Bifurcation diagram and stability for a one-parameter family of planar vector fields

TL;DR

This paper analyzes a specific family of planar quintic systems, refining the parameter range for limit cycle existence, proving hyperbolicity, and exploring stability and polycycles using the Bendixon-Dulac theorem.

Contribution

It provides a new unified proof of previous results, narrows the parameter interval for limit cycles, and establishes hyperbolicity and stability properties.

Findings

Limit cycle exists only for m in (0.547, 0.6)

Limit cycle is hyperbolic

Explicit bounds for basin of attraction of the origin

Abstract

We consider the 1-parameter family of planar quintic systems, $\dot x= y^3-x^3$, $\dot y= -x+my^5$, introduced by A. Bacciotti in 1985. It is known that it has at most one limit cycle and that it can exist only when the parameter $m$ is in $(0.36,0.6)$. In this paper, using the Bendixon-Dulac theorem, we give a new unified proof of all the previous results, we shrink this to $(0.547,0.6)$, and we prove the hyperbolicity of the limit cycle. We also consider the question of the existence of polycycles. The main interest and difficulty for studying this family is that it is not a semi-complete family of rotated vector fields. When the system has a limit cycle, we also determine explicit lower bounds of the basin of attraction of the origin. Finally we answer an open question about the change of stability of the origin for an extension of the above systems.