A Streamlined Analytical Approach for Exact Solutions and Chaotic Dynamics in Two Higher-Dimensional Nonlinear Models

This paper employs a concise method as a streamlined analytical framework for deriving exact analytical solutions with free parameters for the higher-dimensional ZK-BBM equation and the [Formula: see text]-dimensional Bogoyavlenskii system. While structurally related to traditional ansatz-based techniques, this approach reduces the reliance on predefined auxiliary equations or extensive symbolic computation by naturally deducing the governing linear ordinary differential equations during the algebraic balancing process. This offers a direct and manually verifiable derivation mechanism. A set of exact analytical solutions, including bell-shaped bright and kink solitons, are obtained and physically characterized through three-dimensional graphical representations. Dynamical analysis further reveals that these analytical solutions correspond geometrically to the homoclinic or heteroclinic orbits of the unperturbed systems, serving as baseline states for investigating nonlinear wave evolution. By introducing periodic external perturbations, the sensitivity, the coexistence of invariant orbits, and the chaotic behaviors of the corresponding two-dimensional dynamical systems are systematically investigated. The finite-time maximum Lyapunov exponent is used as a quantitative criterion to distinguish between regular quasi-periodic regimes and chaotic states. These results provide detailed insights into complex wave phenomena and the topological dynamics of higher-dimensional nonlinear evolution equations.