# A criticality result for polycycles in a family of quadratic reversible centers

## TL;DR

This paper investigates the criticality of polycycles in a family of quadratic reversible centers, specifically analyzing how many critical periodic orbits can emerge or disappear at the boundary of the period annulus, and identifies parameters with criticality one.

## Contribution

It provides the first proof that a subset of bifurcation parameters in this family has criticality exactly one, advancing understanding of bifurcation phenomena in quadratic reversible centers.

## Key findings

- Identified parameters with criticality equal to one.
- Extended the understanding of bifurcation behavior in quadratic reversible centers.
- Established new results on the criticality of polycycles at bifurcation points.

## Abstract

We consider the family of dehomogenized Loud's centers $X_{\mu}=y(x-1)\partial_x+(x+Dx^2+Fy^2)\partial_y,$ where $\mu=(D,F)\in\mathbb{R}^2,$ and we study the number of critical periodic orbits that emerge or dissapear from the polycycle at the boundary of the period annulus. This number is defined exactly the same way as the well-known notion of cyclicity of a limit periodic set and we call it criticality. The previous results on the issue for the family $\{X_{\mu},\mu\in\mathbb{R}^2\}$ distinguish between parameters with criticality equal to zero (regular parameters) and those with criticality greater than zero (bifurcation parameters). A challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters, which form a set $\Gamma_{B}$ of codimension 1 in $\mathbb{R}^2$. In the present paper we succeed in proving that a subset of $\Gamma_{B}$ has criticality equal to one.