scholarly journals Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control

2008 ◽  
Vol 338 (2) ◽  
pp. 993-1007 ◽  
Author(s):  
Suqi Ma ◽  
Qishao Lu ◽  
Zhaosheng Feng
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Duffing Oscillator ◽  
Van Der Pol ◽  
Double Hopf Bifurcation ◽  
Delay Feedback Control

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.

DOUBLE HOPF BIFURCATION FOR STUART–LANDAU SYSTEM WITH NONLINEAR DELAY FEEDBACK AND DELAY-DEPENDENT PARAMETERS

2007 ◽  
Vol 10 (04) ◽  
pp. 423-448 ◽  
Author(s):  
SUQI MA ◽  
QISHAO LU ◽  
S. JOHN HOGAN
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Resonant Oscillation ◽  
Double Hopf Bifurcation ◽  
Dependent Parameter ◽  
Stable Periodic ◽  
Delay Feedback Control ◽  
Delay Dependent ◽  
Nonlinear Delay

A Stuart–Landau system under delay feedback control with the nonlinear delay-dependent parameter e-pτ is investigated. A geometrical demonstration method combined with theoretical analysis is developed so as to effectively solve the characteristic equation. Multi-stable regions are separated from unstable regions by allocations of Hopf bifurcation curves in (p,τ) plane. Some weak resonant and non-resonant oscillation phenomena induced by double Hopf bifurcation are discovered. The normal form for double Hopf bifurcation is deduced. The local dynamical behavior near double Hopf bifurcation points are also clarified in detail by using the center manifold method. Some states of two coexisting stable periodic solutions are verified, and some torus-broken procedures are also traced.

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.

scholarly journals Double Hopf Bifurcation of a Hematopoietic System with State Feedback Control

2018 ◽  
Vol 10 (4) ◽  
pp. 116
Author(s):  
Suqi Ma
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
State Feedback ◽  
Threshold Value ◽  
State Feedback Control ◽  
Hematopoietic Stem ◽  
Double Hopf Bifurcation ◽  
Periodical Solution ◽  
Varying Time Delay ◽  
Manifold Theory

The dynamics of a system composed of hematopoietic stem cells and its relationship with neutrophils is ubiquitous due to periodic oscillating behavior induce cyclical neutropenia. Underlying the methodology of state feedback control with two time delays, double Hopf bifurcation occurs as varying time delay to reach its threshold value. By applying center manifold theory, the analytical analysis of system exposed the different dynamical feature in the classified regimes near double Hopf point. The novel dynamics as periodical solution and quasi-periodical attractor coexistence phenomena are explored and verified  by numerical simulation.

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.

DOUBLE HOPF BIFURCATION IN DELAYED VAN DER POL–DUFFING EQUATION

2013 ◽  
Vol 23 (01) ◽  
pp. 1350014 ◽  
Author(s):  
YUTING DING ◽  
WEIHUA JIANG ◽  
PEI YU
Keyword(s):  
Hopf Bifurcation ◽  
Critical Point ◽  
Duffing Equation ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Van Der Pol ◽  
Dimensional Parameter ◽  
Center Manifold Reduction ◽  
Double Hopf Bifurcation ◽  
Reduction Methods

In this paper, we study dynamics in delayed van der Pol–Duffing equation, with particular attention focused on nonresonant double Hopf bifurcation. Both multiple time scales and center manifold reduction methods are applied to obtain the normal forms near a double Hopf critical point. A comparison between these two methods is given to show their equivalence. Bifurcations are classified in a two-dimensional parameter space near the critical point. Numerical simulations are presented to demonstrate the applicability of the theoretical results.

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