Normally hyperbolic limit tori near monodromic singularities in 3D polynomial vector fields

TL;DR

This paper studies the maximum number of normally hyperbolic limit tori in 3D polynomial vector fields near monodromic singularities, providing improved lower bounds and recursive estimates for higher degrees.

Contribution

It introduces new lower bounds for the number of limit tori in polynomial vector fields and develops a recursive method to estimate these bounds for higher degrees.

Findings

Established lower bounds: N_h(2) ≥ 3, N_h(3) ≥ 5, N_h(4) ≥ 7, N_h(5) ≥ 13.

Used averaging theory to construct vector fields with multiple nested limit tori.

Extended bounds to higher degrees using a recursive approach inspired by Christopher-Lloyd.

Abstract

We investigate the maximal number $N_h(m)$ of normally hyperbolic limit tori in three-dimensional polynomial vector fields of degree $m$, which extends the classical notion of Hilbert numbers to higher dimensions. Using recent developments in averaging theory, we show the existence of families of vector fields near monodromic singularities, including both Hopf-zero and nilpotent-zero cases, that exhibit multiple nested normally hyperbolic limit tori. This approach allows us to establish improved lower bounds: $N_h(2) \geq 3$, $N_h(3) \geq 5$, $N_h(4) \geq 7$, and $N_h(5) \geq 13$, which are currently the best available in the literature. Furthermore, these bounds are extended using the strict monotonicity of the function $N_h(m)$ and a recursive construction inspired by the Christopher-Lloyd method, leading to new estimates for higher degrees which improves all the previously known results.