The general Li\'enard polynomial system

TL;DR

This paper introduces a geometric bifurcation approach to determine the maximum number of limit cycles in general Li'enard polynomial systems, advancing understanding of Hilbert's sixteenth problem.

Contribution

It provides a novel geometric method to solve the limit cycle problem for arbitrary Li'enard polynomial systems with multiple singular points.

Findings

Determined maximum limit cycles around a singular point.

Extended results to systems with multiple singular points.

Linked findings to Hilbert's sixteenth problem.

Abstract

In this paper, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve first the problem on the maximum number of limit cycles surrounding a unique singular point for an arbitrary polynomial system. Then, by means of the same bifurcationally geometric approach, we solve the limit cycle problem for a general Li\'enard polynomial system with an arbitrary (but finite) number of singular points. This is related to the solution of Hilbert's sixteenth problem on the maximum number and relative position of limit cycles for planar polynomial dynamical systems.