GLOBAL EXISTENCE OF PERIODIC SOLUTIONS IN THE LINEARLY COUPLED MACKEY–GLASS SYSTEM

2011 ◽  
Vol 21 (03) ◽  
pp. 711-724 ◽  
Author(s):  
YANQIU LI ◽  
WEIHUA JIANG
Keyword(s):  
Hopf Bifurcation ◽  
Global Existence ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Glass System ◽  
Positive Equilibrium ◽  
Higher Dimensional ◽  
Distribution Of Eigenvalues ◽  
Existence Of Periodic Solutions ◽  
The Stability

The dynamics of a linearly coupled Mackey–Glass system with delay are investigated. Based on the distribution of eigenvalues, we prove that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and obtain the bifurcation set in the parameter plane. The explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived, using the theories of normal form and center manifold. The global existence of periodic solutions is established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson's criterion for higher dimensional ordinary differential equations due to [Li & Muldowney, 1993].

HOPF BIFURCATION ANALYSIS IN A MACKEY–GLASS SYSTEM

2007 ◽  
Vol 17 (06) ◽  
pp. 2149-2157 ◽  
Author(s):  
JUNJIE WEI ◽  
DEJUN FAN
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Higher Dimensional ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Equation With Delay ◽  
Bendixson Criterion

The dynamics of a Mackey–Glass equation with delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney [1994].

scholarly journals Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Jianming Zhang ◽  
Lijun Zhang ◽  
Chaudry Masood Khalique
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Finite Delay ◽  
The Stability ◽  
Predator System

The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.

scholarly journals The effect of delayed feedback on the dynamics of an autocatalysis reaction–diffusion system

2018 ◽  
Vol 23 (5) ◽  
pp. 749-770 ◽  
Author(s):  
Xin Wei ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Arbitrary Order ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Delayed Feedback ◽  
Positive Equilibrium ◽  
Distribution Of Eigenvalues ◽  
The Stability

This paper deals with an arbitrary-order autocatalysis model with delayed feedback subject to Neumann boundary conditions. We perform a detailed analysis about the effect of the delayed feedback on the stability of the positive equilibrium of the system. By analyzing the distribution of eigenvalues, the existence of Hopf bifurcation is obtained. Then we derive an algorithm for determining the direction and stability of the bifurcation by computing the normal form on the center manifold. Moreover, some numerical simulations are given to illustrate the analytical results. Our studies show that the delayed feedback not only breaks the stability of the positive equilibrium of the system and results in the occurrence of Hopf bifurcation, but also breaks the stability of the spatial inhomogeneous periodic solutions. In addition, the delayed feedback also makes the unstable equilibrium become stable under certain conditions.

scholarly journals Global existence of positive periodic solutions of a general differential equation with neutral type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ming Liu ◽  
Jun Cao ◽  
Xiaofeng Xu
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Global Existence ◽  
Periodic Solutions ◽  
Neutral Type ◽  
Positive Equilibrium ◽  
Positive Periodic Solutions ◽  
General Differential Equation ◽  
Glass Model ◽  
The Stability

AbstractIn this paper, the dynamics of a general differential equation with neutral type are investigated. Under certain assumptions, the stability of positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. And global existence of positive periodic solutions is established by using the global Hopf bifurcation result of Krawcewicz et al. Finally, by taking neutral Nicholson’s blowflies model and neutral Mackey–Glass model as two examples, some numerical simulations are carried out to illustrate the analytical results.

scholarly journals Bifurcation of a Microelectromechanical Nonlinear Coupling System with Delay Feedback

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yanqiu Li ◽  
Wei Duan ◽  
Shujian Ma ◽  
Pengfei Li
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Electromechanical Coupling ◽  
Critical Values ◽  
Nonlinear Coupling ◽  
Coupling System ◽  
Distribution Of Eigenvalues ◽  
The Stability

The dynamics of a kind of electromechanical coupling deformable micromirror device torsion micromirror with delay are investigated. Based on the distribution of eigenvalues, we prove that a sequence of Hopf bifurcation occurs at the equilibrium as the delay increases and obtain the critical values of Hopf bifurcation. Explicit algorithms for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived, using the theories of normal form and center manifold.

Hopf bifurcation analysis in a model of oscillatory gene expression with delay

2009 ◽  
Vol 139 (4) ◽  
pp. 879-895 ◽  
Author(s):  
Junjie Wei ◽  
Chunbo Yu
Keyword(s):  
Gene Expression ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Normal Forms ◽  
Higher Dimensional ◽  
Expression Model ◽  
Oscillatory Gene Expression ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Gene Expression Model

The dynamics of a gene expression model with time delay are investigated. The investigation confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. An explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions has been derived by using the theory of the centre manifold and the normal forms method. The global existence of periodic solutions has been established using a global Hopf bifurcation result by Wu and a Bendixson criterion for higher-dimensional ordinary differential equations due to Li and Muldowney.

Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System

2018 ◽  
Vol 28 (11) ◽  
pp. 1850136 ◽  
Author(s):  
Ben Niu ◽  
Yuxiao Guo ◽  
Yanfei Du
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Immune Cell ◽  
Tumor Treatment ◽  
Delay Effect ◽  
Stable Periodic ◽  
The Stability ◽  
Bifurcation And Stability

Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay [Formula: see text] of immune system where the immune cell has a probability [Formula: see text] in killing tumor cells. We find increasing time delay [Formula: see text] destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability [Formula: see text] on Hopf bifurcation values is also discussed.

scholarly journals Bifurcation Analysis in a Delayed Diffusive Leslie-Gower Model

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Shuling Yan ◽  
Xinze Lian ◽  
Weiming Wang ◽  
Youbin Wang
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Neumann Boundary ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Normal Form Theorem ◽  
The Stability

We investigate a modified delayed Leslie-Gower model under homogeneous Neumann boundary conditions. We give the stability analysis of the equilibria of the model and show the existence of Hopf bifurcation at the positive equilibrium under some conditions. Furthermore, we investigate the stability and direction of bifurcating periodic orbits by using normal form theorem and the center manifold theorem.

Global Hopf Bifurcation in a Phytoplankton–Zooplankton System with Delay and Diffusion

2021 ◽  
Vol 31 (08) ◽  
pp. 2150114
Author(s):  
Ming Liu ◽  
Jun Cao ◽  
Xiaofeng Xu
Keyword(s):  
Hopf Bifurcation ◽  
Global Existence ◽  
Upper And Lower Solutions ◽  
Positive Periodic Solutions ◽  
Global Hopf Bifurcation ◽  
System With Delay ◽  
Local Hopf Bifurcation ◽  
Distribution Of Eigenvalues ◽  
The Stability ◽  
And Diffusion

In this paper, the dynamics of a phytoplankton–zooplankton system with delay and diffusion are investigated. The positivity and persistence are studied by using the comparison theorem and upper and lower solutions method. The stability of steady states and the existence of local Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. And the global existence of positive periodic solutions is established by using the global Hopf bifurcation result given by Wu [1996]. Finally, some numerical simulations are carried out to illustrate the analytical results.

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