On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry

Jaume Llibre; Manuel Ordóñez; Enrique Ponce

doi:10.1016/j.nonrwa.2013.02.004

On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry

Jaume Llibre, Manuel Ordóñez, Enrique Ponce

Department of Mathematics

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

54 Citations (Scopus)

Abstract

Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems. New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results. A relevant application to the McKean piecewise linear model of a single neuron activity is included. © 2013 Elsevier Ltd. All rights reserved.

Original language	English
Pages (from-to)	2002-2012
Journal	Nonlinear Analysis: Real World Applications
Volume	14
Issue number	5
DOIs	https://doi.org/10.1016/j.nonrwa.2013.02.004
Publication status	Published - 1 Oct 2013

Keywords

Liénard equation
Limit cycles
Piecewise linear systems

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abstract = "Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems. New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results. A relevant application to the McKean piecewise linear model of a single neuron activity is included. {\textcopyright} 2013 Elsevier Ltd. All rights reserved.",

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N2 - Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems. New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results. A relevant application to the McKean piecewise linear model of a single neuron activity is included. © 2013 Elsevier Ltd. All rights reserved.

AB - Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems. New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results. A relevant application to the McKean piecewise linear model of a single neuron activity is included. © 2013 Elsevier Ltd. All rights reserved.

KW - Liénard equation

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KW - Piecewise linear systems

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