TAILIEUCHUNG - The cyclicity of the period annulus of a quadratic reversible system with one center of genus one
This paper is concerned with a quadratic reversible and non-Hamiltonian system with one center of genus one. By using the properties of related elliptic integrals and the geometry of some planar curves defined by them, we prove that the cyclicity of the period annulus of the considered system under small quadratic perturbations is two. This verifies Gautier’s conjecture about the cyclicity of the related period annulus. | Turk J Math 35 (2011) , 667 – 685. ¨ ITAK ˙ c TUB doi: The cyclicity of the period annulus of a quadratic reversible system with one center of genus one ∗ Linping Peng and Yannan Sun Abstract This paper is concerned with a quadratic reversible and non-Hamiltonian system with one center of genus one. By using the properties of related elliptic integrals and the geometry of some planar curves defined by them, we prove that the cyclicity of the period annulus of the considered system under small quadratic perturbations is two. This verifies Gautier’s conjecture about the cyclicity of the related period annulus. Key Words: Cyclicity, bifurcation of limit cycles, quadratic perturbations, period annulus, a quadratic reversible system with one center of genus one 1. Introduction and statement of the main result It is well known that the weak Hilbert 16th problem asks for the least upper bound of the number of zeros of the associated Abelian integral. This problem in the quadratic Hamiltonian case has already been solved, that is, the least upper bound of the number of zeros of the Abelian integrals associated with quadratic Hamiltonian systems under quadratic perturbations is two; see [9, 19, 7, 2, 14, 3] and the references therein. The next natural step is to consider quadratic reversible but non-Hamiltonian systems. Form [10], the quadratic reversible systems can be written in the real form x˙ = y + (a + b + 2)x2 − (a + b − 2)y2 , y˙ = −x[1 − 2(a − b)y], () where a, b ∈ R . If c = a − b = 0 , we can make the transformation (x, y, t) → (x/c, y/c, −t), and let a = −(a + b + 2)/(a − b), b = (a + b − 2)/(a − b), then system () becomes the following x˙ = −y + ax2 + by2 , y˙ = x(1 − 2y). () Studies show that the orbital topological properties of quadratic reversible systems under quadratic perturbations are very rich. Most mathematicians working in this field believe that the weak Hilbert 16th problem for quadratic reversible systems is .
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