A family of isochronous foci with Darboux first integral

Jaume Giné and Jaume Llibre

Vol. 218 (2005), No. 2, 343–355

DOI: 10.2140/pjm.2005.218.343

Abstract

We consider the class of polynomial differential equations ẋ = λx − y + P_n(x,y) + P_2n−1(x,y), ẏ = x + λy + Q_n(x,y) + Q_2n−1(x,y) with n ≥ 2, where P_i and Q_i are homogeneous polynomials of degree i. These systems have a focus at the origin if λ≠0, and have either a center or a focus if λ = 0. Inside this class we identify a new subclass of Darboux integrable systems having either a focus or a center at the origin. Under generic conditions such Darboux integrable systems can have at most two limit cycles, and when they exist are algebraic. For the case n = 2 and n = 3 we present new classes of Darboux integrable systems having a focus.

Keywords

integrability, algebraic limit cycle, focus, center

Mathematical Subject Classification 2000

Primary: 34C35, 34D30

Milestones

Received: 11 April 2003

Published: 1 February 2005