On the cyclicity of hyperbolic polycycles

TL;DR

This paper investigates the cyclicity of hyperbolic polycycles in planar vector fields, providing explicit conditions based on hyperbolicity ratios, extending previous results, and demonstrating polynomial perturbations that achieve these bounds.

Contribution

It offers new explicit criteria for the cyclicity of hyperbolic polycycles and improves existing proofs by leveraging recent advances in Dulac map regularity, also exploring polynomial perturbations and applications.

Findings

Explicit conditions for cyclicity based on hyperbolicity ratios

Extension and refinement of previous cyclicity results

Existence of polynomial perturbations achieving cyclicity bounds

Abstract

Let $X$ be a planar smooth vector field with a polycycle $\Gamma^n$ with $n$ sides and all its corners, that are at most $n$ singularities, being hyperbolic saddles. In this paper we study the cyclicity of $\Gamma^n$ in terms of the hyperbolicity ratios of these saddles, giving explicit conditions that ensure that it is at least $k,$ for any $k\leqslant n.$ Our result extends old results and also provides a more accurate proof of the known ones because we rely on some recent powerful works that study in more detail the regularity with respect to initial conditions and parameters of the Dulac map of hyperbolic saddles for families of vector fields. We also prove that when $X$ is polynomial there is a polynomial perturbation (in general with degree much higher that the one of $X$) that attains each of the obtained lower bounds for the cyclicities. Finally, we also study some related inverse problems and provide concrete examples of applications in the polynomial world.