On the structural stability of planar quasihomogeneous polynomial vector fields

TL;DR

This paper characterizes the structural stability of planar quasihomogeneous polynomial vector fields, extending classical results and providing explicit methods to determine stability within these families.

Contribution

It extends the Hartman-Grobmann Theorem to quasihomogeneous vector fields and classifies their topological stability, including cases not covered in previous homogeneous studies.

Findings

Characterization of structurally stable vector fields in $H_{pqm}$.

Explicit criteria to decide stability before vector field computation.

Identification of cases where $H_{pqm}$ is non-empty but contains no stable fields.

Abstract

Denote by $H_{pqm}$ the space of all planar $(p,q)$-quasihomogeneous vector fields of degree $m$ endowed with the coefficient topology. In this paper we characterize the set $\Omega_{pqm}$ of the vector fields in $H_{pqm}$ that are structurally stable with respect to perturbations in $H_{pqm}$, and determine the exact number of the topological equivalence classes in $\Omega_{pqm}$. The characterisation is applied to give an extension of the Hartman-Grobmann Theorem for such family of planar polynomial vector fields. It follows from the main result in this paper that, for a given $X \in H_{pqm}$ we give a explicit method to decide whether it is structurally stable with respect to perturbation in $H_{pqm}$ before finding the vector field induced by $X$ in the Poincar\'e-Lyapunov sphere. This work is an extension and an improvement of the Llibre-Perez-Rodriguez's paper \cite{LRR}, where the homogeneous case was considered. More precisely, if both $p$ and $q$ are odd, the main results of this paper are similar to those of the Llibre-Perez-Rodriguez's paper; if either $p$ or $q$ is odd while the other is even, we present some results which do not appear in the above mentioned paper. For example, one of the interesting results is that there may be triples $(p,q,m)$ such that $H_{pqm}\not=\emptyset$ but $\Omega_{pqm}=\emptyset$, which does not occur in the homogeneous case.