Parametric Sign Inversion and Topological Flipping in Driven Complex Systems: A Mathematical Analogy Across Scales

This work presents a theoretical analysis of parametric sign inversion in nonlinear systems subject to high-frequency driving. Using Floquet averaging, we show that the effective coupling governing macroscopic dynamics is modulated by a Bessel function. As the normalized driving parameter crosses the first root of the zeroth-order Bessel function, the effective interaction changes sign. This sign inversion induces a qualitative change in the topology of the system’s effective phase space. In systems with broken spatial symmetry, this inversion further generates a ratchet-like mechanism, rectifying stochastic fluctuations into directed macroscopic transport. We illustrate this mathematical structure in two distinct model systems: continuous attractor neural networks and the secular dynamics of hierarchical N-body systems. The results highlight a general mechanism by which strong periodic driving can induce topological transitions and directed transport in nonlinear dynamical systems.