A version of Hilbert's 16th problem for 3D polynomial vector fields: Counting isolated invariant tori

D. D. Novaes and P. C. C. R. Pereira

Mathematische Nachrichten, 2025, vol. 298, issue 2, 709-717

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.

Date: 2025
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