Mathematical analysis of a model for thymus infection with discrete and distributed delays

2014 ◽  
Vol 07 (06) ◽  
pp. 1450070 ◽  
Author(s):  
M. Prakash ◽  
P. Balasubramaniam
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Center Manifold Theorem ◽  
Different Types ◽  
Normal Form Method ◽  
Bifurcation And Stability ◽  
Theoretical Results ◽  
Hiv 1

In this paper, the dynamics of mathematical model for infection of thymus gland by HIV-1 is analyzed by applying some perturbation through two different types of delays such as in terms of Hopf bifurcation analysis. Further, the conditions for the existence of Hopf bifurcation are derived by evaluating the characteristic equation. The direction of Hopf bifurcation and stability of bifurcating periodic solutions are determined by employing the center manifold theorem and normal form method. Finally, some of the numerical simulations are carried out to validate the derived theoretical results and main conclusions are included.

scholarly journals Stability and Hopf Bifurcation Analysis of Coupled Optoelectronic Feedback Loops

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Gang Zhu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Stable Equilibrium ◽  
Feedback Loops ◽  
Asymptotically Stable ◽  
Center Manifold Theorem ◽  
Feedback Strength ◽  
Coupling Parameters ◽  
Normal Form Method

The dynamics of a coupled optoelectronic feedback loops are investigated. Depending on the coupling parameters and the feedback strength, the system exhibits synchronized asymptotically stable equilibrium and Hopf bifurcation. Employing the center manifold theorem and normal form method introduced by Hassard et al. (1981), we give an algorithm for determining the Hopf bifurcation properties.

scholarly journals Synchronized Hopf Bifurcation Analysis in a Delay-Coupled Semiconductor Lasers System

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Gang Zhu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Semiconductor Lasers ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Stability Switches ◽  
Center Manifold Theorem ◽  
Coupling Parameters ◽  
Normal Form Method ◽  
The Stability

The dynamics of a system of two semiconductor lasers, which are delay coupled via a passive relay within the synchronization manifold, are investigated. Depending on the coupling parameters, the system exhibits synchronized Hopf bifurcation and the stability switches as the delay varies. Employing the center manifold theorem and normal form method, an algorithm is derived for determining the Hopf bifurcation properties. Some numerical simulations are carried out to illustrate the analysis results.

scholarly journals Hopf Bifurcation in a Delayed SEIQRS Model for the Transmission of Malicious Objects in Computer Network

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Juan Liu
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Computer Network ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Normal Form Method ◽  
Theoretical Results

A delayed SEIQRS model for the transmission of malicious objects in computer network is considered in this paper. Local stability of the positive equilibrium of the model and existence of local Hopf bifurcation are investigated by regarding the time delay due to the temporary immunity period after which a recovered computer may be infected again. Further, the properties of the Hopf bifurcation are studied by using the normal form method and center manifold theorem. Numerical simulations are also presented to support the theoretical results.

scholarly journals Hopf Bifurcation of an Improved SLBS Model under the Influence of Latent Period

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Chunming Zhang ◽  
Wanping Liu ◽  
Jing Xiao ◽  
Yun Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Latent Time ◽  
Normal Form Method ◽  
The Stability ◽  
Theoretical Results

A model applicable to describe the propagation of computer virus is developed and studied, along with the latent time incorporated. We regard time delay as a bifurcating parameter to study the dynamical behaviors including local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when the time delay passes through a sequence of critical values. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is given by using the normal form method and center manifold theorem. Finally, illustrative examples are given to support the theoretical results.

scholarly journals Bifurcation Analysis of a Lotka-Volterra Mutualistic System with Multiple Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Xin-You Meng ◽  
Hai-Feng Huo
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Time Delays ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Multiple Delays ◽  
Center Manifold Theorem ◽  
Normal Form Method ◽  
Three Delays ◽  
Theoretical Results

A class of Lotka-Volterra mutualistic system with time delays of benefit and feedback delays is introduced. By analyzing the associated characteristic equation, the local stability of the positive equilibrium and existence of Hopf bifurcation are obtained under all possible combinations of two or three delays selecting from multiple delays. Not only explicit formulas to determine the properties of the Hopf bifurcation are shown by using the normal form method and center manifold theorem, but also the global continuation of Hopf bifurcation is investigated by applying a global Hopf bifurcation result due to Wu (1998). Numerical simulations are given to support the theoretical results.

scholarly journals Stability and Hopf Bifurcation for a Delayed Computer Virus Model with Antidote in Vulnerable System

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Zizhen Zhang ◽  
Yougang Wang ◽  
Massimiliano Ferrara
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Local Stability ◽  
Center Manifold ◽  
Computer Virus ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Virus Model ◽  
Bifurcation And Stability

A delayed computer virus model with antidote in vulnerable system is investigated. Local stability of the endemic equilibrium and existence of Hopf bifurcation are discussed by analyzing the associated characteristic equation. Further, direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are investigated by using the normal form theory and the center manifold theorem. Finally, numerical simulations are presented to show consistency with the obtained results.

scholarly journals Nonlinear Dynamics of a PI Hydroturbine Governing System with Double Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Hongwei Luo ◽  
Jiangang Zhang ◽  
Wenju Du ◽  
Jiarong Lu ◽  
Xinlei An
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Points ◽  
Bifurcation Theorem ◽  
Center Manifold Theorem ◽  
Governing System ◽  
Normal Form Method ◽  
The Stability ◽  
System Frequency ◽  
Realistic Significance ◽  
Theoretical Results

A PI hydroturbine governing system with saturation and double delays is generated in small perturbation. The nonlinear dynamic behavior of the system is investigated. More precisely, at first, we analyze the stability and Hopf bifurcation of the PI hydroturbine governing system with double delays under the four different cases. Corresponding stability theorem and Hopf bifurcation theorem of the system are obtained at equilibrium points. And then the stability of periodic solution and the direction of the Hopf bifurcation are illustrated by using the normal form method and center manifold theorem. We find out that the stability and direction of the Hopf bifurcation are determined by three parameters. The results have great realistic significance to guarantee the power system frequency stability and improve the stability of the hydropower system. At last, some numerical examples are given to verify the correctness of the theoretical results.

scholarly journals Hopf Bifurcation of a Delayed Ecoepidemic Model with Ratio-Dependent Transmission Rate

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Wanjun Xia ◽  
Soumen Kundu ◽  
Sarit Maitra
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Transmission Rate ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Ratio Dependent ◽  
Theoretical Results

A delayed ecoepidemic model with ratio-dependent transmission rate has been proposed in this paper. Effects of the time delay due to the gestation of the predator are the main focus of our work. Sufficient conditions for local stability and existence of a Hopf bifurcation of the model are derived by regarding the time delay as the bifurcation parameter. Furthermore, properties of the Hopf bifurcation are investigated by using the normal form theory and the center manifold theorem. Finally, numerical simulations are carried out in order to validate our obtained theoretical results.

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.

Stability and Hopf Bifurcation of a Delayed Mutualistic System

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Ruimin Zhang ◽  
Xiaohui Liu ◽  
Chunjin Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Stage Structure ◽  
Center Manifold Theory ◽  
Mutualistic Relationship ◽  
Normal Form Method ◽  
Explicit Formulae ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

In this paper, we study a classic mutualistic relationship between the leaf cutter ants and their fungus garden, establishing a time delay mutualistic system with stage structure. We investigate the stability and Hopf bifurcation by analyzing the distribution of the roots of the associated characteristic equation. By means of the center manifold theory and normal form method, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some numerical simulations are carried out for illustrating the theoretical results.

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