Qualitative planar dynamics with star nodes and homogeneous nonlinearities: beyond Hilbert's $16^{th}$ problem

TL;DR

This paper thoroughly analyzes polynomial planar vector fields with star nodes and homogeneous nonlinearities, extending previous limit cycle studies and providing detailed phase portraits for specific cases.

Contribution

It offers a comprehensive study of such vector fields beyond Hilbert's 16th problem, including explicit phase portraits for quadratic and cubic nonlinearities.

Findings

Extended understanding of limit cycle existence and number

Provided explicit phase portraits for degree 2 and 3 nonlinearities

Generalized results for polynomial vector fields with star nodes

Abstract

This is a full study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part is a homogeneous polynomial of arbitrary degree $n>1$. It extends previous work by other authors that was mainly concerned with the existence and number of limit cycles. The general results are also applied to two classes of examples where the nonlinearities have degrees 2 and 3, for which we provide a set of phase portraits.