The period function for quadratic integrable systems with cubic orbits

Abstract

This paper is concerned with the monotonicity of the period function for families of quadratic systems with centers whose orbits lie on cubic planar curves. It is proved that each such system has a period function with at most one critical point.  2004 Elsevier Inc. All rights reserved.

Criteria to bound the number of critical periods

“…To our knowledge, the key point in almost all the results appearing in the literature dealing with a family of centers with critical periods is that the period function satisfies some kind of Picard-Fuchs differential equation. Let us quote for instance the works of Yulin Zhao for two different families of quadratic centers [24,25] or the papers of Chow and Sanders [4] and Gavrilov [10] on the family of cubic potential centers.…”

The bifurcation set of the period function of the dehomogenized Loud's centers is bounded

“…By using a Picard-Fuchs approach, Yulin Zhao [17,18] describes completely the behaviour of the period function in F = 3/2 and F = 2. Chouikha [3] shows the monotonicity in the straight lines F + 2D = 1 and F = −1 and some segments inside D = −1/2, D = 0, F = 1 and F = 2.…”

On the reversible quadratic centers with monotonic period function

“…Thus, by using a Picard-Fuchs approach, Y. Zhao can completely describe the behaviour of the period function in two straight lines of the parameter plane, namely F = 3/2 in [24] and F = 2 in [25]. There is a number of different authors that have treated the general question of monotonicity of the period function (see [2,3,8,14,17,22] and references there in).…”