An analytical investigation of the Van Der Waals gas system: Dynamics insights into bifurcation, optical pattern along with sensitivity and chaotic analysis

This paper focuses on obtaining exact solutions for the nonlinear Van der Waals gas system using the modified Khater method. Renowned as one of the latest and most precise analytical schemes for nonlinear evolution equations, this method has proven its efficacy by generating diverse solutions for the model under consideration. The governing equation is transformed into an ordinary differential equation through a well-suited wave transformation. This analytical simplification makes it possible to use the provided methods to derive trigonometric, rational, and hyperbolic solutions. To illuminate the physical behavior of the model, graphical plots of selected solutions are presented. By selecting appropriate values for arbitrary factors, this visual representation enhances comprehension of the dynamical system. Furthermore, the system undergoes a certain transformation to become a planar dynamical system, and the bifurcation analysis is examined. Additionally, the sensitivity analysis of the dynamical system is conducted using the Runge–Kutta method to confirm that slight alterations in the initial conditions have minimal impact on the stability of the solution. The presence of chaotic dynamics in the Van der Waals gas system is explored by introducing a perturbed term in the dynamical system. Two and three-dimensional phase profiles are used to illustrate these chaotic behaviors.