Finite-time recurrence network transitivity in the ergodic limiter tokamak map

Hamiltonian maps are a useful tool for modeling the magnetic field configuration in plasma fusion devices, such as tokamaks, particularly for understanding the complex topology created by magnetic perturbations. This work uses recurrence analysis, specifically the concept of recurrence network transitivity, to provide a quantitative characterization of the phase space in the ergodic limiter map, also known as the Ullmann map (UM). We generate recurrence network representations of finite-time example trajectories of this map with different initial conditions and estimate their respective transitivity for different magnetic perturbation strengths, demonstrating that this measure effectively identifies dynamically distinct regions based on their local geometric clustering. We observe marked bands of elevated finite-time transitivity embedded within the chaotic sea, indicating regions of reduced effective transient dimensionality. These highly structured zones in phase space are characteristic of trajectories trapped near partial transport barriers (cantori), making them spatially consistent with sticky dynamics. We also find that trajectories near unstable periodic orbits (UPOs) exhibit exceptionally high transitivity, offering new insights into the local structure of chaotic saddles. This recurrence network-based approach provides a quantitative, data-driven analysis of phase space structures in Hamiltonian systems, being also relevant for the understanding of plasma transport and magnetic confinement in tokamaks.