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

TL;DR

This paper explores how truncating the Gr"unwald--Letnikov memory in a fractional Duffing oscillator affects system coherence, delay complexity, and the spectral structure, revealing non-monotonic memory thresholds and the need for delay-mode representations.

Contribution

It introduces a novel analysis of finite-memory effects as a modification of the dynamical system, including a delay complexity measure and a spectral kernel approximation.

Findings

Memory thresholds depend on forcing, fractional order, and nonlinear sensitivity.

A local characteristic equation for the truncated kernel is derived.

A delay-mode representation captures the spectral structure and coherence loss.

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 full-memory and truncated-memory trajectories, and use it to extract critical truncation thresholds in parameter planes involving the forcing amplitude and the fractional order. The results reveal strongly non-monotonic memory thresholds, 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. A minimal one-delay approximation produces a formal negative delay, indicating that a single causal delay is structurally insufficient. This motivates a positive-delay exponential representation of the finite-memory 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 controls both coherence loss and effective delay complexity.