scholarly journals Dynamical Analysis of a Computer Virus Model with Delays

2016 ◽  
Vol 2016 ◽  
pp. 1-21 ◽  
Author(s):  
Juan Liu ◽  
Carlo Bianca ◽  
Luca Guerrini
Keyword(s):  
Hopf Bifurcation ◽  
Characteristic Root ◽  
Computer Virus ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Virus Model ◽  
Bifurcation And Stability

An SIQR computer virus model with two delays is investigated in the present paper. The linear stability conditions are obtained by using characteristic root method and the developed asymptotic analysis shows the onset of a Hopf bifurcation occurs when the delay parameter reaches a critical value. Moreover the direction of the Hopf bifurcation and stability of the bifurcating period solutions are investigated by using the normal form theory and the center manifold theorem. Finally, numerical investigations are carried out to show the feasibility of 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 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 in a Computer Virus Model with Multistate Antivirus

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Tao Dong ◽  
Xiaofeng Liao ◽  
Huaqing Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Computer Virus ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Form Theory ◽  
Virus Model ◽  
Theoretical Results

By considering that people may immunize their computers with countermeasures in susceptible state, exposed state and using anti-virus software may take a period of time, a computer virus model with time delay based on an SEIR model is proposed. We regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when time delay passes through a sequence of critical value. The linerized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.

scholarly journals Dynamics of a Computer Virus Propagation Model with Delays and Graded Infection Rate

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Zizhen Zhang ◽  
Limin Song
Keyword(s):  
Hopf Bifurcation ◽  
Infection Rate ◽  
Computer Virus ◽  
Propagation Model ◽  
Virus Propagation ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Computer Virus Propagation Model

A four-compartment computer virus propagation model with two delays and graded infection rate is investigated in this paper. The critical values where a Hopf bifurcation occurs are obtained by analyzing the distribution of eigenvalues of the corresponding characteristic equation. In succession, direction and stability of the Hopf bifurcation when the two delays are not equal are determined by using normal form theory and center manifold theorem. Finally, some numerical simulations are also carried out to justify the obtained theoretical results.

scholarly journals Stability and Hopf Bifurcation of a Delayed Epidemic Model of Computer Virus with Impact of Antivirus Software

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Zizhen Zhang ◽  
Ranjit Kumar Upadhyay ◽  
Dianjie Bi ◽  
Ruibin Wei
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Sufficient Conditions ◽  
Computer Virus ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Antivirus Software ◽  
Proposed Model ◽  
Form Theory ◽  
Virus Model

In this paper, we investigate an SLBRS computer virus model with time delay and impact of antivirus software. The proposed model considers the entering rates of all computers since every computer can enter or leave the Internet easily. It has been observed that there is a stability switch and the system becomes unstable due to the effect of the time delay. Conditions under which the system remains locally stable and Hopf bifurcation occurs are found. Sufficient conditions for global stability of endemic equilibrium are derived by constructing a Lyapunov function. Formulae for the direction, stability, and period of the bifurcating periodic solutions are conducted with the aid of the normal form theory and center manifold theorem. Numerical simulations are carried out to analyze the effect of some of the parameters in the system on the dynamic behavior of the system.

scholarly journals Dynamical analysis of a giving up smoking model with time delay

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zizhen Zhang ◽  
Ruibin Wei ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

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.

STABILITY AND HOPF BIFURCATION FOR A THREE-SPECIES REACTION–DIFFUSION PREDATOR–PREY SYSTEM WITH TWO DELAYS

2013 ◽  
Vol 23 (12) ◽  
pp. 1350194
Author(s):  
GAO-XIANG YANG ◽  
JIAN XU
Keyword(s):  
Hopf Bifurcation ◽  
Reaction Diffusion ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Form Theory ◽  
Two Delays ◽  
The Stability ◽  
Bifurcating Periodic Solution

In this paper, a three-species predator–prey system with diffusion and two delays is investigated. By taking the sum of two delays as a bifurcation parameter, it is found that the spatially homogeneous Hopf bifurcation can occur as the sum of two delays crosses a critical value. The direction of Hopf bifurcation and the stability of the bifurcating periodic solution are obtained by employing the center manifold theorem and the normal form theory. In addition, some numerical simulations are also given to illustrate the theoretical analysis.

Stability and Hopf Bifurcation of a Stage-Structured Cannibalism Model with Two Delays

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Tao Zheng ◽  
Long Zhang ◽  
Yantao Luo ◽  
Xinran Zhou ◽  
Hong-Li Li ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Oscillation ◽  
Predator Population ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Lag Effect ◽  
Gestation Time ◽  
Form Theory ◽  
Two Delays

In this work, we consider a stage-structured cannibalism model with two delays. One delay characterizes the lag effect of negative feedback of the prey species, the other has the effect of gestation of the adult predator population. Firstly, criteria for the local stability of feasible equilibria are established. Meanwhile, by choosing delay as a bifurcation parameter, the criteria on the existence of Hopf bifurcation are established. Furthermore, by the normal form theory and center manifold theorem, we derive the explicit formulas determining the properties of periodic solutions. Finally, the theoretical results are illustrated by numerical simulations, from which we can see that the predator’s gestation time delay can make the chaotic phenomenon disappear and maintain periodic oscillation, and that a large feedback time delay of prey can make predators extinct and prey form a periodic solution.

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.

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