Existence of limit cycles for real quadratic differential systems with an invariant cubic

Javier Chavarriga and Isaac A. García

Vol. 223 (2006), No. 2, 201–218

DOI: 10.2140/pjm.2006.223.201

Abstract

This work is part of a wider study of the significance of the existence of invariant algebraic curves for planar polynomial differential systems. The class of real quadratic systems having a cubic invariant algebraic curve is examined. Using affine canonical forms for the members of this class we show that no system of this type has limit cycles except for two cases. For these cases, concrete examples are given with a limit cycle. We also include a simple and short proof on the nonexistence of quadratic systems with an algebraic limit cycle of third degree.

Keywords

polynomial vector fields, limit cycle, Dulac function, invariant algebraic curves

Mathematical Subject Classification 2000

Primary: 34C05

Secondary: 34C23, 37G15

Milestones

Received: 25 May 2001

Revised: 15 November 2001

Accepted: 13 July 2005

Published: 1 February 2006