1, 0). Therefore, it makes sense to consider the case of x-reversible systems separately. The purpose of this paper is to give a complete description of the center set C under the restriction that the vector field is x-reversible, that is to say, the associated foliation by orbits is invariant under the involution (x, y) → (−x, y). Our approach is the following. The invariance under x → −x suggests to introduce the quotient vector field which is still polynomial an
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