ABSTRACT The numerical approximation of nonlinear chaotic differential systems, such as the modified stretch‐twist‐fold (STF) flow and multi‐bond chaotic attractors, presents a significant challenge due to their sensitive dependence on initial conditions and complex dynamics where analytical solutions are unattainable. While many numerical techniques exist, they often struggle to simultaneously achieve the high accuracy, absolute stability and computational efficiency required to reliably capture these intricate behaviors. To address this gap, this study introduces and validates an optimized local linearization hybrid block method (LLHBM). The proposed method uniquely combines Newton‐Raphson and Gauss‐Siedel‐type linearization techniques within a robust framework of the hybrid block methods, integrating linear multistep methods with interpolation and collocation. A key optimization involves selecting the intra‐step points as the normalized extrema of Legendre polynomials, which demonstrably enhances both the accuracy and stability of the numerical scheme. Theoretical analysis confirms that the LLHBM is absolutely stable and achieves a high‐order truncation error of , where is the number of intra‐step points. Through detailed numerical experiments, including analyses of residual and convergence error norms, the study validates the robustness, efficiency and precision of the numerical method in capturing the complex dynamics of the chaotic systems. These results establish the LLHBM as a reliable tool for solving nonlinear chaotic differential equations, offering superior computational efficiency and stability compared to other traditional numerical methods.
Oloniiju et al. (Thu,) studied this question.