scholarly journals Stability and Hopf Bifurcation Analysis in a Delayed Myc/E2F/miR-17-92 Network Involving Interlinked Positive and Negative Feedback Loops

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Guiyuan Wang ◽  
Zhuoqin Yang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Negative Feedback ◽  
Feedback Loop ◽  
Sufficient Conditions ◽  
Center Manifold Theorem ◽  
Negative Feedback Loops ◽  
Feedback Strength ◽  
Normal Form Method ◽  
Delay Differential

MiR-17-92 plays an important role in regulating the levels of the Myc/E2F protein. In this paper, we consider a coupling network between Myc/E2F/miR-17-92 delayed negative feedback loop and Myc/E2F positive feedback loop described by a two-dimensional delay differential equation. Based on linear stability analysis and bifurcation theory, sufficient conditions for stability of equilibria and oscillatory behaviors via Hopf bifurcation are derived when choosing time delay as well as negative feedback strength associated with oscillations as bifurcation parameters, respectively. Furthermore, direction and stability of Hopf bifurcation of time delay are studied by using the normal form method and center manifold theorem. Finally, several numerical simulations are performed to verify the results we obtained.

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 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 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 Hopf Bifurcation Analysis on General Gause-Type Predator-Prey Models with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Shuang Guo ◽  
Weihua Jiang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Normal Form Method ◽  
Predator Prey Models ◽  
The Stability

A class of three-dimensional Gause-type predator-prey model with delay is considered. Firstly, a group of sufficient conditions for the existence of Hopf bifurcation is obtained via employing the polynomial theorem by analyzing the distribution of the roots of the associated characteristic equation. Secondly, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions are determined by applying the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the obtained results.

Stability and Hopf Bifurcation of a Reaction–Diffusion Neutral Neuron System with Time Delay

2017 ◽  
Vol 27 (14) ◽  
pp. 1750214 ◽  
Author(s):  
Tao Dong ◽  
Linmao Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neuron System ◽  
Center Manifold Theorem ◽  
Feedback Strength ◽  
The Stability ◽  
System With Time Delay ◽  
Zero Equilibrium

In this paper, a type of reaction–diffusion neutral neuron system with time delay under homogeneous Neumann boundary conditions is considered. By constructing a basis of phase space based on the eigenvectors of the corresponding Laplace operator, the characteristic equation of this system is obtained. Then, by selecting time delay and self-feedback strength as the bifurcating parameters respectively, the dynamic behaviors including local stability and Hopf bifurcation near the zero equilibrium point are investigated when the time delay and self-feedback strength vary. Furthermore, the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by using the normal form and the center manifold theorem for the corresponding partial differential equation. Finally, two simulation examples are given to verify the theory.

scholarly journals Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease Model with Two Delays and Reinfection

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yanxia Zhang ◽  
Long Li ◽  
Junjian Huang ◽  
Yanjun Liu
Keyword(s):  
Hopf Bifurcation ◽  
Disease Model ◽  
Endemic Equilibrium ◽  
Partial Protection ◽  
Center Manifold Theorem ◽  
Vector Borne Disease ◽  
Two Delays ◽  
Normal Form Method ◽  
Vector Borne ◽  
Delay Differential

In this paper, a vector-borne disease model with two delays and reinfection is established and considered. First of all, the existence of the equilibrium of the system, under different cases of two delays, is discussed through analyzing the corresponding characteristic equation of the linear system. Some conditions that the system undergoes Hopf bifurcation at the endemic equilibrium are obtained. Furthermore, by employing the normal form method and the center manifold theorem for delay differential equations, some explicit formulas used to describe the properties of bifurcating periodic solutions are derived. Finally, the numerical examples and simulations are presented to verify our theoretical conclusions. Meanwhile, the influences of the degree of partial protection for recovered people acquired by a primary infection on the endemic equilibrium and the critical values of the two delays are analyzed.

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 Bifurcation Analysis of a Delayed Worm Propagation Model with Saturated Incidence

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Zizhen Zhang ◽  
Yougang Wang ◽  
Luca Guerrini
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Sufficient Conditions ◽  
Propagation Model ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Worm Propagation ◽  
Center Manifold Theorem ◽  
Saturated Incidence ◽  
Parameter Values

This paper is concerned with a delayed SVEIR worm propagation model with saturated incidence. The main objective is to investigate the effect of the time delay on the model. Sufficient conditions for local stability of the positive equilibrium and existence of a Hopf bifurcation are obtained by choosing the time delay as the bifurcation parameter. Particularly, explicit formulas determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived by using the normal form theory and the center manifold theorem. Numerical simulations for a set of parameter values are carried out to illustrate the analytical 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 in a Kind of Fourth-Order Delay Differential Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-20
Author(s):  
Xiaoqian Cui ◽  
Junjie Wei
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Fourth Order ◽  
Delay Differential Equation ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Stability Switches ◽  
Center Manifold Theorem ◽  
Normal Form Method ◽  
Delay Differential

A kind of fourth-order delay differential equation is considered. Firstly, the linear stability is investigated by analyzing the associated characteristic equation. It is found that there are stability switches for time delay and Hopf bifurcations when time delay cross through some critical values. Then the direction and stability of the Hopf bifurcation are determined, using the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the analytic results.

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