Essential perturbations of polynomial vector fields with a period annulus

TL;DR

This paper defines essential perturbations in polynomial vector fields and analyzes their structure, extending previous results to higher-degree systems and providing explicit formulas for Melnikov functions.

Contribution

It introduces a general framework for essential perturbations and extends Melnikov function analysis to polynomial systems of degree 2 and 3.

Findings

Explicit structure of the k-th Melnikov function for certain polynomial systems

Generalization of previous quadratic system results to higher degrees

Characterization of essential perturbations for polynomial vector fields

Abstract

In this paper we first give the explicit definition of essential perturbation. Secondly, given a perturbation of a particular family of centers of polynomial differential systems of arbitrary degree for which we explicitly know its Poincar\'e--Liapunov constants, we give the structure of its $k$-th Melnikov function. This result generalizes the result obtained by Chicone and Jacobs for perturbations of degree at most two of any center of a quadratic polynomial system. Moreover we study the essential perturbations for all the centers of the differential systems \[ \dot{x} \, = \, -y + P_{\rm d}(x,y), \quad \dot{y} \, = \, x + Q_{\rm d}(x,y), \] where $P_{\rm d}$ and $Q_{\rm d}$ are homogeneous polynomials of degree ${\rm d}$, for ${\rm d}=2$ and ${\rm d}=3$.