Stability of singular limit cycles for Abel equations
Bravo, Jose Luis (Universidad de Extremadura. Departamento de Matemáticas)
Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Fernández, Manuel (Universidad de Extremadura. Departamento de Matemáticas)

Date:	2015
Abstract:	We obtain a criterion for determining the stability of singular limit cycles of Abel equations x = A(t)x3 + B(t)x2 . This stability controls the possible saddle-node bifurcations of limit cycles. Therefore, studying the Hopf-like bifurcations at x = 0, together with the bifurcations at infinity of a suitable compactification of the equations, we obtain upper bounds of their number of limit cycles. As an illustration of this approach, we prove that the family x = at(t−tA )x3 +b(t−tB )x2 , with a, b > 0, has at most two positive limit cycles for any tB , tA .
Grants:	Ministerio de Ciencia y Tecnología MTM 2011-22751 Ministerio de Ciencia y Tecnología MTM 2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-410
Note:	Agraïments: FEDER-Junta Extremadura grant number GR10060
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Abel equation ; Closed solution ; Limit cycles ; Periodic solutions
Published in:	Discrete and continuous dynamical systems. Series A, Vol. 35 Núm. 5 (2015) , p. 1873-1890, ISSN 1553-5231

DOI: 10.3934/dcds.2015.35.1873

Postprint 20 p, 413.0 KB

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