This article presents new exact wave solutions for the generalized (3+1)-dimensional nonlinear wave equation involving gas bubbles. The \((G^{'}/G^2)\) -expansion method is used to find solutions that behave like as periodic, bell-shaped, cubic, and kink ways, as well as solutions that are rational functions. We also provide two-dimensional and three-dimensional plots, as well as corresponding contour plots, using appropriately selected parameters. We further transform the system into a planar dynamical system and analyze the obtained solutions through phase portraits. The qualitative analysis of the evaluated model employs the concepts of bifurcation and chaos. Furthermore, by using an external force, we verify the manifestation of quasi-periodic and chaotic behaviors. We explore a number of different approaches for identifying chaos, including time series and phase patterns in both three and two dimensions. Moreover, we conduct a sensitivity analysis that takes into account various initial conditions. We obtained unique results that demonstrate that the methods we provided for checking soliton solutions and phase drawings for a variety of nonlinear models are effective and useful.
