Some Novel Approaches for Analyzing the Unforced and Forced Duffing–Van der Pol Oscillators

Abstract

In the current investigation, both unforced and forced Duffing–Van der Pol oscillator (DVdPV) oscillators with a strong nonlinearity and external periodic excitations are analyzed and investigated analytically and numerically using some new and improved approaches. The new approach is constructed based on Krylov–Bogoliubov–Metroolsky method (KBMM). One of the most important features of this approach is that we do not need to solve a system of differential equations, but only solve a system of algebraic equations. Moreover, the ease and faster of applying this method gives high-accurate results and this approach is better than many approaches in the literature. This approach is applied for analyzing (un)forced DVdP oscillators. Also, some improvements are made to He’s frequency-amplitude formulation in order to solve unforced DVdP oscillator to obtain high-accurate results. Furthermore, the He’s homotopy perturbation method (He’s HPM) is employed for analyzing unforced DVdP oscillator. The comparison between all mentioned approaches is carried out. The application of our approach is not limited to (un)forced DVdPV oscillators only but can be applied to analyze many higher-order nonlinearity oscillators for any odd power and it gives more accurate results than other approaches. Both used methods and obtained approximations will help many researchers in general and plasma physicists in particular in the interpretation of their results.

Closed-Form Solutions to a Forced Damped Rotational Pendulum Oscillator

“…( 1) and the exact solution (20) to the i.v.p. (19) (linearized form to the i.v.p. ( 1)) are compared with the RK4 numerical approximation to the i.v.p.…”

“…The model of nonlinear oscillators that describes the motion of different types of pendulum oscillators with different rotations and directions is one of the most successful models for describing and modeling many physical and engineering applications such as the wind vibration [11][12][13][14][15][16][17][18][19][20]. To find an exact solution to the dynamical system of the pendulum oscillation is not an easy task, and sometimes it can be impossible due to its nonlinearity.…”

“…However, some numerical methods or analytical approximation methods could be more suitable to get some approximate solutions that may be close enough to the exact solutions [21]. Many authors derived in details the rotating pendulum equation of motion and solved this equation by using different approaches [11][12][13][14][15][16][17][18][19][20]. For instance, He et al [11] used the HPM to derive an analytic solution to the rotating pendulum oscillator.…”

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].…”

Analytical approximations to a generalized forced damped complex Duffing oscillator: multiple scales method and KBM approach

“…) general damped CDO (2) and find some approximation for them using two different approaches, including the hybrid HPM (H-HPM), sometimes also called the KBM method [28][29][30], and the multiple scales method (MSM) [31,32].…”