Controlling of chaos by weak periodic perturbations in Duffing-van der Pol oscillator

Pramana ◽  
1993 ◽  
Vol 41 (4) ◽  
pp. 295-309 ◽  
Author(s):  
S Rajasekar
Keyword(s):  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Periodic Perturbations ◽  
Controlling Of Chaos

CONTROLLING OF CHAOS IN BONHOEFFER- VAN DER POL OSCILLATOR

1992 ◽  
Vol 02 (01) ◽  
pp. 201-204 ◽  
Author(s):  
S. RAJASEKAR ◽  
M. LAKSHMANAN
Keyword(s):  
External Noise ◽  
Chaotic Regime ◽  
Van Der Pol Oscillator ◽  
Control Algorithms ◽  
External Parameter ◽  
Parametric Perturbation ◽  
Van Der Pol ◽  
Controlling Of Chaos ◽  
Adaptive Control Algorithm ◽  
Suppression Of Chaos

In this letter controlling of chaos in Bonhoeffer-van der Pol oscillator is investigated using various different control algorithms. Specifically, using an adaptive control algorithm we analyse the resetting of the system dynamics from a chaotic attractor to a limit cycle attractor. We demonstrate that, starting from a chaotic regime, it is possible to bring the system to a regular regime by means of a small parametric perturbation as well as by the addition of a weak external force. It is then shown that one can eliminate chaotic evolution by stabilizing an unstable periodic orbit by making small adjustments to an external parameter. Finally, we show the suppression of chaos by adding appropriate external noise.

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.

scholarly journals On Periodic Perturbations of Asymmetric Duffing–Van-der-Pol Equation

2014 ◽  
Vol 24 (05) ◽  
pp. 1450061 ◽  
Author(s):  
Albert D. Morozov ◽  
Olga S. Kostromina
Keyword(s):  
Saddle Point ◽  
Limit Cycles ◽  
Small Neighborhood ◽  
Integrable Equation ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Periodic Perturbations ◽  
Autonomous Equation ◽  
Nonautonomous Equation ◽  
Time Periodic

Time-periodic perturbations of an asymmetric Duffing–Van-der-Pol equation close to an integrable equation with a homoclinic "figure-eight" of a saddle are considered. The behavior of solutions outside the neighborhood of "figure-eight" is studied analytically. The problem of limit cycles for an autonomous equation is solved and resonance zones for a nonautonomous equation are analyzed. The behavior of the separatrices of a fixed saddle point of the Poincaré map in the small neighborhood of the unperturbed "figure-eight" is ascertained. The results obtained are illustrated by numerical computations.

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.

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