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

Abstract

Abstract. This paper is concerned with the behaviour of the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namelyIn this paper we show that the bifurcation set of the period function of these centers is contained in the rectangle K = (−7, 2)×(0, 4). More concretely, we prove that if (D, F ) / ∈ K, then the period function of the center is monotonically increasing.

Criteria to bound the number of critical periods

“…For instance, see Lemma 5 in [9], any center with a first integral quadratic in y, i.e., of the form H (x, y) = A(x) + B(x)y + C(x)y 2 , and having an integrating factor only depending on x can be brought to a potential system. Since any reversible quadratic center has this property, Theorem A may contribute to prove Chicone's conjecture [2], which claims that these centers have at most two critical periods (see [14,16,21] for partial results on the issue).…”

The cyclicity and period function of a class of quadratic reversible Lotka–Volterra system of genus one

“…Especially, several authors [8,16,17] have shown that all Lotka-Volterra systems of the formsẋ = x(a − by),ẏ = y(cx − d) have monotone period functions. For more result on monotonicity of period function, see [14,15,19,20], etc., and references therein.…”

Bifurcation of critical periods of polynomial systems

“…valid on C M \ V(B), where the a j and b j are rational functions, and where we have applied the same ideas as were used in deriving (28) from (27). An expression of the form (40) is valid on a neighborhood of each (a * , b * ) ∈ C 2 \ F −1 (V(B)).…”

“…For example monotonicity properties were investigated in [3,5,[8][9][10]16,27,35] and finitude of critical periods in [6,26]. An extensive literature is devoted to isochronicity of centers in systems (1), see for example [3,19,21] and the references they contain.…”