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(Reference retrieved automatically from Web of Science through information on FAPESP grant and its corresponding number as mentioned in the publication by the authors.)

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

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Author(s):
Novaes, Douglas D. [1]
Total Authors: 1
Affiliation:
[1] Univ Estadual Campinas, Dept Matemat, Inst Matemat Estat & Comp Cient IMECC, UNICAMP, Rua Sergio Buarque Holanda 651, BR-13083859 Campinas, SP - Brazil
Total Affiliations: 1
Document type: Journal article
Source: Electronic Journal of Qualitative Theory of Differential Equations; n. 77 2021.
Web of Science Citations: 0
Abstract

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)

FAPESP's process: 18/16430-8 - Global dynamics of nonsmooth differential equations
Grantee:Douglas Duarte Novaes
Support Opportunities: Regular Research Grants
FAPESP's process: 18/13481-0 - Geometry of control, dynamical and stochastic systems
Grantee:Marco Antônio Teixeira
Support Opportunities: Research Projects - Thematic Grants
FAPESP's process: 19/10269-3 - Ergodic and qualitative theories of dynamical systems II
Grantee:Claudio Aguinaldo Buzzi
Support Opportunities: Research Projects - Thematic Grants