Dynamical Systems Analysis of Timelike Geodesics in a Lorentz-Violating Black Hole Spacetime

This paper investigates the global dynamics of timelike geodesics of a spherically symmetric black hole under Lorentz-violating effects governed by parameters λ (scaling exponent) and Υ (Lorentz violation strength). By employing dynamical system techniques, including Poincaré compactification and blow-up methods, we systematically explore finite and infinite equilibrium states of the system derived from a black hole solution with power-law corrections to the Schwarzschild metric. For varying λ (ranging from −2 to 2) and fixed Υ values, we classify the nature of equilibrium states (saddle, center, and node) and analyze their stability. Key findings reveal that the number of equilibrium states increases as λ decreases: two states for λ=2, three for λ=1, four for λ=2/3, and additional configurations for λ=−2. The phase plane diagrams and global dynamics demonstrate distinct topological structures, including attractors at infinity and multi-horizon black hole solutions. Furthermore, degenerate equilibrium states at infinity are resolved through directional blow-ups, elucidating their non-hyperbolic behavior. This study highlights the critical role of Lorentz-violating parameters in shaping the stability and long-term evolution of timelike geodesics, offering new insights into modified black hole physics and spacetime dynamics.