Motion of test particles and topological interpretation of generic rotating regular black holes coupled to non-linear electrodynamics

This research is devoted to investigate the dynamics of test particles and the intricate topological nature of generic rotating Regular Black Holes (RBHs). By applying the Hamilton–Jacobi formalism, we have presented the paths of test particles as they move around the RBHs, graphically. The dynamics of angular momentum and energy of particles in both counter-rotation and co-rotation are studied. As these particles move around RBH, we observe the interplay of forces shaping their journey. We observe how the effective force, effective potential, and Lyapunov exponent change over time. The Lyapunov exponent, a measure of chaos in their motion, evolves, hinting at the stability of their orbits. Moreover, we study the topological properties of a generic rotating RBH and determine their topological numbers, which are sums of winding numbers around defects and probe the fabric of spacetime itself. The winding number is an integer that indicates how many times a curve encircling a defect wraps around the origin. We find that the total topological number is equal to 0 which suggest a system in balance. As the value of degree of nonlinear electrodynamics parameter ($\mu$) increases, the turning points in particle trajectories multiply and presenting the picture of more complexity. In a twist of topology, the interchange of winding numbers can cause a phase transition, reshaping the order parameter space’s topology.