Numerical study of the fractional Van der Pol equation using the modified Atangana-Baleanu-Caputo derivative

This study investigated the nonlinear fractional Van der Pol equation using the modified Atangana-Baleanu-Caputo (mABC) fractional derivative operator. The Van der Pol oscillator is a fundamental model in nonlinear dynamics, with applications spanning electrical circuits, cardiac rhythms, neural activity, mechanical systems, fluid dynamics, control engineering, robotics, and economic cycles. Unlike the classical ABC fractional derivative, the modified operator offers enhanced capability for modeling systems with long-term memory and hereditary properties. The mainobjective of this study is to test the applicability of themodified operator for solving this nonlinear oscillator problem, and to analyze how the fractional order and the damping terminfluence the system dynamics. A predictor-corrector numerical scheme based on Lagrange interpolation was developed and implemented for the coupled fractional system. The key findings demonstrate that the fractional order 𝜉 significantly influences the damping characteristics and oscillatory behavior. The proposed numerical scheme achieves good agreement with the reference methods, including the seventh-order Runge-Kutta (RK7) and three-step Adams-Bashforth (AB3) schemes, with observed quantitative differences attributable to the distinct kernel structures of the fractional operators. The numerical simulations performed in MATLAB R2017b yielded graphical and tabular results. The novelty of this study lies in the application of a mABC derivative to the Van der Pol system.