Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centers

TL;DR

This paper investigates the behavior of Dulac time in saddle-node unfoldings of quadratic centers and demonstrates that certain critical periods do not bifurcate from semi-hyperbolic polycycles.

Contribution

It proves the non-bifurcation of critical periods from semi-hyperbolic polycycles in quadratic centers, advancing understanding of bifurcation phenomena in dynamical systems.

Findings

The derivative of Dulac time tends to negative infinity near specific points.

No bifurcation of critical periods occurs from certain semi-hyperbolic polycycles.

Results are relevant to the bifurcation analysis in Loud's family of quadratic centers.

Abstract

In this paper we consider the unfolding of saddle-node \[ X= \frac{1}{xU_a(x,y)}\Big(x(x^\mu-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \] parametrized by $(\varepsilon,a)$ with $\varepsilon\approx 0$ and $a$ in an open subset $A$ of $\mathbb R^{\alpha},$ and we study the Dulac time $\mathcal T(s;\varepsilon,a)$ of one of its hyperbolic sectors. We prove (Theorem A) that the derivative $\partial_s\mathcal T(s;\varepsilon,a)$ tends to $-\infty$ as $(s,\varepsilon)\to (0^+,0)$ uniformly on compact subsets of $A.$ This result is addressed to study the bifurcation of critical periods in the Loud's family of quadratic centers. In this regard we show (Theorem B) that no bifurcation occurs from certain semi-hyperbolic polycycles.