Tangential center problem for planar Hamiltonian differential equations
Abstract
Consider a planar Hamiltonian differential equation with a nondegenerate center at the origin. The Tangential Center Problem asks for a characterization of polynomial perturbations for which the center persists at first order. This problem plays a fundamental role in the study of the center-focus problem. We propose an effective algorithmic approach to the Tangential Center Problem for planar polynomial Hamiltonian differential equations. Using this approach, we obtain a complete solution of the quadratic Tangential Center Problem. We further apply it to a non-generic family of cubic Hamiltonian differential equations containing the planar Hamiltonian Lotka--Volterra equation. For a general cubic polynomial perturbation, we derive necessary and sufficient conditions for the vanishing of the corresponding first Poincaré--Pontryagin--Melnikov function and characterize all tangential centers.
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BibTeX
@misc{alvarez2026tangential,
title = {{Tangential center problem for planar Hamiltonian differential equations}},
author = {M. J. Alvarez and J. L. Bravo and L. A. Calderon and S. Rebollo-Perdomo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40311},
eprint = {2610.40311},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40311},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tangential center problem for planar Hamiltonian differential equations
AU - M. J. Alvarez
AU - J. L. Bravo
AU - L. A. Calderon
AU - S. Rebollo-Perdomo
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40311
UR - https://arxiv.org/abs/2610.40311
AB - Consider a planar Hamiltonian differential equation with a nondegenerate center at the origin. The Tangential Center Problem asks for a characterization of polynomial perturbations for which the center persists at first order. This problem plays a fundamental role in the study of the center-focus problem. We propose an effective algorithmic approach to the Tangential Center Problem for planar polynomial Hamiltonian differential equations. Using this approach, we obtain a complete solution of the quadratic Tangential Center Problem. We further apply it to a non-generic family of cubic Hamiltonian differential equations containing the planar Hamiltonian Lotka--Volterra equation. For a general cubic polynomial perturbation, we derive necessary and sufficient conditions for the vanishing of the corresponding first Poincaré--Pontryagin--Melnikov function and characterize all tangential centers.
ER -