scholarly journals Bifurcation Analysis of a Lotka-Volterra Mutualistic System with Multiple Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Xin-You Meng ◽  
Hai-Feng Huo
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Time Delays ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Multiple Delays ◽  
Center Manifold Theorem ◽  
Normal Form Method ◽  
Three Delays ◽  
Theoretical Results

A class of Lotka-Volterra mutualistic system with time delays of benefit and feedback delays is introduced. By analyzing the associated characteristic equation, the local stability of the positive equilibrium and existence of Hopf bifurcation are obtained under all possible combinations of two or three delays selecting from multiple delays. Not only explicit formulas to determine the properties of the Hopf bifurcation are shown by using the normal form method and center manifold theorem, but also the global continuation of Hopf bifurcation is investigated by applying a global Hopf bifurcation result due to Wu (1998). Numerical simulations are given to support the theoretical results.

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.

Multiple Bifurcations of Synchronized Oscillators With Delays

2005 ◽  
Author(s):  
Yuan Y. Yuan
Keyword(s):  
Local Stability ◽  
Time Delays ◽  
Center Manifold ◽  
Zero Solution ◽  
Pitchfork Bifurcation ◽  
Bifurcation Theorem ◽  
Center Manifold Theorem ◽  
System A ◽  
Normal Form Method ◽  
Theoretical Results

The synchronized oscillator with two discrete time delays is considered. The local stability of the zero solution of this system is investigated by studying the distributions of the eigenvalues of the system. A complete bifurcation analysis is given by employing the center manifold theorem, normal form method and bifurcation theorem. It is shown that the trivial fixed point may lose stability via a transcritical/pitchfork bifurcation, Hopf bifurcation or Bogdanov-Takens bifurcation. Some numerical simulation examples are given for justify the theoretical 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.

Mathematical analysis of a model for thymus infection with discrete and distributed delays

2014 ◽  
Vol 07 (06) ◽  
pp. 1450070 ◽  
Author(s):  
M. Prakash ◽  
P. Balasubramaniam
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Center Manifold Theorem ◽  
Different Types ◽  
Normal Form Method ◽  
Bifurcation And Stability ◽  
Theoretical Results ◽  
Hiv 1

In this paper, the dynamics of mathematical model for infection of thymus gland by HIV-1 is analyzed by applying some perturbation through two different types of delays such as in terms of Hopf bifurcation analysis. Further, the conditions for the existence of Hopf bifurcation are derived by evaluating the characteristic equation. The direction of Hopf bifurcation and stability of bifurcating periodic solutions are determined by employing the center manifold theorem and normal form method. Finally, some of the numerical simulations are carried out to validate the derived theoretical results and main conclusions are included.

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 Hopf Bifurcation Analysis for a Semiratio-Dependent Predator-Prey System with Two Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Ming Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Two Delays ◽  
Normal Form Method ◽  
Nonmonotonic Functional Response

This paper is concerned with a semiratio-dependent predator-prey system with nonmonotonic functional response and two delays. It is shown that the positive equilibrium of the system is locally asymptotically stable when the time delay is small enough. Change of stability of the positive equilibrium will cause bifurcating periodic solutions as the time delay passes through a sequence of critical values. The properties of Hopf bifurcation such as direction and stability are determined by using the normal form method and center manifold theorem. Numerical simulations confirm our theoretical findings.

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 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 Nonlinear Dynamics of a PI Hydroturbine Governing System with Double Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Hongwei Luo ◽  
Jiangang Zhang ◽  
Wenju Du ◽  
Jiarong Lu ◽  
Xinlei An
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Points ◽  
Bifurcation Theorem ◽  
Center Manifold Theorem ◽  
Governing System ◽  
Normal Form Method ◽  
The Stability ◽  
System Frequency ◽  
Realistic Significance ◽  
Theoretical Results

A PI hydroturbine governing system with saturation and double delays is generated in small perturbation. The nonlinear dynamic behavior of the system is investigated. More precisely, at first, we analyze the stability and Hopf bifurcation of the PI hydroturbine governing system with double delays under the four different cases. Corresponding stability theorem and Hopf bifurcation theorem of the system are obtained at equilibrium points. And then the stability of periodic solution and the direction of the Hopf bifurcation are illustrated by using the normal form method and center manifold theorem. We find out that the stability and direction of the Hopf bifurcation are determined by three parameters. The results have great realistic significance to guarantee the power system frequency stability and improve the stability of the hydropower system. At last, some numerical examples are given to verify the correctness of the theoretical results.

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