We consider a family of planar vector fields that writes as a Liénard system in suitable coordinates. It has a fixed closed invariant curve that often contains periodic orbits of the system. We prove a general result that gives the hyperbolicity of these periodic orbits, and we also study the coexistence of them with other periodic orbits. Our family contains the celebrated Wilson polynomial Liénard equation, as well as all polynomial Liénard systems having hyperelliptic limit cycles. As an illustrative example, we study in more detail a natural 1-parametric extension of Wilson example. It has at least two limit cycles, one of them fixed and algebraic and the other one moving with the parameter, presents a transcritical bifurcation of limit cycles and for a given parameter has a non-hyperbolic double algebraic limit cycle. In order to prove that for some values of the parameter the system has exactly two hyperbolic limit cycles, we use several suitable Dulac functions.

Fixed and moving limit cycles for Liénard equations / Sabatini, Marco; Gasull, Armengol. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - STAMPA. - 2019, 198:6(2019), pp. 1985-2006. [10.1007/s10231-019-00850-z]

Fixed and moving limit cycles for Liénard equations

Sabatini Marco;
2019-01-01

Abstract

We consider a family of planar vector fields that writes as a Liénard system in suitable coordinates. It has a fixed closed invariant curve that often contains periodic orbits of the system. We prove a general result that gives the hyperbolicity of these periodic orbits, and we also study the coexistence of them with other periodic orbits. Our family contains the celebrated Wilson polynomial Liénard equation, as well as all polynomial Liénard systems having hyperelliptic limit cycles. As an illustrative example, we study in more detail a natural 1-parametric extension of Wilson example. It has at least two limit cycles, one of them fixed and algebraic and the other one moving with the parameter, presents a transcritical bifurcation of limit cycles and for a given parameter has a non-hyperbolic double algebraic limit cycle. In order to prove that for some values of the parameter the system has exactly two hyperbolic limit cycles, we use several suitable Dulac functions.
2019
6
Sabatini, Marco; Gasull, Armengol
Fixed and moving limit cycles for Liénard equations / Sabatini, Marco; Gasull, Armengol. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - STAMPA. - 2019, 198:6(2019), pp. 1985-2006. [10.1007/s10231-019-00850-z]
File in questo prodotto:
File Dimensione Formato  
GasSab2018_preprint.pdf

accesso aperto

Descrizione: Preprint dell'Univ. Aut. Barcelona dell'articolo Fixed and moving limit cycles for Liénard equations
Tipologia: Pre-print non referato (Non-refereed preprint)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 405.21 kB
Formato Adobe PDF
405.21 kB Adobe PDF Visualizza/Apri
Gasull-Sabatini2019_Article_FixedAndMovingLimitCyclesForLi.pdf

Solo gestori archivio

Tipologia: Versione editoriale (Publisher’s layout)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 420.46 kB
Formato Adobe PDF
420.46 kB Adobe PDF   Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/233437
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 4
social impact