A Minimal Origin of Nonlinear Scaling: Feedback-Driven Accumulation

Nonlinear behavior is often attributed to complex interactions or system-specific effects. In this work, we show that a simple and general mechanism can give rise to nonlinear scaling: the combination of feedback and temporal accumulation. By removing the leading linear response, we isolate the nonlinear contribution and find that it follows a clear quadratic scaling with respect to both perturbation strength and interaction time, Δφ ∼ h²T². This behavior is not introduced as a higher-order correction, but emerges dynamically as small perturbations repeatedly interact with the evolving system. We test this mechanism across different dynamical models with distinct interaction forms, and find that the same scaling persists. When normalized, results collapse onto a common trajectory, indicating a consistent underlying structure. Control cases further show that removing either feedback or accumulation eliminates the scaling, confirming that both are essential ingredients. These results suggest that nonlinear amplification can arise from a minimal and broadly applicable dynamical process, rather than from detailed system-specific features. This provides a simple perspective on the origin of nonlinear scaling in a wide class of systems.