Upper bounds for the number of limit cycles through linear differential equations

Armengol Gasull and Hector Giacomini

Vol. 226 (2006), No. 2, 277–296

DOI: 10.2140/pjm.2006.226.277

Abstract

Consider the differential equation ẋ = y, ẏ = h₀(x) + h₁(x)y + h₂(x)y² + y³ in the plane. We prove that if a certain solution of an associated linear ordinary differential equation does not change sign, there is an upper bound for the number of limit cycles of the system. The main ingredient of the proof is the Bendixson–Dulac criterion for ℓ-connected sets. Some concrete examples are developed.

Keywords

ordinary differential equation, limit cycle, Bendixson–Dulac criterion, linear ordinary differential equation

Mathematical Subject Classification 2000

Primary: 34C07, 34C05, 34A30, 37C27

Milestones

Received: 20 November 2004

Accepted: 11 January 2005

Published: 1 August 2006