Dynamics of computational waveform: A study of bifurcation, chaos, and sensitivity analysis

This study delves into the extraction of solutions for the fractional-order that describes wave circulation in space-time fractional low-pass electrical transmission (LPET) lines and Drinfel'd-Sokolov-Wilson (DSW) equations. Leveraging the Sardar-subequation scheme, by applying a simple linear fractional transformation, the model equations are transformed into an ordinary differential equation. The use of the Sardar-subequation technique produces a diverse range of traveling waveform for the governing equations. The behavior of the dynamics of select solutions, representing singular and multiple soliton, kink and periodic kink, W-shaped bright soliton, dark-kink soliton and kink-like soliton solutions, is then visually showcased through their two and three-dimensional profiles with the help of computational software Maple and MATLAB. In addition, the dynamical model of the proposed DSW equation is constructed by utilizing the Galilean transformation in order to accomplish our objective. Then, using the concepts of the planar dynamical system, bifurcation, chaos, and sensitivity studies of the aforementioned model are carried out. For the aforementioned model, we find chaotic, quasi-periodic, and periodic behaviors. This research is novel in that it provides new insights into the complex dynamics of the governing model and the variety of waveforms it produces through a comprehensive investigation. By integrating waveform characterization, chaotic behavior, and bifurcation analysis, this study enhances our understanding of the nonlinear behavior of waves in shallow water.