Thermodynamic signatures in black hole geometry and harmonic oscillations with nonlinear electromagnetic fields and phantom global monopole

We investigate the thermodynamic and dynamical properties of charged black holes influenced by nonlinear electromagnetic fields in the presence of a phantom global monopole. Focusing on the thermodynamic topology, we analyze the distribution and stability of topological charges derived from the free energy landscape, revealing a consistent total topological charge characteristic of the Reissner-Nordström black hole class. Extending this approach to photon sphere configurations, we demonstrate that their topological charges remain invariant under variations of the symmetry-breaking energy scale and coupling strengths, reflecting inherent geometric and thermodynamic stability. Additionally, we perform a comprehensive study of quasi-periodic oscillation (QPO) frequencies arising in the spacetime of such black holes, incorporating couplings to both phantom scalar and global monopole sectors. By systematically varying the monopole coupling parameter η , the phantom coupling ζ , and the magnetic charge Q , we elucidate their distinct influences on radial epicyclic, orbital, and periastron precession frequencies. Our results indicate that stronger monopole and phantom couplings suppress epicyclic oscillation amplitudes and shift characteristic frequencies outward, while increased magnetic charge stabilizes orbital motion closer to the black hole by lowering oscillation frequencies. These modifications suggest potential observational signatures in accretion phenomena and X-ray binaries, offering novel avenues to probe beyond-Einstein gravity in strong-field regimes. Future work will extend these analyses to rotating black holes, incorporate higher-order nonlinear electrodynamics effects, and explore implications for gravitational wave emissions.