On the structural instability of non-hyperbolic limit cycles on planar polynomial vector fields

TL;DR

This paper proves that non-hyperbolic limit cycles of odd degree are also structurally unstable in planar polynomial vector fields under Whitney's topology, extending known results from even degree cases.

Contribution

It establishes the structural instability of non-hyperbolic limit cycles of odd degree in polynomial vector fields, filling a gap in the understanding of limit cycle stability.

Findings

Non-hyperbolic limit cycles of odd degree are structurally unstable.

Extends known instability results from even to odd degree polynomial limit cycles.

Uses Whitney's topology to analyze stability properties.

Abstract

It is known that non-hyperbolic limit cycles are structurally unstable in the set of planar smooth and analytical vector fields. In the polynomial case, it is known only that limit cycles of even degree are structurally unstable. In this paper, we prove that non-hyperbolic limit cycles of odd degree are also structurally unstable in the polynomial case, if we consider Whitney's topology.