We study the transition from order to chaos in an effective Hamiltonian three-body system governed by a static, position-dependent effective beyond-Newtonian potential for rotating compact primaries derived from the Fodor–Hoenselaers–Perjés formalism. The model constitutes a perturbed extension of the classical Circular Restricted Three-Body Problem that retains an exact Jacobi integral, incorporating effective beyond-Newtonian multipole corrections while preserving its conservative character. In this formulation, the potential is purely spatial; velocity-dependent gravitomagnetic spin-orbit couplings are not included. This static formulation, having an exact Jacobi integral, thereby enables a systematic large-scale survey of phase space, allowing geometric and orbit-based diagnostics to be applied uniformly to more than ${10}^5$ initial conditions. Equilibrium points are determined using a Newton–Raphson scheme, and the associated basins of convergence are analyzed via the basin entropy ($S_b$) and boundary basin entropy ($S_{bb}$), which quantify global unpredictability and interfacial mixing in phase space. The Smaller Alignment Index (SALI) is employed to probe local dynamical stability and chaotic diffusion, while convergence statistics reveal the role of long-lived transients near fractal basin boundaries. The system displays a distinctly nonmonotonic evolution of dynamical complexity. Both entropy measures and SALI exhibit correlated maxima within an intermediate range of the control parameter $\epsilon$, associated with transitions from regular to strongly chaotic motion driven by changes in the equilibrium structure. The coincidence between the maxima of basin entropy and the peak fraction of chaotic orbits detected by SALI indicates that geometric basin complexity and dynamical instability arise from the same restructuring of the equilibrium configuration of the effective beyond-Newtonian potential. At higher values of the control parameter, the annihilation of equilibrium points alters the potential topology, leading to smoother basin structures and a partial restoration of order. In contrast to earlier studies that focused mainly on equilibrium locations and linear stability, these results establish a quantitative and unified link between potential geometry, equilibrium multiplicity, basin topology, entropy growth, and orbit-based dynamical instability within a well-defined effective Hamiltonian system. They demonstrate how controlled multipole corrections reorganize phase-space structure and govern the onset, amplification, and suppression of chaos in this class of perturbed conservative systems.

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• Chaos
• Classical mechanics
• Astronomical black holes
• Binary stars
• Chaotic systems
• Dynamical systems
• Galaxies
• Numerical simulations in gravitation & astrophysics
• Phase space methods