Number of limit cycles for planar systems with invariant algebraic curves
Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Giacomini, Hector (Centre National de la Recherche Scientifique. Université de Tours. Institut Denis Poisson)

Date:	2023
Abstract:	For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non-existence of periodic orbits not contained in this given algebraic curve. When the method is applied to parametric families of polynomial systems that have limit cycles for some values of the parameters, our result leads to effective algebraic conditions on the parameters that force non-existence of the periodic orbits. As applications we consider several families of quadratic systems: the ones having some quadratic invariant algebraic curve, the known ones having an algebraic limit cycle, a family having a cubic invariant algebraic curve and other ones. For any quadratic system with two invariant algebraic curves we prove a finiteness result for its number of limit cycles that only depends on the degrees of these curves. We also consider some families of cubic systems having either a quadratic or a cubic invariant algebraic curve and a family of Liénard systems. We also give a new and simple proof of the known fact that quadratic systems with an invariant parabola have at most one limit cycle. In fact, what we show is that this result is a consequence of the similar result for quadratic systems with an invariant straight line.
Grants:	Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1617 Agencia Estatal de Investigación CEX2020-001084-M Agencia Estatal de Investigación PID2019-104658GB-I00
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Invariant algebraic curve ; Limit cycle ; Periodic orbit ; Quadratic system ; Cubic system ; Liénard system
Published in:	Qualitative theory of dynamical systems, Vol. 22, Issue 2 (June 2023) , art. 44, ISSN 1662-3592

DOI: 10.1007/s12346-023-00746-7

Available from: 2024-06-30 Postprint

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Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
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