One Upper Estimate on the Number of Limit Cycles of Even Degree Li\'enard Equations in the Focus Case

TL;DR

This paper establishes an explicit upper bound on the number of limit cycles for even-degree Lie9nard equations with a focus singularity at the origin, advancing understanding of their oscillatory behavior.

Contribution

It provides the first explicit upper bound for limit cycles in even-degree Lie9nard equations with a focus, filling a gap in the qualitative theory of these systems.

Findings

Derived an explicit upper bound for limit cycles

Applied to Lie9nard equations of even degree

Enhanced understanding of oscillatory dynamics in these systems

Abstract

We give an explicit upper bound for a number of limit cycles of the Li\'enard equation $\dot{x}=y-F(x)$, $\dot{y}=-x$ of even degree in the case its unique singular point $(0,0)$ is a focus.