scholarly journals Hopf Bifurcation and Stability Analysis of a Congestion Control Model with Delay in Wireless Access Network

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Wen-bo Zhao ◽  
Xiao-ke Sun ◽  
Huicheng Wang
Keyword(s):  
Hopf Bifurcation ◽  
Access Network ◽  
Critical Value ◽  
Wireless Access ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Wireless Access Network ◽  
Form Theory ◽  
The Stability ◽  
Delay Differential

We drive a scalar delay differential system to model the congestion of a wireless access network setting. The Hopf bifurcation of this system is investigated using the control and bifurcation theory; it is proved that there exists a critical value of delay for the stability. When the delay value passes through the critical value, the system loses its stability and a Hopf bifurcation occurs. Furthermore, the direction and stability of the bifurcating periodic solutions are derived by applying the normal form theory and the center manifold theorem. Finally, some examples and numerical simulations are presented to show the feasibility of the theoretical results.

scholarly journals Direction and Stability of Bifurcating Periodic Solutions in a Delay-Induced Ecoepidemiological System

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
N. Bairagi
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Critical Value ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Reproduction Period ◽  
The Stability ◽  
Parameter Values ◽  
Bifurcating Periodic Solution

A SI-type ecoepidemiological model that incorporates reproduction delay of predator is studied. Considering delay as parameter, we investigate the effect of delay on the stability of the coexisting equilibrium. It is observed that there is stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theorem, the explicit formulae which determine the stability and direction of the bifurcating periodic solutions are determined. Computer simulations have been carried out to illustrate different analytical findings. Results indicate that the Hopf bifurcation is supercritical and the bifurcating periodic solution is stable for the considered parameter values. It is also observed that the quantitative level of abundance of system populations depends crucially on the delay parameter if the reproduction period of predator exceeds the critical value.

scholarly journals Dynamic Analysis and Hopf Bifurcation of a Lengyel–Epstein System with Two Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Long Li ◽  
Yanxia Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Stability Method ◽  
The Stability ◽  
Delay Differential

In this paper, a Lengyel–Epstein model with two delays is proposed and considered. By choosing the different delay as a parameter, the stability and Hopf bifurcation of the system under different situations are investigated in detail by using the linear stability method. Furthermore, the sufficient conditions for the stability of the equilibrium and the Hopf conditions are obtained. In addition, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained with the normal form theory and the center manifold theorem to delay differential equations. Some numerical examples and simulation results are also conducted at the end of this paper to validate the developed theories.

scholarly journals Stability and Hopf Bifurcation in a Delayed SEIRS Worm Model in Computer Network

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Computer Network ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Form Theory ◽  
Worm Model ◽  
The Stability

A delayed SEIRS epidemic model with vertical transmission in computer network is considered. Sufficient conditions for local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained by analyzing distribution of the roots of the associated characteristic equation. Furthermore, the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, a numerical example is presented to verify the theoretical analysis.

scholarly journals Dynamical Analysis of a Viral Infection Model with Delays in Computer Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Propagation Model ◽  
Dynamical Analysis ◽  
Infection Model ◽  
Virus Propagation ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
The Stability

This paper is devoted to the study of an SIRS computer virus propagation model with two delays and multistate antivirus measures. We demonstrate that the system loses its stability and a Hopf bifurcation occurs when the delay passes through the corresponding critical value by choosing the possible combination of the two delays as the bifurcation parameter. Moreover, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by means of the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate the obtained results.

scholarly journals Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yuanyuan Chen ◽  
Ya-Qing Bi
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Local Analysis ◽  
Normal Form Theory ◽  
Form Theory ◽  
Vector Borne ◽  
The Stability ◽  
Delay Differential

A delay-differential modelling of vector-borne is investigated. Its dynamics are studied in terms of local analysis and Hopf bifurcation theory, and its linear stability and Hopf bifurcation are demonstrated by studying the characteristic equation. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument.

