On Perturbative Methods for Analyzing Third-Order Forced Van-der Pol Oscillators

Abstract

In this investigation, an (un)forced third-order/jerk Van-der Pol oscillatory equation is solved using two perturbative methods called the Krylov–Bogoliúbov–Mitropólsky method and the multiple scales method. Both the first- and second-order approximations for the unforced and forced jerk Van-der Pol oscillatory equations are derived in detail using the proposed methods. Comparative analysis is performed between the analytical approximations using the proposed methods and the numerical approximations using the fourth-order Runge–Kutta scheme. Additionally, the global maximum error to the analytical approximations compared to the Runge–Kutta numerical approximation is estimated.

Keywords

• jerk oscillator
• third-order non-linear ordinary differential equation
• Van-der Pol equation
• Krýlov–Bogoliúbov–Mitropólsky method
• multiple scales method
• analytical approximations