An Adaptive Residual-Controlled Overlapping Taylor Method for Nonlinear Duffing, Van der Pol, and Mathieu-Type Oscillators

This study presents a new adaptive high-order numerical framework, called the Adaptive Residual-Controlled Overlapping Taylor (ARCOT) method, for the solution of strongly nonlinear oscillatory systems. The proposed approach combines high-order local Taylor expansion, overlapping continuation intervals, residual-controlled adaptivity, and adaptive continuation interval regulation within a unified computational framework. Unlike conventional Taylor continuation methods based on fixed interval propagation, ARCOT continuously monitors the governing equation residual and dynamically adjusts continuation intervals according to the local nonlinear behavior of the solution. Overlapping continuation points are additionally employed to improve long-time numerical stability and suppress cumulative propagation errors. The effectiveness of the proposed methodology is investigated through several benchmark nonlinear oscillators, including Duffing, cubic–quintic, nonlinear Van der Pol, Mathieu-type, and double-well systems. Numerical results obtained using ARCOT are compared with RK4, adaptive RK45, Taylor-based approaches, and available analytical or semi-analytical solutions reported in the literature. The obtained results demonstrate that ARCOT provides highly accurate displacement responses, stable phase-plane trajectories, and reliable long-time integration behavior without noticeable numerical drift. Residual evolution and adaptive continuation histories further confirm that the proposed residual-control mechanism effectively regulates local approximation quality while automatically adapting continuation intervals according to nonlinear response characteristics. The numerical investigations indicate that ARCOT provides a robust, accurate, and flexible computational framework for nonlinear oscillatory systems with considerable extension potential toward fractional, delay, chaotic, and distributed dynamical systems.