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Higher order stroboscopic averaged functions: a general relationship with Melnikov functions

Texto completo
Autor(es):
Novaes, Douglas D. [1]
Número total de Autores: 1
Afiliação do(s) autor(es):
[1] Univ Estadual Campinas, Dept Matemat, Inst Matemat Estat & Comp Cient IMECC, UNICAMP, Rua Sergio Buarque Holanda 651, BR-13083859 Campinas, SP - Brazil
Número total de Afiliações: 1
Tipo de documento: Artigo Científico
Fonte: Electronic Journal of Qualitative Theory of Differential Equations; n. 77 2021.
Citações Web of Science: 0
Resumo

In the research literature, one can find distinct notions for higher order averaged functions of regularly perturbed non-autonomous T-periodic differential equations of the kind x' = epsilon F( t, x, epsilon). By one hand, the classical (stroboscopic) averaging method provides asymptotic estimates for its solutions in terms of some uniquely defined functions gi's, called averaged functions, which are obtained through near-identity stroboscopic transformations and by solving homological equations. On the other hand, a Melnikov procedure is employed to obtain bifurcation functions f(i)'s which controls in some sense the existence of isolated T-periodic solutions of the differential equation above. In the research literature, the bifurcation functions fi's are sometimes likewise called averaged functions, nevertheless, they also receive the name of Poincare-Pontryagin-Melnikov functions or just Melnikov functions. While it is known that f(1) = Tg(1), a general relationship between g(i) and f(i) is not known so far for i >= 2. Here, such a general relationship between these two distinct notions of averaged functions is provided, which allows the computation of the stroboscopic averaged functions of any order avoiding the necessity of dealing with near-identity transformations and homological equations. In addition, an Appendix is provided with implemented Mathematica algorithms for computing both higher order averaging functions. (AU)

Processo FAPESP: 18/16430-8 - Dinâmica global das equações diferenciais não suaves
Beneficiário:Douglas Duarte Novaes
Linha de fomento: Auxílio à Pesquisa - Regular
Processo FAPESP: 18/13481-0 - Geometria de sistemas de controle, sistemas dinâmicos e estocásticos
Beneficiário:Luiz Antonio Barrera San Martin
Linha de fomento: Auxílio à Pesquisa - Temático
Processo FAPESP: 19/10269-3 - Teorias ergódica e qualitativa dos sistemas dinâmicos II
Beneficiário:Claudio Aguinaldo Buzzi
Linha de fomento: Auxílio à Pesquisa - Temático