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

TL;DR

This paper extends Hilbert's 16th Problem to three-dimensional polynomial vector fields by investigating the maximum number of isolated invariant tori, providing lower bounds and construction mechanisms based on averaging methods.

Contribution

It introduces a new mechanism to construct 3D polynomial vector fields with many invariant tori from planar systems, establishing lower bounds for their maximum number.

Findings

Constructed 3D vector fields with many invariant tori from planar systems.

Established a lower bound for the number of invariant tori growing as m^3/128.

Applied methodology to show the growth rate of N(m) with respect to degree m.

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, $3$D polynomial vector fields of a given degree $m$. Here, as an extension of such a problem in the $3$D space, we investigate the number of isolated invariant tori in $3$D polynomial vector fields. In this context, given a natural number $m$, we denote by $N(m)$ the upper bound for the number of isolated invariant tori of $3$D 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 $3$D 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 $N(m)$ lies in the fact that the constructed $3$D 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 $N(m)$ 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 & Lloyd to show that $N(m)$ grows as fast as $m^3/128$. Finally, the above-mentioned problem is also formulated for higher dimensional polynomial vector fields.