The Dynamical Landscape of the Negative-Order (3+1)-Dimensional Calogero–Bogoyavlenskii–Schiff Equation

A new negative-order form of the 3+1-dimensional Calogero–Bogoyavlenskii–Schiff equation is examined in this investigation. This equation plays an important role in accurately describing the thermodynamic properties of mixtures, particularly in chemical engineering applications. Through the use of wave transformations, the model is reduced to a nonlinear differential equation. To obtain exact solutions, two recently developed analytical techniques, namely the Riccati modified extended simple equation method and the generalized projective Riccati equation method, are applied. The solutions derived take the form of hyperbolic, trigonometric, and rational functions, representing different patterns of solitary wave profiles. By assigning specific values to the constants involved, corresponding three-dimensional, two-dimensional, and density plots are generated. In addition, the dynamical system associated with the model is constructed and carefully analyzed through bifurcation analysis, computation of Lyapunov exponents, investigation of quasiperiodic waves, exploration of chaotic behavior, and sensitivity analysis, all illustrated with relevant graphical representations. The study shows that the adopted methods are well suited to the considered model, and the results provide a useful foundation for further research in a wide range of scientific fields.