Essential perturbations of polynomial vector fields with a period annulus

Adriana Buică; Jaume Giné; Maite Grau

doi:10.3934/cpaa.2015.14.1073

Article Contents

2015, Volume 14, Issue 3: 1073-1095. Doi: 10.3934/cpaa.2015.14.1073

This issue Previous Article Asymptotic behavior for the unique positive solution to a singular elliptic problem Next Article Klein-Gordon-Maxwell equations in high dimensions

Essential perturbations of polynomial vector fields with a period annulus

1.
Department of Applied Mathematics, Babeş-Bolyai University, 1 Kogălniceanu str., Cluj-Napoca, 400084
2.
Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida

Received: September 30, 2014

Revised: December 31, 2014

Published: April 2015

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Abstract

Chicone--Jacobs and Iliev found the essential perturbations of quadratic systems when considering the problem of finding the cyclicity of a period annulus. Given a perturbation of a particular family of centers of polynomial differential systems of arbitrary degree for which the expressions of its Poincaré--Liapunov constants are known, we give the structure of its $k$-th Melnikov function. This allows to find the essential perturbations in concrete cases. We study here in detail the essential perturbations for all the centers of the differential systems \begin{eqnarray} \dot{x} = -y + P_{\rm d}(x,y), \quad \dot{y} = x + Q_{d}(x,y), \end{eqnarray} where $P_d$ and $Q_d$ are homogeneous polynomials of degree $d$, for $ d=2$ and $ d=3$.

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Mathematics Subject Classification: Primary: 34C23, 37G15; Secondary: 34C05, 34C25, 34C07.

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