Global planar dynamics with a star node and contracting nonlinearity

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, \({\textbf {Z}}_2\oplus {\textbf {Z}}_2\) symmetric systems and nonlinearities of degree 3, for which we provide complete sets of phase-portraits.