Bifurcation, chaos, and soliton dynamics in the integrable space curve model: A hybrid analytical and artificial neural network approach

This work investigates the integrable Akbota–Gudekli–Kairat–Zhaidary (AGKZ) equation, which belongs to the class of integrable space curves and surfaces. The model exhibits rich nonlinear dynamics and supports a broad spectrum of soliton structures. A Galilean transformation is first employed to derive the corresponding dynamical system. The qualitative behavior is investigated using bifurcation theory within planar dynamical systems. Introducing trigonometric, hyperbolic, elliptic, and Gaussian perturbations, chaotic dynamics are characterized via phase portraits, time series, 3D plots, and multistability analysis. Lyapunov exponents and numerical bifurcation analyses confirm the existence of strange attractors, highlighting the rich nonlinear phenomena uncovered in this study. Exact analytical solutions are constructed using the double G′/G,1/G-expansion method, resulting in multiple classes of soliton solutions expressed in terms of trigonometric, hyperbolic, and rational functions. These include periodic, bright, dark, singular kink, and M-shaped solitons. To evaluate accuracy and stability of obtained solutions, the Levenberg–Marquardt artificial neural network (LM-ANN) technique is employed. This neural framework effectively learns the analytical soliton profiles and verifies them through fitness curves and regression performance metrics, which exhibit high accuracy and rapid convergence. The study presents a unified analytical and neural framework for soliton modeling, contributing to the theoretical advancement of integrable systems, soliton stability, nonlinear wave theory, and computational mathematical physics.