Impulsively Driven Fractional Relaxation: Exact Response, Scaling Laws, Frequency-Domain Signatures, and Memory-Induced Crossover Structures

Abstract

We investigate periodically impulsive fractional relaxation as a minimal model of nonlocal dissipative dynamics under repeated external stimulation. The free relaxation law is classical, but the driven problem introduces an additional timescale whose competition with the intrinsic fractional-memory scale generates nontrivial accumulation and crossover behavior. We derive the exact Laplace-domain and time-domain responses, establish the sparse-forcing scaling of the long-time averaged response, and obtain an analytical crossover interval proportional to the inverse fractional power of the relaxation coefficient. To separate established properties of Mittag--Leffler relaxation from the new effects of forcing, we explicitly compare the unforced, impulsive, and periodically driven settings. Extended long-time simulations verify the algebraic tail, while an independent L1 discretization provides a numerical benchmark for the exact solution. We further connect the relaxation kernel to measurable frequency-domain quantities through the Cole--Cole complex susceptibility and discuss interpretations in dielectric and viscoelastic relaxation, anomalous transport, non-Markovian open systems, and intermittent reinforcement. The resulting framework supplies an analytically solvable reference model for memory accumulation under repeated driving. The present analytical framework therefore provides a bridge between fractional relaxation theory and experimentally accessible time- and frequency-domain observables.