Chaotic behaviour, bifurcation, and stability analysis of the time-fractional phi-four model using the Hirota bilinear form

Abstract

This study investigates the complex dynamics of the time-fractional Phi-four model, a nonlinear PDE that incorporates memory effects through fractional derivatives. To analyze this model, we employ the Hirota bilinear method to derive a variety of exact analytical solutions, including one-wave, two-wave, three-wave, and W-shaped soliton solutions, as well as lump-type solutions such as lumps, one-lump-one-stripe, and one-lump-one-soliton configurations. We conduct a detailed analysis of the system’s chaotic behaviour by calculating Lyapunov exponents, performing sensitivity analysis, and constructing bifurcation diagrams, which reveal transitions between stable, periodic, and chaotic states. The results demonstrate that the fractional-order derivative crucially influences system dynamics by introducing memory effects that can stabilize or destabilize wave propagation. A linear stability analysis confirms the conditions under which these soliton and lump solutions remain structurally stable against perturbations. These findings advance the understanding of nonlinear fractional systems by illustrating their capacity for rich wave interactions and chaotic dynamics.