Fixed and moving limit cycles for Liénard equations
Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Sabatini, Marco (Università di Trento. Dipartimento di Matematica)

Date:	2019
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.
Grants:	Ministerio de Economía y Competitividad MTM2016-77278-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1617
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Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Liénard equation ; Limit cycle ; Bifurcations ; Invariant algebraic curve
Published in:	Annali di matematica pura ed applicata, Vol. 198, Issue 6 (December 2019) , p. 1985-2006, ISSN 1618-1891

DOI: 10.1007/s10231-019-00850-z

Post-print 23 p, 782.4 KB

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