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arXiv · 2401.16484

Surfaces with central configuration and Dulac's problem for a three dimensional isolated Hopf singularity

Nuria Corral·María Martín Vega·Fernando Sanz Sánchez
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Abstract

Let $\xi$ be a real analytic vector field with an elementary isolated singularity at $0\in \mathbb{R}^3$ and eigenvalues $\pm bi,c$ with $b,c\in \mathbb{R}$ and $b\neq 0$. We prove that all cycles of $\xi$ in a sufficiently small neighborhood of $0$, if they exist, are contained in a finite number of subanalytic invariant surfaces entirely composed by a continuum of cycles. In particular, we solve Dulac's problem, i.e. finiteness of limit cycles, for such vector fields.

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Nuria Corral, María Martín Vega, Fernando Sanz Sánchez. 2024-01-29. Surfaces with central configuration and Dulac's problem for a three dimensional isolated Hopf singularity. https://arxiv.org/abs/2401.16484

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