Local and global analysis of the displacement map for some near integrable systems
Braun, Francisco (Universidade Federal de São Carlos. Departamento de Matemática)
Da Cruz, Leonardo Pereira Costa (Universidade de São Paulo. Instituto de Ciências Matemáticas e de Computação)
Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2025
Abstract:	In this paper, we introduce an alternative method for applying averaging theory of orders 1 and 2 in the plane. This is done by combining Taylor expansions of the displacement map with the integral form of the Poincaré-Poyntriagin-Melnikov function. It is known that, to obtain results of order 2 with averaging theory, the first-order averaging function should be identically zero. However, when working with Taylor expansions of the ith-order averaging function, we usually cannot guarantee it is identically zero. We prove that the vanishing of certain coefficients of the Taylor series of the first-order averaging function ensures it is identically zero. We present our reasoning in several concrete examples: a quadratic Lotka-Volterra system, a quadratic Hamiltonian system, the entire family of quadratic isochronous differential systems, and a cubic system. For the latter, we also show that a previous analysis contained in the literature is not correct. In none of the examples is it necessary to precisely calculate the averaging functions.
Grants:	Generalitat de Catalunya 2021/SGR-00113 Agencia Estatal de Investigación PID2022-136613NB-I00 Agencia Estatal de Investigación CEX2020-001084-M
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Periodic solution ; Displacement map ; Smooth differential system
Published in:	Physica D: Nonlinear Phenomena, Vol. 483 (December 2025) , art. 134932, ISSN 0167-2789

DOI: 10.1016/j.physd.2025.134932

Available from: 2027-12-31 Postprint 20 p, 361.3 KB

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