Abstract:We address the classical (degenerate or non-degenerate) center problem posed by Poincaré in the 19th century for monodromic singularities of analytic families of planar vector fields $\mathcal{X}$. We prove that, generically, every analytic center admits a Laurent inverse integrating factor $V$ in weighted polar coordinates. Moreover, we show that when $\mathcal{X}$ has no local curves of zero angular speed, the Poincaré map is analytic. Based on this result, we derive a theoretical procedure to determine parameter constraints within the family that characterize centers without curves of zero angular speed. Applications to nontrivial families that have resisted other methods are also provided.
From: Isaac A. García [view email]
[v1]
Tue, 10 Mar 2026 17:32:40 UTC (22 KB)
[v2]
Wed, 10 Jun 2026 15:57:37 UTC (18 KB)
[v3]
Wed, 1 Jul 2026 17:15:46 UTC (19 KB)
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