Isochronicity conditions for some planar polynomial systems II

TL;DR

This paper identifies 36 new families of isochronous centers in certain planar polynomial systems using the C-algorithm, and provides explicit formulas for linearization when the Urabe function is zero.

Contribution

It extends previous work by classifying new isochronous centers for polynomial systems of degree up to 4, and offers explicit linearization formulas in special cases.

Findings

Found 36 new families of isochronous centers.

Provided explicit linearization formulas when Urabe function is zero.

Extended classification of isochronous centers in polynomial systems.

Abstract

We study the isochronicity of centers at $O\in \mathbb{R}^2$ for systems $$\dot x=-y+A(x,y),\;\dot y=x+B(x,y),$$ where $A,\;B\in \mathbb{R}[x,y]$, which can be reduced to the Li\'enard type equation. When $deg(A)\leq 4$ and $deg(B) \leq 4$, using the so-called C-algorithm we found $36$ new families of isochronous centers. When the Urabe function $h=0$ we provide an explicit general formula for linearization. This paper is a direct continuation of \cite{BoussaadaChouikhaStrelcyn2010} but can be read independantly.