The global bifurcation characteristics of the forced van der Pol oscillator

1996 ◽  
Vol 7 (1) ◽  
pp. 3-19 ◽  
Author(s):  
Jian-Xue Xu ◽  
Jun Jiang
Keyword(s):  
Global Bifurcation ◽  
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.

Global bifurcation structure of a Bonhoeffer-van der Pol oscillator driven by periodic pulse trains

1994 ◽  
Vol 72 (1) ◽  
pp. 55-67 ◽  
Author(s):  
Taishin Nomura ◽  
Shunsuke Sato ◽  
Shinji Doi ◽  
Jose P. Segundo ◽  
Michael D. Stiber
Keyword(s):  
Global Bifurcation ◽  
Van Der Pol Oscillator ◽  
Bifurcation Structure ◽  
Van Der Pol ◽  
Pulse Trains ◽  
Periodic Pulse

Global Bifurcation Analysis of a Duffing–Van der Pol Oscillator with Parametric Excitation

2014 ◽  
Vol 24 (04) ◽  
pp. 1450051 ◽  
Author(s):  
Qun Han ◽  
Wei Xu ◽  
Xiaole Yue
Keyword(s):  
State Space ◽  
Invariant Manifolds ◽  
Global Analysis ◽  
Global Bifurcation ◽  
Van Der Pol Oscillator ◽  
Fractal Boundary ◽  
Van Der Pol ◽  
Cell State ◽  
Boundary Crisis ◽  
Basin Boundaries

In this paper, a composite cell state space is constructed by a multistage division of the continuous phase space. Based on point mapping method, global properties of dynamical systems can be analyzed more accurately and efficiently, and any small regions can be refined for clearly depicting some special basin boundaries in this cell state space. Global bifurcation of a Duffing–Van der Pol oscillator subjected to harmonic parametrical excitation is investigated. Attractors, basins of attraction, basin boundaries, saddles and invariant manifolds have been obtained. As the amplitude of excitation increases, it can be observed that the boundary crisis occurs twice. Then a basin boundary with Wada property appears in the state space and undergoes metamorphosis in the chaotic boundary crisis. At last, two attractors merge into a chaotic one when they simultaneously collide with the chaotic saddle embedded in the fractal boundary. All these results show the effectiveness of the proposed method in global analysis.

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.

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