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 for a Delayed SLBRS Computer Virus Model

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Computer Virus ◽  
Center Manifold Theory ◽  
Dynamical Behaviors ◽  
Virus Model ◽  
Normal Form Method ◽  
Manifold Theory

By incorporating the time delay due to the period that computers use antivirus software to clean the virus into the SLBRS model a delayed SLBRS computer virus model is proposed in this paper. The dynamical behaviors which include local stability and Hopf bifurcation are investigated by regarding the delay as bifurcating parameter. Specially, direction and stability of the Hopf bifurcation are derived by applying the normal form method and center manifold theory. Finally, an illustrative example is also presented to testify our analytical results.

scholarly journals Stability and Hopf Bifurcation in a Prey-Predator System with Disease in the Prey and Two Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Juan Liu
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability ◽  
Predator System

This paper is concerned with a prey-predator system with disease in the prey and two delays. Local stability of the positive equilibrium of the system and existence of local Hopf bifurcation are investigated by choosing different combinations of the two delays as bifurcation parameters. For further investigation, the direction and the stability of the Hopf bifurcation are determined by using the normal form method and center manifold theorem. Finally, some numerical simulations are given to support the theoretical analysis.

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 SIQR Computer Virus Model with Time Delay

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Computer Virus ◽  
Center Manifold Theory ◽  
Virus Prevalence ◽  
Virus Model ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

A delayed SIQR computer virus model is considered. It has been observed that there exists a critical value of delay for the stability of virus prevalence by choosing the delay as a bifurcation parameter. Furthermore, the properties of the Hopf bifurcation such as direction and stability are investigated by using the normal form method and center manifold theory. Finally, some numerical simulations for supporting our theoretical results are also performed.

scholarly journals Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Recurrent Neural Network ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Form Theory ◽  
Two Delays ◽  
The Stability ◽  
Manifold Theory

A four-dimensional recurrent neural network with two delays is considered. The main result is given in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the zero equilibrium and existence of the Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. In particular, explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form theory and center manifold theory. Some numerical examples are also presented to verify the theoretical analysis.

scholarly journals Stability and Hopf Bifurcation of a Computer Virus Model with Infection Delay and Recovery Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Haitao Song ◽  
Qiaochu Wang ◽  
Weihua Jiang
Keyword(s):  
Hopf Bifurcation ◽  
Virus Infection ◽  
Periodic Orbits ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Computer Virus ◽  
Stable Periodic ◽  
Virus Model ◽  
Manifold Theory ◽  
Bifurcation And Stability

A computer virus model with infection delay and recovery delay is considered. The sufficient conditions for the global stability of the virus infection equilibrium are established. We show that the time delay can destabilize the virus infection equilibrium and give rise to Hopf bifurcations and stable periodic orbits. By the normal form and center manifold theory, the direction of the Hopf bifurcation and stability of the bifurcating periodic orbits are determined. Numerical simulations are provided to support our theoretical conclusions.

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.

Stability and Hopf Bifurcation in a Three-Species Food Chain System With Harvesting and Two Delays

2014 ◽  
Vol 9 (2) ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Food Chain ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Chain System ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability

In this paper, we analyze the dynamics of a delayed food chain system with harvesting. Sufficient conditions for the local stability of the positive equilibrium and for the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation. Formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Finally, numerical simulation results are presented to validate the theoretical analysis.

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 Hopf Bifurcation and Dynamic Analysis of an Improved Financial System with Two Delays

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
G. Kai ◽  
W. Zhang ◽  
Z. Jin ◽  
C. Z. Wang
Keyword(s):  
Nonlinear Dynamics ◽  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Chaotic System ◽  
Financial System ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

The complex chaotic dynamics and multistability of financial system are some important problems in micro- and macroeconomic fields. In this paper, we study the influence of two-delay feedback on the nonlinear dynamics behavior of financial system, considering the linear stability of equilibrium point under the condition of single delay and two delays. The system undergoes Hopf bifurcation near the equilibrium point. The stability and bifurcation directions of Hopf bifurcation are studied by using the normal form method and central manifold theory. The theoretical results are verified by numerical simulation. Furthermore, one feature of the proposed financial chaotic system is that its multistability depends extremely on the memristor initial condition and the system parameters. It is shown that the nonlinear dynamics of financial chaotic system can be significantly changed by changing the values of time delays.

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