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Hopf Bifurcation Analysis in a Lorenz Equation with Extended Delay Feedback

Author: GuoYuZuo
Tutor: HanBo
School: Harbin Institute of Technology
Course: Applied Mathematics
Keywords: Lorenz system Delayed Feedback Hopf bifurcation Periodic solution Chaos
CLC: O231
Type: Master's thesis
Year: 2011
Downloads: 28
Quote: 0
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Abstract


Delayed feedback control method has become an important class of methods in the control areas , a growing number of examples show that considering the current and historical status of the controller design can effectively play the role of feedback . Study delay feedback control system for some inherently chaotic phenomenon , has become an important issue in the nonlinear dynamics of chaos . The main content of this study is the analysis of the differential equation model with extended delay feedback control branch . In addition , for the study of this type of feedback control of chaotic phenomena role Lorenz system is selected as the basic model for analysis . First, the problem will be transformed . Through analysis , the following conclusions : with extended delay feedback control system of differential equations can be transformed into retarded functional differential equations with infinite delay or Neutral Differential Equations . Secondly, local Lorentz equation with extended delay feedback control . Neutral Functional Differential Equations linearization of the characteristic equation roots distribution analysis gives a sufficient condition for the stability of the equilibrium point of the system results Hopf bifurcation , and then found that the presence of the system stability switch delay conclusions. Another numerical simulation shows accompanied by changes in the stability of the equilibrium point , the the Lorentz equation chaos phenomenon was into switch -like appear . Kinetic properties , center manifold theory and norms method in order to get the system in the vicinity of the branch point of Hopf bifurcation branch direction and stability bifurcation formula . Finally , the results of the numerical simulation to support the existing theoretical results given instance .

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