Dynamical characteristic of wave solutions of the fractional (4+1)-dimensional Fokas equation

Nonlinear fractional differential equations are superior for modeling complex dynamic systems with memory effects, yet the wave solution dynamics and complex behaviors of the fractional [Formula: see text]-dimensional ([Formula: see text]-d) Fokas equation remain to be further explored. To address this issue, this study adopts the logistic method to construct the exact wave solutions of the aforementioned equation, and visualizes the physical structures of the solutions via MATLAB. We systematically investigate the system’s dynamic characteristics including phase portraits, bifurcation behaviors, sensitivity to initial conditions and parameter perturbations, as well as chaotic dynamics under noise-induced periodic perturbations. Additionally, the lump–Weierstrass elliptic function simultaneous solutions of the model are derived by virtue of bilinear transformation. The results reveal the regulatory effect of the fractional-order [Formula: see text] on the nonlinear oscillation of the solution, clarify the bifurcation rules of the system with parameter changes, and confirm the system’s high sensitivity and noise-induced chaotic behavior; the combination of lump and Weierstrass elliptic function only weakly perturbs the soliton structure and enriches the solution types. This research deepens the understanding of the fractional [Formula: see text]-d Fokas equation’s dynamic mechanisms, and the logistic method employed is proven direct and effective, which can be extended to the study of other fractional-order differential equations.