| Full text | |
| Author(s): |
Novaes, D. D.
;
Pereira, P. C. C. R.
Total Authors: 2
|
| Document type: | Journal article |
| Source: | Mathematische Nachrichten; v. 298, n. 2, p. 9-pg., 2024-12-30. |
| Abstract | |
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) | |
| FAPESP's process: | 20/14232-4 - Bifurcation of invariant tori of differential systems via higher order averaging theory |
| Grantee: | Pedro Campos Christo Rodrigues Pereira |
| Support Opportunities: | Scholarships in Brazil - Doctorate |
| FAPESP's process: | 19/10269-3 - Ergodic and qualitative theories of dynamical systems II |
| Grantee: | Claudio Aguinaldo Buzzi |
| Support Opportunities: | Research Projects - Thematic 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: | 22/09633-5 - Averaging theory for studying invariant tori and periodic behavior in differential equations and inclusions |
| Grantee: | Douglas Duarte Novaes |
| Support Opportunities: | Regular Research Grants |