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

2025/07/23 by Arakaki, Lucas Queiroz, Gouveia, Luiz F. S., Novaes, Douglas D.
#34C23 #34C29 #34C45 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.17932

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) ≥ 3, N_h(3) ≥ 5, N_h(4) ≥ 7, and N_h(5) ≥ 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.

Citations

• Normal Hyperbolicity in Secondary Hopf Bifurcations
• A version of Hilbert's 16th Problem for 3D polynomial vector fields: Counting isolated invariant tori
• Zero-Hopf bifurcation in the general Van der Pol-Duffing equation
• Higher order averaging theory for finding periodic solutions via Brouwer degree

Related

• Strict increase in the number of normally hyperbolic limit tori in 3D polynomial vector fields
• A version of Hilbert's 16th Problem for 3D polynomial vector fields: Counting isolated invariant tori
• Towards sharp regularity: Full-dimensional tori in C^∞ vector fields over \mathbbT^∞
• Logarithmic vector fields and hyperbolicity
• Some variants of the generalized Borel Theorem and applications