scholarly journals Bifurcation Analysis in a Two-Dimensional Neutral Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Ming Liu ◽  
Xiaofeng Xu
Keyword(s):  
Periodic Solutions ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Neutral Form ◽  
Global Hopf Bifurcation ◽  
Normal Form Method ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Manifold Theory

The dynamics of a 2-dimensional neural network model in neutral form are investigated. We prove that a sequence of Hopf bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using normal form method and center manifold theory. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. Finally, some numerical simulations are carried out to support the analytic results.

Bifurcation Analysis for Neural Networks in Neutral Form

2016 ◽  
Vol 26 (06) ◽  
pp. 1650100 ◽  
Author(s):  
Hong-Bing Chen ◽  
Xiao-Ke Sun
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Neutral Form ◽  
Theoretical Predictions ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory

In this paper, a system of neural networks in neutral form with time delay is investigated. Further, by introducing delay [Formula: see text] as a bifurcation parameter, it is found that Hopf bifurcation occurs when [Formula: see text] is across some critical values. The direction of the Hopf bifurcations and the stability are determined by using normal form method and center manifold theory. Next, the global existence of periodic solution is established by using a global Hopf bifurcation result. Finally, an example is given to support the theoretical predictions.

scholarly journals Dynamics of a Delayed Model for the Transmission of Malicious Objects in Computer Network

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Periodic Solutions ◽  
Center Manifold ◽  
Computer Network ◽  
Center Manifold Theory ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory ◽  
Delayed Model ◽  
Theoretical Results

An SEIQRS model for the transmission of malicious objects in computer network with two delays is investigated in this paper. We show that possible combination of the two delays can affect the stability of the model and make the model bifurcate periodic solutions under some certain conditions. For further investigation, properties of the periodic solutions are studied by using the normal form method and center manifold theory. Finally, some numerical simulations are given to justify the theoretical results.

Stability and Hopf Bifurcation of a Delayed Mutualistic System

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Ruimin Zhang ◽  
Xiaohui Liu ◽  
Chunjin Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Stage Structure ◽  
Center Manifold Theory ◽  
Mutualistic Relationship ◽  
Normal Form Method ◽  
Explicit Formulae ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

In this paper, we study a classic mutualistic relationship between the leaf cutter ants and their fungus garden, establishing a time delay mutualistic system with stage structure. We investigate the stability and Hopf bifurcation by analyzing the distribution of the roots of the associated characteristic equation. By means of the center manifold theory and normal form method, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some numerical simulations are carried out for illustrating the theoretical results.

Bifurcation Analysis of Phytoplankton and Zooplankton Interaction System with Two Delays

2020 ◽  
Vol 30 (03) ◽  
pp. 2050039
Author(s):  
Zhichao Jiang ◽  
Jiangtao Dai ◽  
Tongqian Zhang
Keyword(s):  
Periodic Solutions ◽  
Center Manifold ◽  
Threshold Values ◽  
Center Manifold Theory ◽  
Local Hopf Bifurcation ◽  
Two Delays ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results ◽  
Positivity And Boundedness

In this paper, the system of describing the interactions between poisonous phytoplankton and zooplankton is presented. It focuses on the effects of two delays on the dynamic behavior of the system. At first, the properties of solutions including positivity and boundedness are given. Next, the stability of equilibria and the existence of local Hopf bifurcation are established when delays change and cross some threshold values. Especially, the existence of global periodic solutions is discussed when the two delays are equal. Furthermore, the implicit algorithm is derived for deciding the properties of the branching periodic solutions by using center manifold theory. Some numerical simulations are performed for supporting the theoretical results. Finally, some conclusions are given.

scholarly journals Stability and Hopf Bifurcation in a Modified Holling-Tanner Predator-Prey System with Multiple Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang ◽  
Juan Liu
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Local Stability ◽  
Center Manifold ◽  
Multiple Delays ◽  
Center Manifold Theory ◽  
Predator Prey ◽  
Prey System ◽  
Existence Of Periodic Solutions ◽  
Manifold Theory

A modified Holling-Tanner predator-prey system with multiple delays is investigated. By analyzing the associated characteristic equation, the local stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays are established. Direction and stability of the periodic solutions are obtained by using normal form and center manifold theory. Finally, numerical simulations are carried out to substantiate the analytical results.

STABILITY AND BIFURCATION ANALYSIS IN A DELAYED PREDATOR-PREY SYSTEM

2009 ◽  
Vol 02 (04) ◽  
pp. 483-506 ◽  
Author(s):  
ZHICHAO JIANG ◽  
WENZHI ZHANG ◽  
DONGSHENG HUO
Keyword(s):  
Local Stability ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Predator Prey ◽  
Prey System ◽  
Stability And Bifurcation Analysis ◽  
Characteristic Values ◽  
The Stability ◽  
Manifold Theory

A delayed ratio-dependent one-predator and two-prey system with Michaelis–Menten type functional response is investigated. We show the existence of nonnegative equilibria under some appropriated conditions. Criteria for local stability, instability of nonnegative equilibria are obtained. The existence of Hopf bifurcations at the endemic equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the normal form and the center manifold theory. At last, some numerical simulations to support the analytical conclusions are carried out.

ON HOPF BIFURCATION OF A DELAYED PREDATOR–PREY SYSTEM WITH DIFFUSION

2013 ◽  
Vol 23 (02) ◽  
pp. 1350023 ◽  
Author(s):  
JIANXIN LIU ◽  
JUNJIE WEI
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Predator Prey ◽  
Prey System ◽  
Normal Form Method ◽  
Manifold Theory

A delayed predator–prey system with diffusion and Dirichlet boundary conditions is considered. By regarding the growth rate a of prey as a main bifurcation parameter, we show that Hopf bifurcation occurs when the parameter a is varied. Then, by using the center manifold theory and normal form method, an explicit algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcating periodic solutions is derived.

scholarly journals Global Hopf Bifurcation Analysis for a Time-Delayed Model of Asset Prices

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Ying Qu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Asset Prices ◽  
Asset Markets ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Global Hopf Bifurcation ◽  
The Stability ◽  
Manifold Theory ◽  
Delayed Model

A time-delayed model of speculative asset markets is investigated to discuss the effect of time delay and market fraction of the fundamentalists on the dynamics of asset prices. It proves that a sequence of Hopf bifurcations occurs at the positive equilibriumv, the fundamental price of the asset, as the parameters vary. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined using normal form method and center manifold theory. Global existence of periodic solutions is established combining a global Hopf bifurcation theorem with a Bendixson's criterion for higher-dimensional ordinary differential equations.

STABILITY ANALYSIS OF A PREDATOR-PREY MODEL

2012 ◽  
Vol 05 (01) ◽  
pp. 1250007 ◽  
Author(s):  
ZHICHAO JIANG ◽  
ZHAOZHUANG GUO ◽  
YUEFANG SUN
Keyword(s):  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Characteristic Values ◽  
The Stability ◽  
Manifold Theory

In this paper, a time-delayed predator-prey system is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.

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.

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