scholarly journals Stability Analysis of a Fractional-Order SEIR-KS Computer Virus-Spreading Model with Two Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Zhufeng Wang ◽  
Xiaoqian Nie ◽  
Maoxin Liao
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Computer Virus ◽  
Bifurcation Theorem ◽  
Different Types ◽  
Two Delays ◽  
Virus Model ◽  
The Stability ◽  
Virus Spreading

In this paper, the stability and Hopf bifurcation of a fractional-order model of the Susceptible-Exposed-Infected-Kill Signals Recovered (SEIR-KS) computer virus with two delays are studied. The sufficient conditions for solving the stability and the occurrence of Hopf bifurcation of the system are established by using Laplace transform, stability theory, and Hopf bifurcation theorem of fractional-order differential systems. The research shows that time delays and fractional order q have an important effect on the stability and the emergence of Hopf bifurcation of the fractional computer virus model. In addition, the validity of the theoretical analysis is verified by selecting appropriate system parameters for numerical simulation and the biological correlation of the equilibrium point is discussed. The results show that the bifurcation point of the model increases with the decrease in the model fractional order q. Under the same fractional order q, the effects of different types of delays on bifurcation points are obviously different.

scholarly journals Hopf Bifurcation Analysis for a Computer Virus Model with Two Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Computer Virus ◽  
Positive Equilibrium ◽  
Local Hopf Bifurcation ◽  
Two Delays ◽  
Virus Model ◽  
Normal Form Method ◽  
Manifold Theory

This paper is concerned with a computer virus model with two delays. Its dynamics are studied in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the positive equilibrium and existence of the local Hopf bifurcation are obtained by regarding the possible combinations of the two delays as a bifurcation parameter. Furthermore, explicit formulae for determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are obtained by using the normal form method and center manifold theory. Finally, some numerical simulations are presented to support the theoretical results.

scholarly journals Stability and Hopf Bifurcation Analysis for a Computer Virus Propagation Model with Two Delays and Vaccination

2017 ◽  
Vol 2017 ◽  
pp. 1-17
Author(s):  
Zizhen Zhang ◽  
Yougang Wang ◽  
Dianjie Bi ◽  
Luca Guerrini
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Computer Virus ◽  
Propagation Model ◽  
Virus Propagation ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Two Delays ◽  
The Stability ◽  
Computer Virus Propagation Model

A further generalization of an SEIQRS-V (susceptible-exposed-infectious-quarantined-recovered-susceptible with vaccination) computer virus propagation model is the main topic of the present paper. This paper specifically analyzes effects on the asymptotic dynamics of the computer virus propagation model when two time delays are introduced. Sufficient conditions for the asymptotic stability and existence of the Hopf bifurcation are established by regarding different combination of the two delays as the bifurcation parameter. Moreover, explicit formulas that determine the stability, direction, and period of the bifurcating periodic solutions are obtained with the help of the normal form theory and center manifold theorem. Finally, numerical simulations are employed for supporting the obtained analytical 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 Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Global Hopf Bifurcation ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Prey System ◽  
Two Delays ◽  
Functional Differential ◽  
The Stability

A modified Leslie-Gower predator-prey system with two delays is investigated. By choosingτ1andτ2as bifurcation parameters, we show that the Hopf bifurcations occur when time delay crosses some critical values. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of the periodic solutions.

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.

Stability Analysis of Computer Virus Model System in Networks

2013 ◽  
Vol 278-280 ◽  
pp. 2033-2038 ◽  
Author(s):  
Shao Ting Ge ◽  
Gong You Tang ◽  
Xue Yang ◽  
Qi Lei Xu ◽  
Hao Yu ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Computer Virus ◽  
Model System ◽  
Virus Model ◽  
The Stability ◽  
Conditions Of Stability ◽  
The Mathematical Model ◽  
Disease Free

This paper considers stability analysis of a discrete-time computer virus model in networks. The disease-free equilibrium and the disease equilibrium are first derived from the mathematical model. Then the sufficient condition of stability for the disease-free equilibrium is obtained by the first Lyapunov method. And the sufficient conditions of stability for the disease equilibrium are given by disc theorem. Simulation results demonstrate the effectiveness of the stability conditions.

