Transition Geometry and Hysteresis in Predictive-Regime Space: Asymmetric Paths Through Model-Validity Boundaries

When a dynamical system crosses from linear-dominant to nonlinear-dominant behavior, the trajectory through predictive-regime space need not be symmetric: forward onset and reverse recovery may follow distinct geometric paths. We study this asymmetry systematically using a supervisory model-validity monitoring framework based on online Volterra-kernel model selection. By driving four canonical nonlinear systems—the logistic map, Lorenz system, Rössler system, and Hénon map—through bidirectional parameter sweeps across known bifurcations, we measure hysteresis in the monitoring framework’s composite boundary index and its constituent risk components. We find strong forward/reverse asymmetry in systems with sharp bifurcations, where the monitoring framework’s internal state retains memory from the chaotic regime and delays recovery of low index values during reverse sweeps. Component decomposition reveals that transition onset ordering is system-dependent and that some components exhibit anti-correlated behavior during transitions. Metastability analysis at boundary parameters identifies genuine regime flipping near bifurcation cascades. Extension to multi-channel monitoring shows that inter-channel coherence itself exhibits hysteresis, and that mixed signal-plus-noise channel configurations correctly suppress contributions from uninformative channels. These results elevate hysteresis from an engineering convenience for threshold-jitter suppression to a diagnostic observable that encodes information about transition sharpness, regime memory, and component-level dynamics.