scholarly journals Solutions of the Force-Free Duffing-van der Pol Oscillator Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Muhammad Jamil ◽  
Syed Anwar Ali ◽  
Nadeem Alam Khan
Keyword(s):  
Approximate Solution ◽  
Numerical Solution ◽  
Approximate Method ◽  
Governing Equation ◽  
Laplace Transformation ◽  
Large Time ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Alternative Framework ◽  
Runge Kutta

A new approximate method for solving the nonlinear Duffing-van der pol oscillator equation is proposed. The proposed scheme depends only on the two components of homotopy series, the Laplace transformation and, the Padé approximants. The proposed method introduces an alternative framework designed to overcome the difficulty of capturing the behavior of the solution and give a good approximation to the solution for a large time. The Runge-Kutta algorithm was used to solve the governing equation via numerical solution. Finally, to demonstrate the validity of the proposed method, the response of the oscillator, which was obtained from approximate solution, has been shown graphically and compared with that of numerical solution.

NUMERICAL SOLUTION FOR DUFFING-VAN DER POL OSCILLATOR VIA BLOCK METHOD

2020 ◽  
Vol 10 (1) ◽  
pp. 1857-8365
Author(s):  
A. F. Nurullah ◽  
M. Hassan ◽  
T. J. Wong ◽  
L. F. Koo
Keyword(s):  
Numerical Solution ◽  
Van Der Pol Oscillator ◽  
Block Method ◽  
Van Der Pol

An Approximate Solution for Period-1 Motions in a Periodically Forced Van Der Pol Oscillator

2014 ◽  
Vol 9 (3) ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Approximate Solution ◽  
Fourier Series ◽  
Analytical Solutions ◽  
Amplitude Spectrum ◽  
Van Der Pol Oscillator ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Approximate Analytical Solutions ◽  
Series Expression

In this paper the approximate analytical solutions of period-1 motion in the periodically forced van der Pol oscillator are obtained by the generalized harmonic balance (HB) method. Such an approximate solution of periodic motion is given by the Fourier series expression, and the convergence of such an expression is guaranteed by the Fourier series theory of periodic functions. The approximate solution is different from traditional, approximate solution because the number of total harmonic terms (N) is determined by the precision of harmonic amplitude quantity level, set by the investigator (e.g., AN≤ɛ and ɛ=10-8). The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis of the coefficient dynamical systems of the Fourier series expressions of periodic solutions, and numerical illustrations of period-1 motions are compared to verify the analytical solutions of periodic motions. The trajectories and analytical harmonic amplitude spectrum for stable and unstable periodic motions are presented. The harmonic amplitude spectrum shows the harmonic term effects on periodic motions, and one can directly know which harmonic terms contribute on periodic motions and the convergence of the Fourier series expression is clearly illustrated.

scholarly journals Phase Fitted And Amplification Fitted Of Runge-Kutta-Fehlberg Method Of Order 4(5) For Solving Oscillatory Problems

2020 ◽  
Vol 17 (2(SI)) ◽  
pp. 0689
Author(s):  
Mohammed Salih ◽  
Fudziah Ismail ◽  
Norazak Senu
Keyword(s):  
Periodic Solution ◽  
Approximate Solution ◽  
Numerical Solution ◽  
Present Method ◽  
Phase Lag ◽  
Real Solution ◽  
Oscillatory Solutions ◽  
Dispersion Error ◽  
Recent Method ◽  
Runge Kutta

In this paper, the proposed phase fitted and amplification fitted of the Runge-Kutta-Fehlberg method were derived on the basis of existing method of 4(5) order to solve ordinary differential equations with oscillatory solutions. The recent method has null phase-lag and zero dissipation properties. The phase-lag or dispersion error is the angle between the real solution and the approximate solution. While the dissipation is the distance of the numerical solution from the basic periodic solution. Many of problems are tested over a long interval, and the numerical results have shown that the present method is more precise than the 4(5) Runge-Kutta-Fehlberg method.

scholarly journals He’s Max-Min Approach for Coupled Cubic Nonlinear Equations Arising in Packaging System

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Jun Wang
Keyword(s):  
Computer Simulation ◽  
Approximate Solution ◽  
Numerical Solution ◽  
Nonlinear Equations ◽  
High Accuracy ◽  
Packaging System ◽  
Novel Method ◽  
Runge Kutta

He’s inequalities and the Max-Min approach are briefly introduced, and their application to a coupled cubic nonlinear packaging system is elucidated. The approximate solution is obtained and compared with the numerical solution solved by the Runge-Kutta algorithm yielded by computer simulation. The result shows a great high accuracy of this method. The research extends the application of He’s Max-Min approach for coupled nonlinear equations and provides a novel method to solve some essential problems in packaging engineering.

scholarly journals Classical and Higher Order Runge-Kutta Methods with Nonlinear Shooting Technique for Solving Van der Pol (VdP) Equation

2018 ◽  
Vol 66 (1) ◽  
pp. 43-47
Author(s):  
Farhana Ahmed Simi ◽  
Goutam Saha
Keyword(s):  
Exact Solution ◽  
Perturbation Method ◽  
Numerical Solution ◽  
Accurate Result ◽  
Research Work ◽  
Single Step ◽  
The Other ◽  
Van Der Pol ◽  
Shooting Technique ◽  
Runge Kutta

The goal of the research work is to examine the improvement of numerical solution of VdP equation. The well-known VdP equation is governed by the second order nonlinear ODE and then solved numerically using the classical Runge-Kutta (RK) method, RK-Fehlberg method of order five, Verner method of order eight and Cash-Karp method of order six with nonlinear shooting technique. In this work, numerical simulations have been carried out using NVdP code which is written in MATHEMATICA. Also, the accuracy and efficiency of the solution of VdP equation using different RK methods with nonlinear shooting technique has been investigated. For analysis of accuracy, the approximate exact solution obtained by perturbation method is used for the comparison. It is observed that all the different RK methods give accurate result of the VdP equation. But the classical RK method shows slightly better performance than the other single step techniques. Dhaka Univ. J. Sci. 66(1): 43-47, 2018 (January)

SPECTRAL AND CORRELATION ANALYSIS OF SPIRAL CHAOS

2003 ◽  
Vol 03 (02) ◽  
pp. L213-L221 ◽  
Author(s):  
VADIM S. ANISHCHENKO ◽  
TATJANA E. VADIVASOVA ◽  
ANDREY S. KOPEIKIN ◽  
GALINA I. STRELKOVA ◽  
JÜRGEN KURTHS
Keyword(s):  
Diffusion Coefficient ◽  
Large Time ◽  
Effective Diffusion ◽  
Small Time ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Time Intervals ◽  
Exponential Law ◽  
Amplitude Fluctuations ◽  
Different Time Scales

We study numerically the behavior of the autocorrelation function (ACF) and the power spectrum of spiral attractors without and in the presence of noise. It is shown that the ACF decays exponentially and has two different time scales. The rate of the ACF decrease is defined by the amplitude fluctuations on small time intervals, i.e., when τ < τ cor , and by the effective diffusion coefficient of the instantantaneous phase on large time intervals. It is also demonstrated that the ACF in the Poincare map also decreases according to the exponential law exp (- λ+ k), where λ+ is the positive Lyapunov exponent. The obtained results are compared with the theory of fluctuations for the Van der Pol oscillator.

Sign in / Sign up

Export Citation Format

Share Document