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.

Power series approach to nonlinear oscillators

“…This work has many applications in different field such as physics Ref. 13 and financial market Ref. 14, and we can make this for many other applied areas that use oscillating behavior.…”

“…5 Several important oscillators with important applications in physics and engineering have attracted the attention of many researches. [12][13][14][15][16][17] For example, the excited spring pendulum has been studied well using the so-called multi-scale method, in which the authors obtained approximate solutions of the equations of motion of the system. 12,13 Furthermore, the Van-der Pol and Duffing oscillators have been considered in many works.…”

“…[12][13][14][15][16][17] For example, the excited spring pendulum has been studied well using the so-called multi-scale method, in which the authors obtained approximate solutions of the equations of motion of the system. 12,13 Furthermore, the Van-der Pol and Duffing oscillators have been considered in many works. [13][14][15][16][17] In these references, one can see that the Van-der Pol oscillator was analyzed using some perturbation techniques to obtain approximate solution to its dynamical equations.…”

“…12,13 Furthermore, the Van-der Pol and Duffing oscillators have been considered in many works. [13][14][15][16][17] In these references, one can see that the Van-der Pol oscillator was analyzed using some perturbation techniques to obtain approximate solution to its dynamical equations. 14,15 On the other hand, the Duffing oscillator was also investigated in Refs.…”

Multiple scales method for analyzing a forced damped rotational pendulum oscillator with gallows

“…The Multiple Scales Method (MSM) and Krýlov-Bogoliúbov-Mitropólsky method (KBMM) were employed to provide approximate solutions for a time Delay Duffing-Helmholtz equation [25]. Furthermore, both KBMM and MSM were used for analyzing and solving several nonlinear oscillators with strong nonlinearity [26][27][28][29][30][31][32].…”