Isochronicity conditions for some planar polynomial systems II

Magali Bardet (1) , Islam Boussaada (2) , A. Raouf Chouikha , Jean-Marie Strelcyn (2, 3)

1 LITIS - Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes
2 LMRS - Laboratoire de Mathématiques Raphaël Salem
3 LAGA - Laboratoire Analyse, Géométrie et Applications

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 Liénard 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 [I. Boussaada, A.R. Chouikha, J-M. Strelcyn, Isochronicity conditions for some planar polynomial systems, To appear in Bulletins des Sciences Mathématiques, 2010] but can be read independantly.

Mots clés: lin- earizability; polynomial systems; center; isochronicity; Lie ́nard type equation; Urabe function; lin- earizability.

Domaines: Analyse classique [math.CA]; Systèmes dynamiques [math.DS]

Magali Bardet, Islam Boussaada, A. Raouf Chouikha, Jean-Marie Strelcyn. Isochronicity conditions for some planar polynomial systems II. Bulletin des Sciences Mathématiques, 2011, 135 (2), pp.230-249. ⟨10.1016/j.bulsci.2010.12.003⟩. ⟨hal-00533275⟩