Financial Frequency Combs

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.