Global planar dynamics with a star node and contracting nolinearity

TL;DR

This paper provides a comprehensive classification of planar polynomial vector fields with a star node and contracting nonlinearities, revealing invariant circles and detailed phase portraits for various symmetric and degree-specific cases.

Contribution

It introduces a complete invariant for the dynamics of such systems, extending previous work by classifying phase portraits beyond limit cycle analysis.

Findings

Existence of invariant circles in the studied systems

Complete classification of phase portraits for specific nonlinearities

Application of results to symmetric and degree-3 polynomial systems

Abstract

This is a complete study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part is a contracting homogeneous polynomial. The contracting nonlinearity provides the existence of an invariant circle and allows us to obtain a classification through a complete invariant for the dynamics, extending previous work by other authors that was mainly concerned with the existence and number of limit cycles. The general results are also applied to some classes of examples: definite nonlinearities, $\ZZ_2\oplus\ZZ_2$ symmetric systems and nonlinearities of degree 3, for which we provide complete sets of phase portraits.