Quartic rigid systems in the plane and in the Poincar\'e sphere

TL;DR

This paper investigates a specific family of planar and spherical rigid systems defined by polynomial functions, analyzing their centers, singular points, and limit cycles to understand their qualitative behavior.

Contribution

It introduces and studies a new class of quartic rigid systems with no rotatory parameters, providing insights into their global phase portrait on the plane and Poincaré sphere.

Findings

Characterization of centers and singular points.

Analysis of limit cycles in the family.

Extension of the system to the Poincaré sphere.

Abstract

We consider the planar family of rigid systems of the form $x'=-y+xP(x,y), y'=x+yP(x,y)$, where $P$ is any polynomial with monomials of degree one and three. This is the simplest non-trivial family of rigid systems with no rotatory parameters. The family can be compactified to the Poincar\'e sphere such that the vector field along the equator is not identically null. We study the centers, singular points and limit cycles of that family on the plane and on the sphere.