Strict increase in the number of normally hyperbolic limit tori in 3D polynomial vector fields

TL;DR

The paper proves that the maximum number of normally hyperbolic limit tori, $N_h(m)$, in 3D polynomial vector fields, increases strictly with the degree: if finite, $N_h(m+1)\ge N_h(m)+1$. The authors develop explicit, verifiable criteria for torus bifurcation near Hopf–Zero equilibria using a Jordan-form linear part, circumventing higher-order normal-form computations. They derive a robust two-parameter averaging framework and Melnikov analysis to detect torus birth via Neimark–Sacker bifurcation, and demonstrate a constructive perturbation approach that preserves existing tori while creating a new one. Consequently, they establish a mechanism for incremental growth in the number of normally hyperbolic limit tori with increasing degree, linking the 3D torus dynamics to the Hilbert problem’s spirit in higher dimensions.

Abstract

The second part of Hilbert's 16th problem concerns determining the maximum number $H(m)$ of limit cycles that a planar polynomial vector field of degree $m$ can exhibit. A natural extension to the three-dimensional space is to study the maximum number $N(m)$ of limit tori that can occur in spatial polynomial vector fields of degree $m$. In this work, we focus on normally hyperbolic limit tori and show that the corresponding maximum number $N_h(m)$, if finite, increases strictly with $m$. More precisely, we prove that $N_h(m+1) \geqslant N_h(m) + 1$. Our proof relies on the torus bifurcation phenomenon observed in spatial vector fields near Hopf-Zero equilibria. While conditions for such bifurcations are typically expressed in terms of higher-order normal form coefficients, we derive explicit and verifiable criteria for the occurrence of a torus bifurcation assuming only that the linear part of the unperturbed vector field is in Jordan normal form. This approach circumvents the need for intricate computations involving higher-order normal forms.

Figures (2)

• Figure 1: Numerical simulations of several trajectories of system \ref{['eq:ex1']} for $(\mu,\varepsilon) = (0.05, 0.05)$. The bottom panel shows the Poincaré map defined on the section $y = 0$, while the top panel shows some views of the phase portrait of \ref{['eq:ex1']} rescaled via $(x,y,z) \mapsto \varepsilon(x,y,z)$.
• Figure 2: Schematic of the proof of Lemma \ref{['Lema:intersecB']}.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Theorem 3: CanNov20JDE and Novaes2025
• Theorem 4
• proof
• Lemma 5
• proof