A Modified Analytical Data-Mapping Framework for Symmetric Multiscale Soliton and Chaotic Dynamics

The (3 + 1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff equation, a model that describes long-wave interactions and has numerous applications in mathematics, engineering, and physics, is examined in this work. First, a wave transformation is used to reduce the equation to lower dimensions. The modified Khater method is then used to derive different types of solitary wave solutions, such as chirped, kink, periodic, and kink-bright types. By allocating suitable constant parameters, 3D, 2D, and contour plots are created to demonstrate the physical behavior of these solutions. Phase portraits are used to qualitatively analyze the undisturbed planar system using bifurcation theory. The system is then perturbed by an external force, resulting in chaotic dynamics. Chaos in the system is confirmed using multiple diagnostic tools, including time series plots, Poincaré sections, chaotic attractors, return maps, bifurcation diagrams, power spectra, and Lyapunov exponents. The stability of the model is further investigated with varying initial conditions. A bidirectional scatter plot technique, which efficiently reveals overlapping regions using data point distributions, is presented for comparing solution behaviors. Overall, this work offers useful tools for advancing applied mathematics research as well as a deeper understanding of nonlinear wave dynamics.