On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities

Jaume Llibre; Enrique Ponce; Francisco Torres

doi:https://doi.org/10.1088/0951-7715/21/9/013

On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities

Jaume Llibre, Enrique Ponce, Francisco Torres

Department of Mathematics

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

43 Citations (Scopus)

Abstract

In this paper we study the non-existence and the uniqueness of limit cycles for the Liénard differential equation of the form x″ - f(x)x′ + g(x) = 0 where the functions f and g satisfy xf(x) > 0 and xg(x) > 0 for x ≠ 0 but can be discontinuous at x = 0. In particular, our results allow us to prove the non-existence of limit cycles under suitable assumptions, and also prove the existence and uniqueness of a limit cycle in a class of discontinuous Liénard systems which are relevant in engineering applications. © 2008 IOP Publishing Ltd and London Mathematical Society.

Original language	English
Pages (from-to)	2121-2142
Journal	Nonlinearity
Volume	21
DOIs	https://doi.org/10.1088/0951-7715/21/9/013
Publication status	Published - 1 Sep 2008

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title = "On the existence and uniqueness of limit cycles in Li{\'e}nard differential equations allowing discontinuities",

abstract = "In this paper we study the non-existence and the uniqueness of limit cycles for the Li{\'e}nard differential equation of the form x″ - f(x)x′ + g(x) = 0 where the functions f and g satisfy xf(x) > 0 and xg(x) > 0 for x ≠ 0 but can be discontinuous at x = 0. In particular, our results allow us to prove the non-existence of limit cycles under suitable assumptions, and also prove the existence and uniqueness of a limit cycle in a class of discontinuous Li{\'e}nard systems which are relevant in engineering applications. {\textcopyright} 2008 IOP Publishing Ltd and London Mathematical Society.",

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N2 - In this paper we study the non-existence and the uniqueness of limit cycles for the Liénard differential equation of the form x″ - f(x)x′ + g(x) = 0 where the functions f and g satisfy xf(x) > 0 and xg(x) > 0 for x ≠ 0 but can be discontinuous at x = 0. In particular, our results allow us to prove the non-existence of limit cycles under suitable assumptions, and also prove the existence and uniqueness of a limit cycle in a class of discontinuous Liénard systems which are relevant in engineering applications. © 2008 IOP Publishing Ltd and London Mathematical Society.

AB - In this paper we study the non-existence and the uniqueness of limit cycles for the Liénard differential equation of the form x″ - f(x)x′ + g(x) = 0 where the functions f and g satisfy xf(x) > 0 and xg(x) > 0 for x ≠ 0 but can be discontinuous at x = 0. In particular, our results allow us to prove the non-existence of limit cycles under suitable assumptions, and also prove the existence and uniqueness of a limit cycle in a class of discontinuous Liénard systems which are relevant in engineering applications. © 2008 IOP Publishing Ltd and London Mathematical Society.

U2 - https://doi.org/10.1088/0951-7715/21/9/013

DO - https://doi.org/10.1088/0951-7715/21/9/013

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EP - 2142

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