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.

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.

Dynamics of Two Van Der Pol Oscillators Coupled via a Bath

2003 ◽  
Author(s):  
Erika Camacho ◽  
Richard Rand ◽  
Howard Howland
Keyword(s):  
Software Package ◽  
Bifurcation Theory ◽  
Floquet Theory ◽  
Van Der Pol Oscillator ◽  
Direct Connection ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Existence And Stability ◽  
Van Der Pol Oscillators ◽  
The Stability

In this work we study a system of two van der Pol oscillators, x and y, coupled via a “bath” z: x¨−ε(1−x2)x˙+x=k(z−x)y¨−ε(1−y2)y˙+y=k(z−y)z˙=k(x−z)+k(y−z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

scholarly journals Anticontrol of Hopf Bifurcation and Control of Chaos for a Finance System through Washout Filters with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Huitao Zhao ◽  
Mengxia Lu ◽  
Junmei Zuo
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Feedback Control ◽  
Price Index ◽  
Financial System ◽  
Control Of Chaos ◽  
Control Laws ◽  
Washout Filter ◽  
And Control

A controlled model for a financial system through washout-filter-aided dynamical feedback control laws is developed, the problem of anticontrol of Hopf bifurcation from the steady state is studied, and the existence, stability, and direction of bifurcated periodic solutions are discussed in detail. The obtained results show that the delay on price index has great influences on the financial system, which can be applied to suppress or avoid the chaos phenomenon appearing in the financial system.

Dynamics of distributed-order hyperchaotic complex van der Pol oscillators and their synchronization and control

2020 ◽  
Vol 135 (1) ◽  
Author(s):  
Gamal M. Mahmoud ◽  
Ahmed A. Farghaly ◽  
Tarek M. Abed-Elhameed ◽  
Shaban A. Aly ◽  
Ayman A. Arafa
Keyword(s):  
Van Der Pol ◽  
Van Der Pol Oscillators ◽  
And Control ◽  
Distributed Order

Stochastic Stability of Mathieu-Van Der Pol System with Delayed Feedback Control

2011 ◽  
Vol 105-107 ◽  
pp. 132-138
Author(s):  
Chang Shui Feng ◽  
Shuang Lin Chen
Keyword(s):  
Time Delay ◽  
Lyapunov Exponent ◽  
Feedback Control ◽  
Lyapunov Stability ◽  
Original System ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Control Force ◽  
Van Der Pol ◽  
Time Delayed Feedback

The asymptotic Lyapunov stability with probability one of Mathieu-Van der Pol system with time-delayed feedback control under wide-band noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original system, and the asymptotic Lyapunov stability with probability one of the original system can be determined approximately by using the Lyapunov exponent. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. The theoretical results are well verified through digital simulation.

Sign in / Sign up

Export Citation Format

Share Document