A note on Hilbert 16th problem

Gasull, Armengol; Santana, Paulo

doi:10.1090/proc/17116

Proc. Amer. Math. Soc.

2024

DOI: 10.1090/proc/17116

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A note on Hilbert 16th problem

Armengol Gasull,

Paulo Santana

Abstract: Let H ( n ) \mathcal {H}(n) be the maximum number of limit cycles that a planar polynomial vector field of degree n n can have. In this paper we prove that H ( n ) \mathcal {H}(n) is realizable by structurally stable vector fields with only hyperbolic limit cycles and that it is a strictly increasing function whenever it is finite.

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2024

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

Novaes,

Pereira

2024

Mathematische Nachrichten

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Hilbert's 16th problem, about the maximum number of limit cycles of planar polynomial vector fields of a given degree , 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 . 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 , we denote by the upper bound for the number of isolated invariant tori of 3D polynomial vector fields of degree . 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 of normally hyperbolic invariant tori from a given planar differential vector field with hyperbolic limit cycles. The strength of our mechanism in studying the number 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 in terms of lower bounds for the number of hyperbolic limit cycles of planar polynomial vector fields of degree . Based on this last result, we apply a methodology due to Christopher and Lloyd to show that grows as fast as . Finally, the above‐mentioned problem is also formulated for higher dimensional polynomial vector fields.

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

Novaes,

Pereira

2024

Mathematische Nachrichten

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A note on Hilbert 16th problem

Cited by 1 publication

References 14 publications

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

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

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