Self-similarity of stochastic magnetic field lines near the X-point

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Abstract
The properties of magnetic field line trajectories near a separatrix are studied and the renormalization invariance of the Hamiltonian of the system near the X‐point is considered in relation to the perturbation amplitude. To describe the footprint of intersections of the trajectories with an arbitrarily positioned plane, the shifted separatrix map is derived and applied to the analysis of the magnetic field lines dynamics in the stochastic layer and to the magnetic footprint. The numerical simulations confirm the renormalization invariance of the field line equations, surface of section of the field line trajectories, and magnetic footprint obtained by the shifted separatrix map for a simple Hamiltonian system, which is topologically equivalent to the single null poloidal divertor of a tokamak.