Investigating the Impact of Fractional Parameters on Stability and Bifurcation in the Time-Fractional Swift–Hohenberg Model

This paper investigates the effect of fractional parameters on chaos and pattern formation in the fractional-in-time complex Swift–Hohenberg model. A novel higher-order numerical scheme is developed, combining a Formula: see textth-order Grünwald–Letnikov discretization for the Caputo time-fractional derivative with a nine-point finite difference approximation for spatial operators, achieving fourth-order spatial accuracy and incorporating a short-memory principle for computational efficiency. Linear stability and Turing bifurcation analysis reveal that the fractional order Formula: see text nonmonotonically modulates both the complexity measures and the Turing instability threshold, enabling multiple stability switches and transitions to globally unstable regimes. Numerical simulations demonstrate that Formula: see text acts as a bifurcation parameter, inducing complex pattern selection and chaotic dynamics, with the proposed scheme effectively capturing rich spatiotemporal behaviors across various parameter regimes.