In this paper, we investigated the (3+1)-dimensional nonlinear wave equation in gas-bubble-containing liquid, which can delineate the dynamic behaviors of waves in the mechanics of fluids. It plays a crucial role in explaining the propagation of weakly nonlinear waves, which are commonly encountered in natural sciences, medicine, fluid mechanics, and engineering. Based on the bifurcation method, the dynamical behavior of the reduced system was performed through the phase portrait. By using different parametric conditions, new soliton wave solutions were derived, including periodic wave-type, periodic breaking wave-type, solitary wave-type, and unbounded wave-type solutions. Additionally, the visualizations of the obtained exact solutions were employed to present the wave behavior of the equation. It is of great importance for understanding the inherent evolution laws of nonlinear waves in bubbly media within fluid dynamics. To ensure stability for the solutions, a sensitivity analysis was performed within a framework for a dynamical system, as well as sensitivities for parameters to the points of equilibrium. The orbital structures are discussed. Furthermore, the chaotic behavior of the equation under external periodic forcing was also investigated, and the existence of periodic motions was verified via Lyapunov exponents. Compared with the literature, for the first time, we adopt the bifurcation method of dynamical systems to solve the equation and obtained novel exact solutions.