A truncated Hirota-type approach for wave structures and chaotic dynamics in the time-fractional modified Liouville equation

This paper investigates the time-fractional modified Liouville equation with exponential nonlinearity through a truncated Hirota-type analytical approach. By applying a Hirota-type dependent-variable transformation together with a controlled truncation of the exponential term, the governing equation is converted into an auxiliary nonlinear formulation suitable for constructing analytical wave structures. Based on this framework, several localized structures, including one-wave, two-wave, and three-wave soliton-like solutions, lump-type solutions, and one-lump–one-stripe interaction patterns, are obtained. The characteristics of these wave structures are illustrated using three-dimensional surface plots, contour plots, and two-dimensional profiles, revealing their spatial distributions and evolution. To explore the influence of external forcing, a periodically perturbed form of the model is reduced to a nonlinear dynamical system through a travelling-wave transformation. The resulting system is analyzed using phase portraits, bifurcation diagrams, time-series analysis, Poincaré sections, return maps, and Outcome-basin analysis to investigate transitions between regular and irregular dynamical regimes. Furthermore, a local stability analysis is performed by introducing perturbations around the travelling-wave solutions, demonstrating the influence of the system parameters on solution stability. The results reveal that the truncated time-fractional modified Liouville model supports diverse wave structures and complex nonlinear dynamics, including chaotic behaviors under periodic excitation. This study provides insights into the analytical structures and dynamical characteristics of fractional nonlinear systems described by exponential-type nonlinearities.