Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: General setting

Bifurcations of Zeros in Translated Families of Functions and Applications

“…On the other hand, in our study of critical points of the period function of Loud systems [6,7,9,10,13] we gave a conjectural bifurcation diagram. We could not prove it in full generality due in part to some phenomena of breaking of separatrices of the polycycle bounding the period annulus.…”

“…F −1 and applying Theorem A of [13] we obtain T − (s) = T 00 + T 01 s r + T 101 sω ᾱ (s) + T 100 s + F ∞ 3/2−δ (s) for every δ > 0. On the other hand, half of the Dulac time of the finite singular point at (−1/D, 0) has the form D, F ) be the breaking parameter of the connexion, where U(D, F ) > 0 by (9).…”

The criticality of reversible quadratic centers at the outer boundary of its period annulus

“…Let us remark at this point that if Chicone's conjecture about the number of critical periodic orbits of the quadratic centers is true then Crit (Π ν0 , X ν0 ), X ν 2 for all ν ∈ R 2 , see (2). In this paper, by applying our recent results from [29,30,31], we prove the following:…”

“…Our main theorem follows by obtaining the asymptotic expansion to fourth order and computing its coefficients, which are not polynomial in ν but transcendental (more concretely, they are hypergeometric functions). To this end we strongly rely on the tools that we develop in our recent papers [29,30,31]. The results that we obtain in the present paper can be viewed conceptually as the analogue for the outer boundary of the work carried out by Chicone and Jacobs in [6] on the bifurcation of critical periodic orbits from the inner boundary of the quadratic centers.…”

Simultaneous Bifurcation of Limit Cycles and Critical Periods

“…The greatest difficulty to deal with is the fact that the outer boundary of the period annulus changes together with the parameters inside a fixed family, see for example [15,33]. Hence, the usual perturbation techniques are not useful and new tools need to be developed ( [27]). As we have described, the maximal number of zeros of the derivative of the period function under perturbation (in some fixed class) is known as the criticality of the center.…”