ON THE STUDY OF DELAY FEEDBACK CONTROL AND ADAPTIVE SYNCHRONIZATION NEAR SUB-CRITICAL HOPF BIFURCATION

2008 ◽  
Vol 19 (01) ◽  
pp. 169-185 ◽  
Author(s):  
DIBAKAR GHOSH ◽  
PAPRI SAHA ◽  
A. ROY CHOWDHURY
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Coupling Parameter ◽  
Nonlinear Dynamical System ◽  
Chaotic Regime ◽  
Adaptive Synchronization ◽  
Unstable Periodic Orbits ◽  
Nonlinear Dynamical ◽  
Form Analysis ◽  
Delay Feedback Control

The effect of delay feedback control and adaptive synchronization is studied near sub-critical Hopf bifurcation of a nonlinear dynamical system. Previously, these methods targeted the nonlinear systems near their chaotic regime but it is shown here that they are equally applicable near the branch of unstable solutions. The system is first analyzed from the view point of bifurcation, and the existence of Hopf bifurcation is established through normal form analysis. Hopf bifurcation can be either sub-critical or super-critical, and in the former case, unstable periodic orbits are formed. Our aim is to control them through a delay feedback approach so that the system stabilizes to its nearest stable periodic orbit. At the vicinity of the sub-critical Hopf point, adaptive synchronization is studied and the effect of the coupling parameter and the speed factor is analyzed in detail. Adaptive synchronization is also studied when the system is in the chaotic regime.

scholarly journals Delay Feedback Control of the Lorenz-Like System

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Qin Chen ◽  
Jianguo Gao
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Control Method ◽  
Functional Differential Equations ◽  
Variable Parameter ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Feedback Control Method ◽  
Delay Feedback Control

We choose the delay as a variable parameter and investigate the Lorentz-like system with delayed feedback by using Hopf bifurcation theory and functional differential equations. The local stability of the positive equilibrium and the existence of Hopf bifurcations are obtained. After that the direction of Hopf bifurcation and stability of periodic solutions bifurcating from equilibrium is determined by using the normal form theory and center manifold theorem. In the end, some numerical simulations are employed to validate the theoretical analysis. The results show that the purpose of controlling chaos can be achieved by adjusting appropriate feedback effect strength and delay parameters. The applied delay feedback control method in this paper is general and can be applied to other nonlinear chaotic systems.

BIFURCATION ANALYSIS OF A NFDE ARISING FROM MULTIPLE-DELAY FEEDBACK CONTROL

2011 ◽  
Vol 21 (03) ◽  
pp. 759-774 ◽  
Author(s):  
BEN NIU ◽  
JUNJIE WEI
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Feedback Control ◽  
Center Manifold ◽  
Amplitude Equation ◽  
Neutral Functional Differential Equation ◽  
Multiple Delay ◽  
Delay Feedback Control ◽  
Functional Differential ◽  
The Stability

We study the stability and Hopf bifurcation of a neutral functional differential equation (NFDE) which is transformed from an amplitude equation with multiple-delay feedback control. By analyzing the distribution of the eigenvalues, the stability and existence of Hopf bifurcation are obtained. Furthermore, the direction and stability of the Hopf bifurcation are determined by using the center manifold and normal form theories for NFDEs. Finally, we carry out some numerical simulations to illustrate the results.

Control of Large-Scale Dynamical Systems via Vector Lyapunov Functions

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Feedback Control ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Large Scale ◽  
Nonlinear Dynamical System ◽  
Control Vector ◽  
Nonlinear Dynamical ◽  
Vector Lyapunov Functions

This chapter introduces the notion of a control vector Lyapunov function as a generalization of control Lyapunov functions, showing that asymptotic stabilizability of a nonlinear dynamical system is equivalent to the existence of a control vector Lyapunov function. These control vector Lyapunov functions are used to develop a universal decentralized feedback control law for a decentralized nonlinear dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. The chapter also describes the connections between the notion of vector dissipativity and optimality of the proposed decentralized feedback control law. The proposed control framework is then used to construct decentralized controllers for large-scale nonlinear dynamical systems with robustness guarantees against full modeling uncertainty.

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