Critical periods in planar polynomial centers near a maximum number of cusps

We provide the best lower bound for the number of critical periods of planar polynomial centers known up to now. The new lower bound is obtained in the Hamiltonian class and considering a single period annulus. This lower bound doubles the previous one from the literature, and we end up with at leas...

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Bibliographic Details
Published in:Journal of Differential Equations 2024-01, Vol.380, p.181-197
Main Authors: De Maesschalck, Peter, Torregrosa, Joan
Format: Article
Language:English
Subjects:
Best lower bound
Critical periods
Degree n vector field
Hamiltonian vector fields
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Summary:We provide the best lower bound for the number of critical periods of planar polynomial centers known up to now. The new lower bound is obtained in the Hamiltonian class and considering a single period annulus. This lower bound doubles the previous one from the literature, and we end up with at least n2−2 (resp. n2−2n−1) critical periods for planar polynomial systems of odd (resp. even) degree n. Key idea is the perturbation of a vector field with many cusp equilibria, whose construction is by itself a nontrivial construction that uses elements of catastrophe theory.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2023.10.034