Bifurcation–Orbit classification, chaos, and nonlinear wave dynamics in (2+1)-dimensional complex coupled dispersionless system: An improved Cham method

This study investigates the dynamical behavior and exact nonlinear wave structures of an integrable (2+1)-dimensional complex coupled dispersionless system, which serves as a model for understanding nonlinear wave interactions in diverse physical phenomena such as fluid dynamics and plasma physics. The system is reduced to a planar dynamical system using a traveling wave transformation, which is analyzed through bifurcation theory. Different orbit types are classified, leading to exact analytical solutions expressed in terms of hyperbolic and Jacobi elliptic functions, including bell-shaped and periodic wave structures. Sensitivity analysis reveals the system’s response to variations in initial conditions and parameters, highlighting its sensitivity and potential instability. Chaotic dynamics are explored by introducing trigonometric, elliptic, and Gaussian perturbations, with phase portraits, time series, 3D plots, return maps, and multistability analysis highlighting complex behaviors. Lyapunov exponents confirm the presence of strange attractors. Furthermore, the improved Cham method is applied to derive exact analytical solutions in the form of trigonometric, hyperbolic, and exponential functions, enabling a systematic exploration of anti-bell-shaped wave behavior. The combined framework of bifurcation analysis, chaos theory, and analytical method provides a comprehensive understanding of nonlinear wave dynamics. The results demonstrate the model’s relevance to nonlinear optical fibers, revealing new soliton structures, wave patterns, and spatial symmetry effects with potential applications in optical systems.