On the Chebyshev property for a new family of functions

Abstract

Preprin

Periodic orbits from second order perturbation via rational trigonometric integrals

“…In the last proof, an alternative study for the number of zeros of F i (r), defined in (20), can also be made following [12]. In this reference, functions of type (20) have been considered and it is proved that they have as many zeros as the number of monomials minus one.…”

“…In this section we perturb systems S 1 , S 2 and S 3 given in (12). For these families, there are birational changes of variables, (u, v) = ϕ(x, y), so that system (11) is written as…”

“…Let F 1 and F 2 be the first and second Poincaré-PontryaginMelnikov functions associated with the quadratic perturbation of each differential equation in (12). The maximum number of zeros of F 1 , taking into account their multiplicity, is N 1 (n) and of F 2 (when…”

“…The proof is obtained by a case by case study. That is, taking (P 0 , Q 0 ) in (11) as the corresponding one in each family of (12) and considering perturbations of degrees n = 0, 1, 2.…”

“…Consider a perturbed system as (11) for each differential equation given in (12). Then each family, using up to a second order study in ε, has no more than two hyperbolic periodic orbits under quadratic perturbations and this number is sharp.…”