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

Abstract

Hilbert's 16th problem, about the maximum number of limit cycles of planar polynomial vector fields of a given degree $m$, 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 $m$. 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 $m$, we denote by $N\left(m\right)$ the upper bound for the number of isolated invariant tori of 3D polynomial vector fields of degree $m$. 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 $H$ of normally hyperbolic invariant tori from a given planar differential vector field with $H$ hyperbolic limit cycles. The strength of our mechanism in studying the number $N\left(m\right)$ 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 $N\left(m\right)$ in terms of lower bounds for the number of hyperbolic limit cycles of planar polynomial vector fields of degree $[m/2]-1$. Based on this last result, we apply a methodology due to Christopher and Lloyd to show that $N\left(m\right)$ grows as fast as $m^3/128$. Finally, the above-mentioned problem is also formulated for higher dimensional polynomial vector fields.