Thermodynamic Topology and Periodic Orbits of Ayón–Beato–García Black Hole Coupled With a Cloud of Strings

Abstract

In this work, we systematically investigate the Ayón–Beato–García (ABG) regular black hole coupled with a cloud of strings from two distinct yet complementary perspectives: macroscopic thermodynamic topology and microscopic strong-field orbital dynamics. First, employing Duan's topological mapping theory via the generalized thermodynamic potential $F$, we classify the black hole's phase structure. We observe two branches of the inverse temperature $\tau$ curve with associated winding numbers $w_1=+1$ and $w_2=-1$, yielding a globally invariant topological class of $W=0$. We find that this topological number remains unaffected by variations in the magnetic monopole charge $g$ and the string cloud parameter $a$. Second, we analyze the geodesic motion of massive test particles to model Extreme Mass-Ratio Inspiral (EMRI) systems. We rigorously determine the bounding radii, demonstrating that an increasing magnetic charge regularizes the core and pulls the Innermost Stable Circular Orbit (ISCO) inward, whereas the string cloud parameter pushes it outward. By mapping the $(L,E)$ phase space, we classify complex zoom-whirl periodic trajectories using the rational frequency ratio $q$ and the topological triplet $(z,w,v)$. Finally, utilizing the adiabatic approximation and the quadrupole emission formula, we generate time-domain gravitational waveforms ($h_{+},h_{\times }$). Our results indicate that the spacetime deviations induced by $g$ and $a$ manifest as measurable cumulative phase advances and temporal delays, respectively. These distinct amplitude and phase modulations provide a framework for probing regular black holes with future space-based gravitational wave observatories.