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arXiv · 2610.40311

Tangential center problem for planar Hamiltonian differential equations

M. J. Alvarez·J. L. Bravo·L. A. Calderon·S. Rebollo-Perdomo
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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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M. J. Alvarez, J. L. Bravo, L. A. Calderon, S. Rebollo-Perdomo. 2026-10-29. Tangential center problem for planar Hamiltonian differential equations. https://arxiv.org/abs/2610.40311

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