| Texto completo | |
| Autor(es): |
Novaes, D. D.
;
Pereira, P. C. C. R.
Número total de Autores: 2
|
| Tipo de documento: | Artigo Científico |
| Fonte: | Mathematische Nachrichten; v. 298, n. 2, p. 9-pg., 2024-12-30. |
| Resumo | |
Hilbert's 16th problem, about the maximum number of limit cycles of planar polynomial vector fields of a given degree m$m$, has been one of the most important driving forces for new developments in the qualitative theory of vector fields. Increasing the dimension, one cannot expect the existence of a finite upper bound for the number of limit cycles of, for instance, 3D polynomial vector fields of a given degree m$m$. Here, as an extension of such a problem in the 3D space, we investigate the number of isolated invariant tori in 3D polynomial vector fields. In this context, given a natural number m$m$, we denote by N(m)$N(m)$ the upper bound for the number of isolated invariant tori of 3D polynomial vector fields of degree m$m$. Based on a recently developed averaging method for detecting invariant tori, our first main result provides a mechanism for constructing 3D differential vector fields with a number H$H$ of normally hyperbolic invariant tori from a given planar differential vector field with H$H$ hyperbolic limit cycles. The strength of our mechanism in studying the number N(m)$N(m)$ lies in the fact that the constructed 3D differential vector field is polynomial provided that the given planar differential vector field is polynomial. Accordingly, our second main result establishes a lower bound for N(m)$N(m)$ in terms of lower bounds for the number of hyperbolic limit cycles of planar polynomial vector fields of degree [m/2]-1$[m/2]-1$. Based on this last result, we apply a methodology due to Christopher and Lloyd to show that N(m)$N(m)$ grows as fast as m3/128$m<^>3/128$. Finally, the above-mentioned problem is also formulated for higher dimensional polynomial vector fields. (AU) | |
| Processo FAPESP: | 20/14232-4 - Bifurcação de toros invariantes em sistemas diferenciais via teoria da média de ordem superior |
| Beneficiário: | Pedro Campos Christo Rodrigues Pereira |
| Modalidade de apoio: | Bolsas no Brasil - Doutorado |
| Processo FAPESP: | 19/10269-3 - Teorias ergódica e qualitativa dos sistemas dinâmicos II |
| Beneficiário: | Claudio Aguinaldo Buzzi |
| Modalidade de apoio: | Auxílio à Pesquisa - Temático |
| Processo FAPESP: | 18/13481-0 - Geometria de sistemas de controle, sistemas dinâmicos e estocásticos |
| Beneficiário: | Marco Antônio Teixeira |
| Modalidade de apoio: | Auxílio à Pesquisa - Temático |
| Processo FAPESP: | 22/09633-5 - Teoria da média no estudo de toros invariantes e comportamento periódico em equações e inclusões diferenciais |
| Beneficiário: | Douglas Duarte Novaes |
| Modalidade de apoio: | Auxílio à Pesquisa - Regular |