Bifurcation Control for a Kind of Non-Autonomous System with Time Delay

2010 ◽  
Vol 34-35 ◽  
pp. 1752-1756
Author(s):  
Chang Zhao Qian ◽  
Zhi Wen Wang ◽  
Chuang Wen Dong ◽  
Yang Liu
Keyword(s):  
Time Delay ◽  
Perturbation Method ◽  
Time Delays ◽  
Autonomous System ◽  
Primary Resonance ◽  
Stable Region ◽  
Bifurcation Equation ◽  
Van Der Pol ◽  
Average Equation ◽  
System With Time Delay

A forced van der Pol system with two time-delays is studied. The central aim is analyzing primary resonance of this system. Perturbation method is used to obtain the average equation and bifurcation equation with time-delays. Based on the average equation, the stable region of this system is discussed. Based on the bifurcation equation, the multivalued property of response amplitude is studied. The result indicates that this system can be well controlled with time delays.

Bifurcation Analysis of Parametrically Excited Moving String With Aerodynamic Forces

2012 ◽  
Author(s):  
Changzhao Qian ◽  
Changping Chen ◽  
Liming Dai
Keyword(s):  
Bifurcation Analysis ◽  
Parametric Analysis ◽  
Aerodynamic Force ◽  
Equation Of Motion ◽  
Stable Region ◽  
Second Law ◽  
Bifurcation Equation ◽  
Aerodynamic Forces ◽  
Average Equation ◽  
Prototypical Model

Considering the large deformation of the string, A prototypical model of a elastic moving string with aerodynamic forces is studied. The equation of motion is obtained by Newton’s second law. Then the Perturbation method is used to obtain the average equation and bifurcation equation. Based on the average equation, the stable region of this system is discussed. Based on the bifurcation equation, the multivalued property of response amplitude is studied. At last, the flutter effects of aerodynamic force are discussed by the parametric analysis.

scholarly journals Double Hopf Bifurcation with Strong Resonances in Delayed Systems with Nonlinerities

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
J. Xu ◽  
K. W. Chung
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Nonlinear Systems ◽  
Efficient Method ◽  
Cubic Nonlinearity ◽  
Van Der Pol ◽  
Delayed Systems ◽  
Double Hopf Bifurcation ◽  
Codimension Two ◽  
System With Time Delay

An efficient method is proposed to study delay-induced strong resonant double Hopf bifurcation for nonlinear systems with time delay. As an illustration, the proposed method is employed to investigate the 1 : 2 double Hopf bifurcation in the van der Pol system with time delay. Dynamics arising from the bifurcation are classified qualitatively and expressed approximately in a closed form for either square or cubic nonlinearity. The results show that 1 : 2 resonance can lead to codimension-three and codimension-two bifurcations. The validity of analytical predictions is shown by their consistency with numerical simulations.

Multiple Bifurcations and Complex Responses of Nonlinear Time-Delay Oscillators

2021 ◽  
Author(s):  
Xiaochen Mao ◽  
Fuchen Lei ◽  
Xingyong Li ◽  
Weijie Ding ◽  
Tiantian Shi
Keyword(s):  
Time Delay ◽  
Time Delays ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Electronic Circuit ◽  
Dynamic Features ◽  
Nonlinear Circuit ◽  
Amplitude Death ◽  
Van Der Pol ◽  
Dynamical Behaviors

Abstract In this paper, the dynamical properties of multiple van der Pol-Duffing oscillators with time delays are studied. The amplitude death and bifurcation curves in the parameter plane are determined by using the space decomposition method. Different patterns of bifurcated solutions are given on the basis of the symmetric bifurcation theory. The properties of bifurcated solutions are shown by using the norm forms on the center manifold. The interactions of bifurcations are discussed and their dynamical behaviors are shown. An electronic circuit platform is implemented by means of nonlinear circuit and time delay circuit. The revealed behaviors of the circuit reach an agreement with the obtained results. It is shown that the nonlinearity and time delays have great effects on the system performance and can induce interesting and abundant dynamic features.

scholarly journals Applying higher order stochastic averaging method to the Van der Pol system with timedelay under random excitation

2019 ◽  
Vol 2 (2) ◽  
pp. 102-109
Author(s):  
Hao Ngoc Duong ◽  
Anh Dong Nguyen ◽  
Dung Quang Nguyen
Keyword(s):  
Time Delay ◽  
Averaging Method ◽  
Order Approximation ◽  
Original System ◽  
Stochastic Averaging ◽  
Random Excitation ◽  
Stochastic Averaging Method ◽  
Higher Order ◽  
Van Der Pol ◽  
System With Time Delay

The paper investigated the Van der Pol system with time-delay under random excitation by the higher stochastic averaging method. The original system was expressed in terms without time-delay under the assumption that the state variabled of the system were slowly varying processed. Then the higher stochastic averaging method was applied on the approximation system. By this technique, the analytical expression of the stationary probability density function for the Van der Pol system with time-delay under random excitation was showed in higher order approximation for the first time. Effects of the parameter time-delay on the system’s response were investigated. The analytical results were suited well to numerical ones obtained by Monte-Carlo method. It was also showed that the higher order averaging solution was better than the one obtained by the traditional stochastic averaging method.

