scholarly journals Stability and Hopf Bifurcation Analysis of a Nutrient-Phytoplankton Model with Delay Effect

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xinhong Pan ◽  
Min Zhao ◽  
Chuanjun Dai ◽  
Yapei Wang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Center Manifold Theory ◽  
Delay Effect ◽  
Delay Differential System ◽  
The Stability ◽  
Delay Differential ◽  
Manifold Theory ◽  
Two Equilibria

A delay differential system is investigated based on a previously proposed nutrient-phytoplankton model. The time delay is regarded as a bifurcation parameter. Our aim is to determine how the time delay affects the system. First, we study the existence and local stability of two equilibria using the characteristic equation and identify the condition where a Hopf bifurcation can occur. Second, the formulae that determine the direction of the Hopf bifurcation and the stability of periodic solutions are obtained using the normal form and the center manifold theory. Furthermore, our main results are illustrated using numerical simulations.

scholarly journals Stability and Hopf bifurcation periodic orbits in delay coupled Lotka-Volterra ring system

2019 ◽  
Vol 17 (1) ◽  
pp. 962-978
Author(s):  
Rina Su ◽  
Chunrui Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Ring System ◽  
Center Manifold Theory ◽  
Dihedral Groups ◽  
The Stability ◽  
Delay Differential ◽  
Manifold Theory

Abstract In this paper, we consider a class of delay coupled Lotka-Volterra ring systems. Based on the symmetric bifurcation theory of delay differential equations and representation theory of standard dihedral groups, properties of phase locked periodic solutions are given. Moreover, the direction and the stability of the Hopf bifurcation periodic orbits are obtained by using normal form and center manifold theory. Finally, the research results are verified by numerical simulation.

scholarly journals Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yuanyuan Chen ◽  
Ya-Qing Bi
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Local Analysis ◽  
Normal Form Theory ◽  
Form Theory ◽  
Vector Borne ◽  
The Stability ◽  
Delay Differential

A delay-differential modelling of vector-borne is investigated. Its dynamics are studied in terms of local analysis and Hopf bifurcation theory, and its linear stability and Hopf bifurcation are demonstrated by studying the characteristic equation. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument.

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.

Stability and Bifurcation Analysis in a Food Chain Model with Delay

2011 ◽  
Vol 110-116 ◽  
pp. 3382-3388
Author(s):  
Zhang Li
Keyword(s):  
Hopf Bifurcation ◽  
Food Chain ◽  
Center Manifold ◽  
Chain Model ◽  
Center Manifold Theory ◽  
Stability And Bifurcation Analysis ◽  
Existence And Stability ◽  
The Stability ◽  
Manifold Theory ◽  
Food Chain Model

In this paper, we investigate a delayed three-species food chain model. The existence and stability of equilibria are obtained. A 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.

scholarly journals Hopf Bifurcation Analysis for the Modified Rayleigh Price Model with Time Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yanhui Zhai ◽  
Haiyun Bai ◽  
Ying Xiong ◽  
Xiaona Ma
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Center Manifold ◽  
Center Manifold Theory ◽  
Price Model ◽  
Stability Switches ◽  
Normal Form Theory ◽  
Form Theory ◽  
Manifold Theory

This paper mainly modifies and further develops the Reyleigh price model. By modifying the basic Reyleigh model, we can more accurately illustrate the economic phenomena with price varying. First, we research the dynamics of the modified Reyleigh model with time delay. By employing the normal form theory and center manifold theory, we obtain some testable results on these issues. The conclusion confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. Finally, some numerical simulations are given to illustrate the effectiveness of our 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 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.

STABILITY AND HOPF BIFURCATION OF A DELAY DIFFERENTIAL SYSTEM IN MICROBIAL CONTINUOUS CULTURE

2009 ◽  
Vol 02 (03) ◽  
pp. 321-328 ◽  
Author(s):  
XIAOFANG LI ◽  
RONGNING QU ◽  
ENMIN FENG
Keyword(s):  
Hopf Bifurcation ◽  
Continuous Culture ◽  
Differential System ◽  
Center Manifold ◽  
Continuous Fermentation ◽  
Transversality Condition ◽  
Discrete Time Delay ◽  
Delay Differential System ◽  
The Stability ◽  
Delay Differential

Introducing discrete time delay into the model for producing 1, 3-propanediol by microbial continuous fermentation, the stability and Hopf bifurcation of a delay differential system for microorganisms in continuous culture are considered in this paper, including the changing regularity of bifurcation value and oscillating period. Algebraic criteria for absolute stability, as well as the transversality condition for Hopf bifurcation of this kind system are obtained. Explicit algorithm for determining the direction of Hopf bifurcation and the stability of periodic solution is derived, using the theory of normal form and center manifold. Finally, numerical simulations show the effectiveness of our results. The pictures of periodic solutions and phase planes with specified parameters suggest that our results can qualitatively describe oscillatory phenomena occurring in experiments.

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.

Numerical Approximation of Hopf Bifurcation for Tumor-Immune System Competition Model with Two Delays

2013 ◽  
Vol 5 (2) ◽  
pp. 146-162
Author(s):  
Jing-Jun Zhao ◽  
Jing-Yu Xiao ◽  
Yang Xu
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Numerical Approximation ◽  
Center Manifold ◽  
Competition Model ◽  
Center Manifold Theory ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory

AbstractThis paper is concerned with the Hopf bifurcation analysis of tumor-immune system competition model with two delays. First, we discuss the stability of state points with different kinds of delays. Then, a sufficient condition to the existence of the Hopf bifurcation is derived with parameters at different points. Furthermore, under this condition, the stability and direction of bifurcation are determined by applying the normal form method and the center manifold theory. Finally, a kind of Runge-Kutta methods is given out to simulate the periodic solutions numerically. At last, some numerical experiments are given to match well with the main conclusion of this paper.

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