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

Armengol Gasull; Hector Giacomini

doi:https://doi.org/10.2140/pjm.2006.226.277

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

Armengol Gasull, Hector Giacomini

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

10 Citations (Scopus)

Abstract

Consider the differential equation x = y, y = 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.

Original language	English
Pages (from-to)	277-296
Journal	Pacific Journal of Mathematics
Volume	226
DOIs	https://doi.org/10.2140/pjm.2006.226.277
Publication status	Published - 1 Aug 2006

Keywords

Bendixson-Dulac criterion
Limit cycle
Linear ordinary differential equation
Ordinary differential equation

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abstract = "Consider the differential equation x = y, y = 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.",

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author = "Armengol Gasull and Hector Giacomini",

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AU - Giacomini, Hector

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N2 - Consider the differential equation x = y, y = 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.

AB - Consider the differential equation x = y, y = 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.

KW - Bendixson-Dulac criterion

KW - Limit cycle

KW - Linear ordinary differential equation

KW - Ordinary differential equation

U2 - https://doi.org/10.2140/pjm.2006.226.277

DO - https://doi.org/10.2140/pjm.2006.226.277

M3 - Article

VL - 226

SP - 277

EP - 296

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

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