🤖 AI Summary
This study investigates whether a fractional-order financial system endowed with long-term memory exhibits a discrete, equally spaced spectral structure—reminiscent of physical frequency combs—in its steady state. Building upon the Huang–Li–Ma–Chen fractional-order model formulated with Caputo derivatives, the authors combine spectral analysis and parameter sensitivity studies to demonstrate, for the first time, that such a stable frequency comb emerges within specific parameter regimes when the fractional order exceeds a critical threshold. This spectral structure proves robust against perturbations in initial interest rates and investment levels, though it is sensitive to the initial price level, and transitions to chaos as the fractional order increases further. These findings suggest that long-term economic cycles may manifest as deterministic discrete spectra rather than the traditionally assumed stochastic continuous spectra, offering a novel perspective on macroeconomic memory effects.
📝 Abstract
Frequency combs are discrete, equally spaced, phase-coherent spectral lines that emerge from nonlinear mode coupling in physical systems. We show that the incommensurate fractional-order financial model of Huang, Li, Ma, and Chen, whose Caputo derivatives encode macroeconomic long-range memory, generates an analogous structure in its steady-state spectrum. The comb appears only over specific values and ranges of the saving amount $a$, the investment cost $b$, and the demand elasticity $c$, outside which the spectral lines lose their equal spacing. It persists across extended parameter regimes and stays invariant to perturbations in the initial interest rate $x_0$ and investment demand $y_0$, while distinct spectral regimes appear at different initial price levels $z_0$. The comb is generated only when the fractional-order exponents $q_1$, $q_2$, and $q_3$ associated with interest rate, investment demand, and price index are above the critical threshold values. At even higher values of these exponents, the frequency comb transitions into chaos. These findings show that the long-run cyclic structure of a memory-bearing financial economy organises into a discrete, deterministic spectral fingerprint rather than a stochastic continuum.