2022/12/22 by Douglas D. Novaes, Novaes, Douglas D., Pedro C. C. R. Pereira +1 · 3 citations
Mathematics · Physics and Astronomy · #34C23 #34C29 #34C45 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2212.12006
openalex publication_date 2022/12/22 · openalex created_date 2023/01/04 · openalex updated_date 2026/07/28
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(m) 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(m) 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(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 m3/128. Finally, the above-mentioned problem is also formulated for higher dimensional polynomial vector fields.