STABILITY AND BIFURCATION ANALYSIS ON A DELAYED NEURAL NETWORK MODEL

2005 ◽  
Vol 15 (09) ◽  
pp. 2883-2893 ◽  
Author(s):  
XIULING LI ◽  
JUNJIE WEI
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Network Model ◽  
Neural Network Model ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Delayed Neural Network ◽  
The Stability

A simple delayed neural network model with four neurons is considered. Linear stability of the model is investigated by analyzing the associated characteristic equation. It is found that Hopf bifurcation occurs when the sum of four delays varies and passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. An example is given and numerical simulations are performed to illustrate the obtained results. Meanwhile, the bifurcation set is provided in the appropriate parameter plane.

Stability and Hopf Bifurcation in a Delayed SIS Epidemic Model with Double Epidemic Hypothesis

2018 ◽  
Vol 19 (6) ◽  
pp. 561-571
Author(s):  
Jiangang Zhang ◽  
Yandong Chu ◽  
Wenju Du ◽  
Yingxiang Chang ◽  
Xinlei An
Keyword(s):  
Hopf Bifurcation ◽  
Epidemic Model ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Characteristic Roots ◽  
Sis Epidemic Model ◽  
Form Theory ◽  
The Stability ◽  
Sis Epidemic

AbstractThe stability and Hopf bifurcation of a delayed SIS epidemic model with double epidemic hypothesis are investigated in this paper. We first study the stability of the unique positive equilibrium of the model in four cases, and we obtain the stability conditions through analyzing the distribution of characteristic roots of the corresponding linearized system. Moreover, we choosing the delay as bifurcation parameter and the existence of Hopf bifurcation is investigated in detail. We can derive explicit formulas for determining the direction of the Hopf bifurcation and the stability of bifurcation periodic solution by center manifold theorem and normal form theory. Finally, we perform the numerical simulations for justifying the theoretical results.

scholarly journals Hopf Bifurcation Analysis of a Synthetic Drug Transmission Model with Time Delays

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Zizhen Zhang ◽  
Fangfang Yang ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Transmission Model ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Synthetic Drug ◽  
The Stability

This paper is concerned with the Hopf bifurcation of a synthetic drug transmission model with two delays. Firstly, some sufficient conditions of delay-induced bifurcation for such a model are captured by using different combinations of the two delays as the bifurcation parameter. Secondly, based on the center manifold theorem and normal form theory, some sufficient conditions determining properties of the Hopf bifurcation such as the direction and the stability are established. Finally, to underline the effectiveness of the obtained results, some numerical simulations are ultimately addressed.

scholarly journals Stability and Hopf Bifurcation of Delayed Predator-Prey System Incorporating Harvesting

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Fengying Wei ◽  
Lanqi Wu ◽  
Yuzhi Fang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Theoretical Predictions ◽  
Form Theory ◽  
The Stability

A kind of delayed predator-prey system with harvesting is considered in this paper. The influence of harvesting and delay is investigated. Our results show that Hopf bifurcations occur as the delayτpasses through critical values. By using of normal form theory and center manifold theorem, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are obtained. Finally, numerical simulations are given to support our theoretical predictions.

scholarly journals Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped Cobb-Douglas Production Function

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Massimiliano Ferrara ◽  
Luca Guerrini ◽  
Giovanni Molica Bisci
Keyword(s):  
Hopf Bifurcation ◽  
Perturbation Method ◽  
Production Function ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

Matsumoto and Szidarovszky (2011) examined a delayed continuous-time growth model with a special mound-shaped production function and showed a Hopf bifurcation that occurs when time delay passes through a critical value. In this paper, by applying the center manifold theorem and the normal form theory, we obtain formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Moreover, Lindstedt’s perturbation method is used to calculate the bifurcated periodic solution, the direction of the bifurcation, and the stability of the periodic motion resulting from the bifurcation.

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