scholarly journals A New Design Method for PI-PD Control of Unstable Fractional-Order System with Time Delay

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Min Zheng ◽  
Tao Huang ◽  
Guangfeng Zhang
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Graphical Method ◽  
Stability Boundary ◽  
Fractional Order System ◽  
Stable Region ◽  
Order System ◽  
Pd Controller ◽  
The Stability ◽  
System With Time Delay

In this paper, a practical PI-PD controller parameter tuning method is proposed, which uses the incenter of the triangle and the Fermat point of the convex polygon to optimize the PI-PD controller. Combined with the stability boundary locus method, the PI-PD controller parameters that can ensure stability for the unstable fractional-order system with time delay are obtained. Firstly, the parameters of the inner-loop PD controller are determined by the centre coordinates of the CSR in the kd−kf plane. Secondly, a new graphical method is used to calculate the parameters of the PI controller, in which Fermat points in the CSR of (kp−ki) plane are selected. Furthermore, the method is extended to uncertain systems, and the PI-PD controller parameters are obtained by using the proposed method through common stable region of all stable regions. The proposed graphical method not only ensures the stability of the closed-loop system but also avoids the complicated optimization calculations. The superior control performance of this method is illustrated by simulation.

Synchronization of Chaotic Systems with Time Delays via Periodically Intermittent Control

2017 ◽  
Vol 26 (09) ◽  
pp. 1750139 ◽  
Author(s):  
Chao Liu ◽  
Zheng Yang ◽  
Dihua Sun ◽  
Xiaoyang Liu ◽  
Wanping Liu
Keyword(s):  
Time Delay ◽  
Time Delays ◽  
Chaotic Systems ◽  
Control Period ◽  
Feasible Region ◽  
Intermittent Control ◽  
Periodically Intermittent Control ◽  
First Order ◽  
Response System ◽  
System With Time Delay

This paper investigates the exponential synchronization of chaotic systems with time delays via periodically intermittent control. Under the new differential inequality, some novel synchronization criteria are derived. In contrast to the existing works, the proposed results are less conservative because they can obtain more precise synchronized rate under the identical control conditions and remove the restrictions on the control period (or the control width) and the time delay. By using special parameters, the feasible region D(ξ), which guarantees the response system synchronizes with the drive system with synchronized rate 0.5ξ, is obtained. The Lu chaotic attractor and a first-order chaotic system with time delay are presented to demonstrate the effectiveness of the proposed results.

Resonance behavior in coupled Van der Pol harmonic oscillators with controllers and delayed feedback

2020 ◽  
pp. 107754632093818
Author(s):  
Ashraf T EL-Sayed
Keyword(s):  
Frequency Response ◽  
Time Delays ◽  
Degrees Of Freedom ◽  
Primary Resonance ◽  
Delayed Feedback ◽  
Harmonic Oscillators ◽  
Feedback Signal ◽  
Response Curves ◽  
Internal Resonances ◽  
Van Der Pol

It has been revealed in the proposed work that a pair of delay positive position feedback control can lessen the vibration response of double Van der Pol oscillators with external forces. We also studied the effects of both the control and the delayed feedback signal gains to illustrate the low vibration amplitudes. The averaging perturbation process has been used to consider the frequency-response equations of amplitudes and modulation phases at the primary resonance and one-to-one internal resonances. According to the perturbation solutions for the four-degrees-of-freedom system, we presented the frequency response curves that were periodic in the time delays. The stability analysis presented in this study has shown optimum stable ranges. If the time delays increase, the steady-state amplitudes of the oscillator’s system will periodically result in few stable regions and more unstable ones. The numerical simulation has been introduced to check the analytical approximation. It was also found to be almost identical after presenting the comparison of the results.

scholarly journals Response of the Duffing-Van der Pol Oscillator under Position Feedback Control with Two Time Delays

2011 ◽  
Vol 18 (1-2) ◽  
pp. 377-386
Author(s):  
Xinye Li ◽  
Huabiao Zhang ◽  
Lijuan Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Trivial Solution ◽  
Time Delays ◽  
Linear Feedback ◽  
Nonlinear Feedback ◽  
Nontrivial Solutions ◽  
Van Der Pol ◽  
Position Feedback ◽  
The Stability

In this paper, the dynamics of Duffing-van der Pol oscillators under linear-plus-nonlinear position feedback control with two time delays is studied analytically and numerically. By the averaging method, together with truncation of Taylor expansions for those terms with time delay, the slow-flow equations are obtained from which the trivial and nontrivial solutions can be found. It is shown that the trivial solution can be stabilized by appropriate gain and time delay in linear feedback although it loses its stability via Hopf bifurcation and results in periodic solution for uncontrolled systems. And the stability of the trivial solution is independent of nonlinear feedback. Different from the case of the trivial solution, the stability of nontrivial solutions is also associated with nonlinear feedback besides linear feedback. Non-trivial solutions may lose their stability via saddle-node or Hopf bifurcation and the resulting response of the system may be quasi-periodic or chaotic. The feedback gains and time delays have great effects on the amplitude of the periodic solutions and their bifurcation control. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.

Sign in / Sign up

Export Citation Format

Share Document