Directional Transport Basins and Final-State Selection in the Damped-Driven Pendulum

The damped-driven pendulum is a classical nonlinear oscillator whose periodic, quasiperiodic, and chaotic responses are well known. The present paper studies a complementary global question: how does a fixed forcing-phase slice of the initial-condition plane select the longtime direction of rotational transport? The nondimensional pendulum equation is integrated over two-dimensional grids of initial angle and initial velocity. After a transient interval is removed, each trajectory is classified by its mean rotation number over a sampling interval, giving a discrete partition into positive rotation, negative rotation, and libration or nearly zero net rotation. A 300-point parameter screen over damping coefficient and forcing amplitude is used to identify nontrivial regimes in which the initial-condition plane contains substantial fractions of both positive- and negative-rotation outcomes. Four representative cases are then analyzed in detail using transport-basin maps, continuous mean-rotation-number maps, basinfraction statistics, finite-resolution boundary metrics, and representative trajectories. The selected regimes show coherent curved basins of opposite rotational direction rather than isolated single-trajectory behavior. The transition from positive-rotation-dominated to negative-rotationdominated basin fractions across the selected parameter sequence demonstrates that the dominant long-time direction in a fixed phase slice can change while both transport outcomes remain present. Grid-resolution convergence, sampling-window convergence, and time-step checks for representative interior points support the numerical stability of the reported basin fractions and transport labels over the tested settings. The results do not claim a rigorous proof of fractal or Wada basin boundaries; instead, they provide a reproducible finite-resolution characterization of directional final-state predictability in a familiar dissipative nonlinear oscillator. The basinmap approach complements standard phase portraits, Poincaré sections, and Lyapunov-type diagnostics by showing how an ensemble of possible initial conditions partitions into different long-time transport states.