scholarly journals Delayed Feedback Control and Bifurcation Analysis of an Autonomy System

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Zhen Wang ◽  
Huitao Zhao ◽  
Xiangyu Kong
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic Oscillation ◽  
Stable State ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Stable Periodic Orbit ◽  
Stable Periodic ◽  
Functional Differential ◽  
Time Delayed Feedback ◽  
Zero Equilibrium

An autonomy system with time-delayed feedback is studied by using the theory of functional differential equation and Hassard’s method; the conditions on which zero equilibrium exists and Hopf bifurcation occurs are given, the qualities of the Hopf bifurcation are also studied. Finally, several numerical simulations are given; which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable state or a stable periodic orbit.

Delayed Feedback Control and Bifurcation Analysis in a Chaotic Chemostat System

2015 ◽  
Vol 25 (06) ◽  
pp. 1550087 ◽  
Author(s):  
Zhichao Jiang ◽  
Wanbiao Ma
Keyword(s):  
Center Manifold ◽  
Chaotic Oscillation ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Stable Steady State ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Stable Periodic ◽  
Form Theory ◽  
The Stability

In this paper, the effect of delay on a nonlinear chaotic chemostat system with delayed feedback is investigated by regarding delay as a parameter. At first, the stability of the positive equilibrium and the existence of Hopf bifurcations are obtained. Then an explicit algorithm for determining the direction and the stability of the bifurcating periodic solutions is derived by using the normal form theory and center manifold argument. Finally, some numerical simulation examples are given, which indicate that the chaotic oscillation can be converted into a stable steady state or a stable periodic orbit when delay passes through certain critical values.

scholarly journals Bifurcation Analysis for a Kind of Nonlinear Finance System with Delayed Feedback and Its Application to Control of Chaos

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Rongyan Zhang
Keyword(s):  
Center Manifold ◽  
Chaotic Oscillation ◽  
Delayed Feedback ◽  
Stable Steady State ◽  
Normal Form Theory ◽  
Finance System ◽  
Stable Periodic ◽  
Form Theory ◽  
The Stability ◽  
Time Delayed Feedback

A kind of nonlinear finance system with time-delayed feedback is considered. Firstly, by employing the polynomial theorem to analyze the distribution of the roots to the associate characteristic equation, the conditions of ensuring the existence of Hopf bifurcation are given. Secondly, by using the normal form theory and center manifold argument, we derive the explicit formulas determining the stability, direction, and other properties of bifurcating periodic solutions. Finally, we give several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.

Criticality of Hopf Bifurcation in Precision Motion Stage With PID and Time-Delayed Feedback Controls

2020 ◽  
Author(s):  
Sunit K. Gupta ◽  
Jiamin Wang ◽  
Oumar R. Barry
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Delayed Feedback ◽  
State Feedback Control ◽  
Lumped Parameter Model ◽  
Delayed Feedback Control ◽  
Use Of Time ◽  
Motion Stage ◽  
Precision Motion ◽  
Time Delayed Feedback

Abstract The use of precision motion stages is very popular among advanced manufacturing and machining industries. However, the performance of these motion stages is usually undermined by friction-induced vibration. In this paper, we propose the use of time-delayed feedback control to minimize the undesirable effects of friction-induced vibrations. The use of time-delayed feedback control is well established in the literature; however, the use of time-delayed feedback control in PID controlled motion-stages has not been explored yet. Here, we consider a lumped parameter model of the PID controlled precision motion stage with a linear time-delayed state feedback control. The dynamical friction in the systemis modeled using the LuGre model. Stability and nonlinear analysis of the system are carried out using analytical methods. The stability analysis reveals the existence of multiple stability lobes and codimension-2 Hopf points for a given choice of system parameters. Also, the nature of Hopf bifurcation is determined by using the method of multiple scales. We observe the existence of both subcritical and supercritical Hopf bifurcations in the system, depending on the choice of control parameters. This observation implies that the nonlinearity in the system could both be stabilizing or destabilizing in nature.

BIFURCATION DYNAMICS IN CONTROL SYSTEMS

1999 ◽  
Vol 09 (01) ◽  
pp. 287-293 ◽  
Author(s):  
GUANRONG CHEN ◽  
JIALIANG LU ◽  
BRENT NICHOLAS ◽  
SWATIPRAKASH M. RANGANATHAN
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Control Systems ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Main Concern ◽  
Feedback Strategy ◽  
Feedback Control Systems ◽  
Time Delayed Feedback ◽  
Control Purpose

This paper is to report the observation that when the popular time-delayed feedback strategy is used for control purpose, it may actually create unwanted bifurcations. Hopf bifurcation created by delayed feedback control is the main concern of this article, but some other types of bifurcations are also observed to exist in such delayed-feedback control systems. The observations are illustrated by computer simulations.

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