Unfolding of a Quadratic Integrable System with a Homoclinic Loop

Four Limit Cycles From Perturbing Quadratic Integrable Systems by Quadratic Polynomials

“…ε=0 is a reversible integrable system. It has been noted that in all the cases considered in [23,24,25,26], the parameters a and c were chosen as a = − 3, c = − 2, but with b = 1 in [23]; b = − 1 in [24], b ∈ (− ∞, −1) ∪ (−1, 0) in [25], and b ∈ (0, 2) in [26]. In these papers, complete analysis on the perturbation parameters was carried out with the aid of Poincaré transformation and the Picard-Fuchs equation, but it needed to fix all (or most of) the parameters a, b and c. This way it may miss opportunity to find more limit cycles, such as possible existence of 4 limit cycles.…”

Quadratic perturbations of a quadratic reversible center of genus one

“…Substituting (32) into the third equality in (26), we get Further differentiating (16) with respect to h, we get…”

“…To our knowledge, known results are very limited. Let us list here some papers concerning the quadratic perturbations from the reversible systems: [3] for the isochronous centers; [17] for the unbounded heteroclinic loop; [6,11,12,16,18] for system (3) with a = −3 and different b; [2] with a = −4 and a = 2, 0 < b < 2; [5] with a = −1/2, 0 < b < 2; [15] with a = −3/2, b ∈ (−∞, 0] ∪ 2.…”

Bounding the number of zeros of certain Abelian integrals

“…For instance, in some papers (e.g. [3,15,21]) the authors study the geometrical properties of the so-called centroid curve using that it verifies a Riccati equation (which is itself deduced from a Picard-Fuchs system). In other papers (e.g.…”