Non-equivalence between the Melnikov and the averaging methods for nonsmooth differential systems
Guo, Zhifei (Sichuan University. School of Mathematics)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2022
Abstract:	It is known that for smooth differential systems in the plane R2 the Melnikov and the averaging methods for studying the limit cycles produce the same results. Here we prove that this is not the case for nonsmooth differential systems in the plane. More precisely, we prove that the linear system x˙=y, y˙=−x, can produce at most 5 crossing limit cycles using the averaging theory of first order and also produce at most 5 crossing limit cycles using the averaging theory of second order, when it is perturbed by discontinuous piecewise polynomials of two pieces separated by the cubic curve y=x3, and having in each piece a quadratic polynomial differential system. While using the Melnikov theory up to second order these discontinuous piecewise differential systems already produce 7 crossing limit cycles having in each piece a linear polynomial differential system. Note that the class of the linear polynomial differential systems is contained into the class of quadratic polynomial differential systems.
Grants:	Agencia Estatal de Investigación PID2019-104658GB-I00 European Commission 777911
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Limit cycles ; The averaging method ; Discontinuous piecewise differential systems
Published in:	Qualitative theory of dynamical systems, Vol. 21, Issue 4 (December 2022) , art. 114, ISSN 1662-3592

DOI: 10.1007/s12346-022-00643-5

Postprint 12 p, 350.7 KB

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