Entrained oscillations in a system of mutually coupled van der pol oscillators with time delay in the coupling paths

1987 ◽  
Vol 113 (3) ◽  
pp. 503-527
Author(s):  
A. Kouda ◽  
S. Mori
Keyword(s):  
Time Delay ◽  
Van Der Pol ◽  
Van Der Pol Oscillators

Death Behavior of Mean-Field Coupled van der Pol Oscillators Induced by Time-Delayed Feedback

2020 ◽  
Vol 30 (03) ◽  
pp. 2050040
Author(s):  
Qin Guo ◽  
Xige Yang ◽  
Jiankang Liu ◽  
Wei Xu
Keyword(s):  
Time Delay ◽  
Mean Field ◽  
Delayed Feedback ◽  
Van Der Pol ◽  
Coupled Nonlinear Oscillators ◽  
Oscillatory State ◽  
Feedback Strength ◽  
Death State ◽  
Van Der Pol Oscillators ◽  
Transition Points

Explosive death in coupled nonlinear oscillators has been an active area of extensive research in nonlinear dynamics in the recent decades. Depending on proper choice of network topology, coupling scenarios, and feedback strength, explosive death can be revealed. In this work, for the first time, we report the effect of delayed feedback on the death behavior in an ensemble of identical mean-field coupled van der Pol oscillators. In both systems with or without time delay, the normalized amplitude exhibits an abrupt transition between the oscillatory state and the death state. Intriguingly, the presence of time delay in the coupling may induce the normalized amplitude of all oscillators in the network to experience a step-like descent with small jumps in approaching the death state, pulling back the forward and backward transition points. The backward transition point has been explicitly obtained, which is confirmed by the numerical results.

Time Delay Control for Two van der Pol Oscillators

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Attilio Maccari
Keyword(s):  
Time Delay ◽  
Vibration Control ◽  
Periodic Solutions ◽  
Original System ◽  
Excitation Amplitude ◽  
Van Der Pol ◽  
Flow Equations ◽  
Technical Requirements ◽  
Stable Periodic ◽  
Van Der Pol Oscillators

A method for time delay vibration control of the principal and fundamental resonances of two nonlinearly coupled van der Pol oscillators is investigated Using the asymptotic perturbation method, four slow-flow equations on the amplitude and phase of the oscillators are obtained. Their fixed points correspond to a two-period quasi-periodic phase-locked motion for the original system. In the system without control, stable periodic solutions (if any) exist only for fixed values of amplitude and phase and depend on the system parameters and excitation amplitude. In many cases, the amplitudes of these solutions do not correspond to the technical requirements. On the contrary, it is demonstrated that, if vibration control terms are added, stable two-period quasi-periodic solutions with arbitrarily chosen amplitudes can be accomplished. Therefore, an effective vibration control is possible if appropriate time delay and feedback gains are chosen.

scholarly journals The Principal Parametric Resonance of Coupled van der Pol Oscillators under Feedback Control

2012 ◽  
Vol 19 (3) ◽  
pp. 365-377 ◽  
Author(s):  
Xinye Li ◽  
Huabiao Zhang ◽  
Lijuan He
Keyword(s):  
Time Delay ◽  
Feedback Control ◽  
Parametric Resonance ◽  
Relaxation Oscillation ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Flow Equations ◽  
Principal Parametric Resonance ◽  
Van Der Pol Oscillators ◽  
And Control

The principal parametric resonance of two van der Pol oscillators under coupled position and velocity feedback control with time delay is investigated analytically and numerically on the assumption that only one of the two oscillators is parametrically excited and the feedback control is linear. The slow-flow equations are obtained by the averaging method and simplified by truncating the first term of Taylor expansions for those terms with time delay. It is found that nontrivial solutions corresponding to periodic motions exist only for one oscillator if no feedback control is applied although the two oscillators are nonlinearly coupled. Based on Levenberg-Marquardt method, the effects of excitation and control parameters on the amplitude of periodic solutions of the system are graphically given. It can be seen that both of the two oscillators can be excited in periodic vibration with proper feedback. However, the amplitudes of the periodic vibrations are independent of the sign of feedback gains. In addition, the influence of time delay on the response of the system is periodic. In terms of numerical simulations, it is shown that both of the two oscillators can also have quasi-periodic motions, periodic motions about a new equilibrium position and other complex motions such as relaxation oscillation when feedback control is considered.

Asymptotic Wave Solutions for the Model of a Medium with Van Der Pol Oscillators

2014 ◽  
Vol 59 (9) ◽  
pp. 932-938
Author(s):  
V.A. Danylenko ◽  
◽  
S.I. Skurativskyi ◽  
I.A. Skurativska ◽  
◽  
...  
Keyword(s):  
Van Der Pol ◽  
Wave Solutions ◽  
Van Der Pol Oscillators

scholarly journals Modal synchronization of coupled bistable van der Pol oscillators

2021 ◽  
Vol 143 ◽  
pp. 110555
Author(s):  
I.B. Shiroky ◽  
O.V. Gendelman
Keyword(s):  
Van Der Pol ◽  
Van Der Pol Oscillators

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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