Non-degenerate unfolding conjecture for homoclinic tangencies of Beltrami fields

Let XB(R3)X\in\mathscr B(\mathbb R^3) be a Beltrami field, and let a periodic saddle orbit of XX have a homoclinic tangency. A homoclinic tangency is non-degenerately unfolded in B(R3)\mathscr B(\mathbb R^3) when it admits a non-degenerate unfolding by a family of Beltrami fields in B(R3)\mathscr B(\mathbb R^3). Non-degenerate unfolding conjecture. Every homoclinic tangency for a periodic saddle orbit of a Beltrami field XB(R3)X\in\mathscr B(\mathbb R^3) unfolds non-degenerately in B(R3)\mathscr B(\mathbb R^3). This is proposed as the Beltrami counterpart of the known result for smooth volume-preserving vector fields, and would clarify the generic dynamical behavior of Beltrami fields near homoclinic tangencies. The supplied text does not give a resolution.

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Primary source

Pierre Berger, Anna Florio and Daniel Peralta-Salas, “Steady Euler flows on R^3 with wild and universal dynamics”, arXiv:2202.02848 (2022).

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