Analytical solutions of the combined Kairat-Ⅱ-Ⅹ equation: a dynamical perspective on bifurcation, chaos, energy, and sensitivity

In this study, we apply the $ (m+\frac{1}{\Phi'}) $-expansion and modified extended tanh function (METHF) methods to investigate the exact solutions of the new Kairat-Ⅱ-Ⅹ model. Using these methods, new exact solutions are derived for the proposed model. The hyperbolic, periodic, and singular forms of exact solutions are among those obtained. The propagating behaviors of wave solutions are depicted using three-dimensional (3D), contour, and two-dimensional (2D) surfaces, providing comprehensive visualizations of the wave dynamics. Physical meanings of the chosen solutions are determined through these simulations. Traveling wave solutions of nonlinear partial differential equations can be obtained using the techniques presented in this paper since they have been shown to be dependable, strong, and effective. Furthermore, the phase portrait has been thoroughly analyzed according to the equilibrium points, and various scenarios have been visualized using these portraits. With the use of time series graphs, Poincaré maps, and 3D and 2D plots, the impact of the perturbation term has been thoroughly investigated. The system's periodic, quasi-periodic, and chaotic structures are clearly illustrated by these depictions. The results enhance the understanding of the new Kairat-Ⅱ-Ⅹ equation's dynamic structure and how it applies to real-world events.