Exploring Nonlinear Dynamics and Chaos in the Modified Korteweg–de Vries–Zakharov–Kuznetsov Equation with NARX Neural Networks

This work examines the nonlinear dynamics of a generalized Korteweg–de Vries–Zakharov–Kuznetsov equation, a model that appears in plasma physics, shallow water flows, and nonlinear wave propagation. By applying a solitary-wave transformation, the governing partial differential equation is reduced to an autonomous dynamical system, enabling a direct study of its phase portraits and equilibrium behavior. Stability of the fixed points is assessed through Jacobian matrices and eigenvalue classification, revealing parameter regimes that admit saddle states, centers, and oscillatory structures. The system’s richer behavior is explored by varying key parameters, with phase-space trajectories exhibiting periodic, quasiperiodic, and irregular wave patterns. To probe the onset of complexity, we employ several diagnostic tools, including time-series evolution, Lyapunov exponents, bifurcation analysis, sensitivity tests, and Poincaré sections, which together indicate transitions to chaotic motion. The resulting dynamics are further captured using a nonlinear autoregressive neural network, which accurately reproduces the observed trajectories. The combination of analytical and computational perspectives provides a clear framework for understanding this generalized equation and offers a practical approach for investigating other nonlinear systems with a similar structure.