scholarly journals Stability and Hopf Bifurcation Analysis for a Gause-Type Predator-Prey System with Multiple Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Juan Liu ◽  
Changwei Sun ◽  
Yimin Li
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Multiple Delays ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
Two Delays ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with a Gause-type predator-prey system with two delays. Firstly, we study the stability and the existence of Hopf bifurcation at the coexistence equilibrium by analyzing the distribution of the roots of the associated characteristic equation. A group of sufficient conditions for the existence of Hopf bifurcation is obtained. Secondly, an explicit formula for determining the stability and the direction of periodic solutions that bifurcate from Hopf bifurcation is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the main theoretical results.

scholarly journals Bifurcation of a Fractional-Order Delayed Malware Propagation Model in Social Networks

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Changjin Xu ◽  
Maoxin Liao ◽  
Peiluan Li
Keyword(s):  
Hopf Bifurcation ◽  
Network Security ◽  
Fractional Order ◽  
Rapid Development ◽  
Sufficient Conditions ◽  
Propagation Model ◽  
The Internet ◽  
Propagation Models ◽  
Malware Propagation ◽  
The Stability

In recent years, with the rapid development of the Internet and the Internet of Things, network security is urgently needed. Malware becomes a major threat to network security. Thus, the study on malware propagation model plays an important role in network security. In the past few decades, numerous researchers put up various kinds of malware propagation models to analyze the dynamic interaction. However, many works are only concerned with the integer-order malware propagation models, while the investigation on fractional-order ones is very few. In this paper, based on the earlier works, we will put up a new fractional-order delayed malware propagation model. Letting the delay be bifurcation parameter and analyzing the corresponding characteristic equations of considered system, we will establish a set of new sufficient conditions to guarantee the stability and the existence of Hopf bifurcation of fractional-order delayed malware propagation model. The study shows that the delay and the fractional order have important effect on the stability and Hopf bifurcation of considered system. To check the correctness of theoretical analyses, we carry out some computer simulations. At last, a simple conclusion is drawn. The derived results of this paper are completely innovative and play an important guiding role in network security.

scholarly journals Periodic Oscillatory Phenomenon in Fractional-Order Neural Networks Involving Different Types of Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Nengfa Wang ◽  
Changjin Xu ◽  
Zixin Liu
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Hopf Bifurcation ◽  
Fractional Order ◽  
Network Systems ◽  
Different Types ◽  
Variable Substitution ◽  
The Stability ◽  
Distribution Of Roots ◽  
Oscillatory Phenomenon

This research is chiefly concerned with the stability and Hopf bifurcation for newly established fractional-order neural networks involving different types of delays. By means of an appropriate variable substitution, equivalent fractional-order neural network systems involving one delay are built. By discussing the distribution of roots of the characteristic equation of the established fractional-order neural network systems and selecting the delay as bifurcation parameter, a novel delay-independent bifurcation condition is derived. The investigation verifies that the delay is a significant parameter which has an important influence on stability nature and Hopf bifurcation behavior of neural network systems. The computer simulation plots and bifurcation graphs effectively illustrate the reasonableness of the theoretical fruits.

scholarly journals Double Hopf bifurcation of a diffusive predator–prey system with strong Allee effect and two delays

2021 ◽  
Vol 26 (1) ◽  
pp. 72-92
Author(s):  
Yuying Liu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Allee Effect ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Double Hopf Bifurcation ◽  
Prey System ◽  
Two Delays ◽  
Strong Allee Effect ◽  
The Stability ◽  
Two Parameters

In this paper, we consider a diffusive predator–prey system with strong Allee effect and two delays. First, we explore the stability region of the positive constant steady state by calculating the stability switching curves. Then we derive the Hopf and double Hopf bifurcation theorem via the crossing directions of the stability switching curves. Moreover, we calculate the normal forms near the double Hopf singularities by taking two delays as parameters. We carry out some numerical simulations for illustrating the theoretical results. Both theoretical analysis and numerical simulation show that the system near double Hopf singularity has rich dynamics, including stable spatially homogeneous and inhomogeneous periodic solutions. Finally, we evaluate the influence of two parameters on the existence of double Hopf bifurcation.

Sign in / Sign up

Export Citation Format

Share Document