Bifurcation analysis of the Van der Pol oscillator

2018 ◽  
Vol 11 (85) ◽  
pp. 4245-4252
Author(s):  
Diana Marcela Devia Narvaez ◽  
German Correa Velez ◽  
Diego Fernando Devia Narvaez
Keyword(s):  
Bifurcation Analysis ◽  
Van Der Pol Oscillator ◽  
Van Der Pol

BIFURCATION SCENARIO OF A THREE-DIMENSIONAL VAN DER POL OSCILLATOR

1993 ◽  
Vol 03 (02) ◽  
pp. 399-404 ◽  
Author(s):  
T. SÜNNER ◽  
H. SAUERMANN
Keyword(s):  
Bifurcation Diagram ◽  
Bifurcation Analysis ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Global Bifurcation ◽  
Dynamical Behaviour ◽  
Van Der Pol Oscillator ◽  
Two Dimensional ◽  
Van Der Pol ◽  
Dimensional Models

Nonlinear self-excited oscillations are usually investigated for two-dimensional models. We extend the simplest and best known of these models, the van der Pol oscillator, to a three-dimensional one and study its dynamical behaviour by methods of bifurcation analysis. We find cusps and other local codimension 2 bifurcations. A homoclinic (i.e. global) bifurcation plays an important role in the bifurcation diagram. Finally it is demonstrated that chaos sets in. Thus the system belongs to the few three-dimensional autonomous ones modelling physical situations which lead to chaotic behavior.

Dynamics of a Simplified van der Pol Oscillator Revisited

2007 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arun Rajendran
Keyword(s):  
Bifurcation Analysis ◽  
Local Stability ◽  
Van Der Pol Oscillator ◽  
Nonsmooth Dynamics ◽  
Mapping Technique ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Chaotic Motions ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

In this paper, the dynamic characteristics of a simplified van der Pol oscillator are investigated. From the theory of nonsmooth dynamics, the structures of periodic and chaotic motions for such an oscillator are developed via the mapping technique. The periodic motions with a certain mapping structures are predicted analytically for m-cycles with n-periods. Local stability and bifurcation analysis for such motions are carried out. The (m:n)-periodic motions are illustrated. The further investigation of the stable and unstable periodic motions in such a system should be completed. The chaotic motion based on the Levinson donuts should be further discussed.

Analytical Solutions of Period-1 Motions in a Periodically Forced van der Pol Oscillator

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Eigenvalue Analysis ◽  
Van Der Pol Oscillator ◽  
Comprehensive Understanding ◽  
Van Der Pol ◽  
Stability And Bifurcation ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcation Analysis ◽  
The Stability

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 balanced method. The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis, and numerical illustrations of periodic-1 solutions are given to verify the approximate motion. This investigation provides more accurate solutions of period-1 motions in the van der pol oscillator for a better and comprehensive understanding of motions in such an oscillator.

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

scholarly journals Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang ◽  
Yuancen Wang
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Van Der Pol Oscillator ◽  
Order System ◽  
Damping Force ◽  
Van Der Pol ◽  
Restoring Force ◽  
White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

Sign in / Sign up

Export Citation Format

Share Document