scholarly journals Hopf Bifurcation Analysis for the Model of the Chemostat with One Species of Organism

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Haiyun Bai ◽  
Yanhui Zhai
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Center Manifold ◽  
Stability Switches ◽  
Chemostat Model ◽  
Explicit Formulas ◽  
Normal Form Theory ◽  
Form Theory ◽  
The Stability

We research the dynamics of the chemostat model with time delay. The conclusion confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. By using the normal form theory and center manifold method, we derive the explicit formulas determining the stability and direction of bifurcating periodic solutions. Finally, some numerical simulations are given to illustrate the effectiveness of our results.

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.

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.

scholarly journals Stability and Bifurcation for a Delayed Diffusive Two-Zooplankton One-Phytoplankton Model with Two Different Functions

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-24
Author(s):  
Xin-You Meng ◽  
Li Xiao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Center Manifold ◽  
Functional Differential Equations ◽  
Bifurcation Parameter ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Functional Differential

In this paper, a diffusion two-phytoplankton one-zooplankton model with time delay, Beddington–DeAnglis functional response, and Holling II functional response is proposed. First, the existence and local stability of all equilibria of such model are studied. Then, the existence of Hopf bifurcation of the corresponding model without diffusion is given by taking time delay as the bifurcation parameter. Next, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated by using the normal form theory and center manifold theorem. Furthermore, due to the local bifurcation theory of partial functional differential equations, Hopf bifurcation of the model is investigated by considering time delay as the bifurcation parameter. The explicit formulas to determine the properties of Hopf bifurcation are given by the method of the normal form theory and center manifold theorem for partial functional differential equations. Finally, some numerical simulations are performed to check out our theoretical results.

Local Stability and Hopf Bifurcation Analysis of the Arneodo’s System

2011 ◽  
Vol 130-134 ◽  
pp. 2550-2557
Author(s):  
Yi Jing Liu ◽  
Zhi Shu Li ◽  
Xiao Mei Cai ◽  
Ya Lan Ye
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Theoretical Analysis ◽  
Local Stability ◽  
Hopf Bifurcations ◽  
Explicit Formulas ◽  
Normal Form Theory ◽  
Numerical Examples ◽  
Form Theory ◽  
The Stability

The chaotic behaviors of the Arneodo’s system are investigated in this paper. Based on the Arneodo's system characteristic equation, the equilibria of the system and the conditions of Hopf bifurcations are obtained, which shows that Hopf bifurcations occur in this system. Then using the normal form theory, we give the explicit formulas which determine the stability of bifurcating periodic solutions and the direction of the Hopf bifurcation. Finally, some numerical examples are employed to demonstrate the effectiveness of the theoretical analysis.

The Stability and Hopf Bifurcation for a Predator-Prey Model with Delays

2014 ◽  
Vol 926-930 ◽  
pp. 3314-3317
Author(s):  
Hong Bing Chen
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory ◽  
The Stability

In this paper, a predator–prey model with discrete and distributed delays is investigated. the direction of Hopf bifurcation as well as stability of periodic solution are studied. The method which we used is the normal form theory and center manifold. At last, an example showed the feasibility of results.

Hopf Bifurcation Analysis for a Novel Hyperchaotic System

2012 ◽  
Vol 8 (1) ◽  
Author(s):  
Kejun Zhuang
Keyword(s):  
Numerical Simulation ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Local Stability ◽  
Center Manifold ◽  
Hyperchaotic System ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
The Stability

The paper mainly focuses on a novel hyperchaotic system. The local stability of equilibrium is analyzed and existence of Hopf bifurcation is established. Moreover, formulas for determining the stability and direction of bifurcating periodic solutions are derived by center manifold theorem and normal form theory. Finally, numerical simulation is given to illustrate the theoretical analysis.

Bifurcation analysis of modeling biodegradation of Microcystins

2019 ◽  
Vol 12 (03) ◽  
pp. 1950028
Author(s):  
Keying Song ◽  
Wanbiao Ma ◽  
Zhichao Jiang
Keyword(s):  
Time Delay ◽  
Normal Form ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Form Theory ◽  
The Stability

In this paper, a model with time delay describing biodegradation of Microcystins (MCs) is investigated. Firstly, the stability of the positive equilibrium and the existence of Hopf bifurcations are obtained. Furthermore, an explicit algorithm for determining the direction and the stability of the bifurcating periodic solutions is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the applications of the results.

scholarly journals Delay-Induced Oscillations in a Competitor-Competitor-Mutualist Lotka-Volterra Model

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Changjin Xu
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Computer Simulations ◽  
Center Manifold ◽  
Volterra Model ◽  
Normal Form Theory ◽  
Biological Meaning ◽  
Theoretical Predictions ◽  
Form Theory ◽  
The Stability

This paper deals with a competitor-competitor-mutualist Lotka-Volterra model. A series of sufficient criteria guaranteeing the stability and the occurrence of Hopf bifurcation for the model are obtained. Several concrete formulae determine the properties of bifurcating periodic solutions by applying the normal form theory and the center manifold principle. Computer simulations are given to support the theoretical predictions. At last, biological meaning and a conclusion are presented.

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 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.

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