Phase trajectories and Chaos theory for dynamical demonstration and explicit propagating wave formation

This paper is subjected to study the nonlinear integrable model which is the (3+1)-dimensional Boussinesq equation which has a lot of applications in engineering and modern sciences. To find and examine the analytical exact solitary wave solutions of (3+1)-dimensional Boussinesq equation, a modified generalized exponential rational functional method is exerted. As a result, waves, singular periodic, hyperbolic, and trigonometric type solutions are obtained. These acquired solutions are more innovative and encouraging to researchers in their endeavor to study physical marvels. To illustrate how some selected exact solutions propagate, the graphical representation in 2D, Contour, and 3D of those solutions is provided with various parametric values. The considered equation is additionally transformed into the planar dynamical structure by applying the Galilean transformation. All potential phase portraits of the dynamical system are investigated using the theory of bifurcation. The Hamiltonian function of the dynamical system of differential equations is established to see that, the system is conservative over time. The presentation of energy levels through graphics provides valuable insights, and it demonstrates that the model has solutions that can be expressed in closed form. The periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D, and time series are also observable once the dynamical system is subjected to an external force. Meanwhile, the sensitivity of the derived solutions is carefully examined for a range of initial conditions.