Gr\"unwald--Letnikov Memory Truncation in a Fractional Duffing Oscillator: Coherence Loss and Effective Delay Complexity

Abstract

We investigate the dynamical and analytical consequences of truncating the Gr\"unwald--Letnikov memory term in a fractional Duffing oscillator. The truncated memory is treated not merely as a computational approximation, but as a finite-memory modification of the underlying dynamical system. We define a coherence-loss time from direct comparisons between the full-history discrete GL reference and its truncated-memory counterpart, and use it to extract critical memory horizons in parameter planes involving the forcing amplitude and the fractional order. The results reveal strongly non-monotonic critical memory horizons, showing that the retained memory required to preserve coherence depends on the forcing regime, the fractional order, and the nonlinear sensitivity of the dynamics. We also derive a local characteristic equation for the truncated GL kernel and show that it admits a local delay-type interpretation. In particular, a low-order matching yields an effective representation in terms of an instantaneous contribution plus a delayed exponential term, providing a causal local surrogate of the finite-memory kernel. This local spectral viewpoint motivates a positive-delay exponential representation of the truncated kernel. The minimum number of positive-delay modes required to reach a prescribed spectral accuracy defines an operational delay-complexity measure, \(r_{\min}\). Overall, the truncated GL kernel emerges as an intermediate object between distributed fractional memory and delay-type dynamics, with a local spectral structure that is associated with the observed coherence loss and provides an operational diagnostic of effective delay complexity.