scholarly journals Effective Control and Bifurcation Analysis in a Chaotic System with Distributed Delay Feedback

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Wei Tan ◽  
Jianguo Gao ◽  
Wenjun Fan
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic System ◽  
Center Manifold ◽  
Distributed Delay ◽  
Effective Control ◽  
Asymptotically Stable ◽  
Bifurcation Parameter ◽  
Normal Form Theorem ◽  
The Stability ◽  
Variation Tendency

We discuss the dynamic behavior of a new Lorenz-like chaotic system with distributed delayed feedback by the qualitative analysis and numerical simulations. It is verified that the equilibria are locally asymptotically stable whenα∈(0,α0)and unstable whenα∈(α0,∞); Hopf bifurcation occurs whenαcrosses a critical valueα0by choosingαas a bifurcation parameter. Meanwhile, the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Furthermore, regardingαas a bifurcation parameter, we explore variation tendency of the dynamics behavior of a chaotic system with the increase of the parameter valueα.

scholarly journals Bifurcation Analysis in a Delayed Diffusive Leslie-Gower Model

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Shuling Yan ◽  
Xinze Lian ◽  
Weiming Wang ◽  
Youbin Wang
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Neumann Boundary ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Normal Form Theorem ◽  
The Stability

We investigate a modified delayed Leslie-Gower model under homogeneous Neumann boundary conditions. We give the stability analysis of the equilibria of the model and show the existence of Hopf bifurcation at the positive equilibrium under some conditions. Furthermore, we investigate the stability and direction of bifurcating periodic orbits by using normal form theorem and the center manifold theorem.

scholarly journals Effect of Prey Refuge and Harvesting on Dynamics of Eco-epidemiological Model with Holling Type III

2021 ◽  
Vol 3 (1) ◽  
pp. 16-25
Author(s):  
Adin Lazuardy Firdiansyah
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Solutions ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Bifurcation Parameter ◽  
Prey Refuge ◽  
Interesting Phenomenon ◽  
Type Iii ◽  
Interior Equilibrium ◽  
The Stability

In this research, we formulate and analyze an eco-epidemiology model of the modified Leslie-Gower model with Holling type III by incorporating prey refuge and harvesting. In the model, we find at most six equilibrium where three equilibrium points are unstable and three equilibrium points are locally asymptotically stable. Furthermore, we find an interesting phenomenon, namely our model undergoes Hopf bifurcation at the interior equilibrium point by selecting refuge as the bifurcation parameter. Moreover, we also conclude that the stability of all populations occurs faster when the harvesting rate increases.  In the end, several numerical solutions are presented to check the analytical results.

scholarly journals Dynamics analysis of a delayed virus model with two different transmission methods and treatments

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Tongqian Zhang ◽  
Junling Wang ◽  
Yuqing Li ◽  
Zhichao Jiang ◽  
Xiaofeng Han
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Reproduction Number ◽  
Dynamics Analysis ◽  
Asymptotically Stable ◽  
Bifurcation Parameter ◽  
Local Hopf Bifurcation ◽  
Virus Model ◽  
The Stability ◽  
The Basic Reproduction Number

AbstractIn this paper, a delayed virus model with two different transmission methods and treatments is investigated. This model is a time-delayed version of the model in (Zhang et al. in Comput. Math. Methods Med. 2015:758362, 2015). We show that the virus-free equilibrium is locally asymptotically stable if the basic reproduction number is smaller than one, and by regarding the time delay as a bifurcation parameter, the existence of local Hopf bifurcation is investigated. The results show that time delay can change the stability of the endemic equilibrium. Finally, we give some numerical simulations to illustrate the theoretical findings.

scholarly journals Effect of Prey Refuge and Harvesting on Dynamics of Eco-epidemiological Model with Holling Type III

2021 ◽  
Vol 1 (1) ◽  
pp. 16-25
Author(s):  
Adin Lazuardy Firdiansyah
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Solutions ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Bifurcation Parameter ◽  
Prey Refuge ◽  
Interesting Phenomenon ◽  
Type Iii ◽  
Interior Equilibrium ◽  
The Stability

In this research, we formulate and analyze an eco-epidemiology model of the modified Leslie-Gower model with Holling type III by incorporating prey refuge and harvesting. In the model, we find at most six equilibrium where three equilibrium points are unstable and three equilibrium points are locally asymptotically stable. Furthermore, we find an interesting phenomenon, namely our model undergoes Hopf bifurcation at the interior equilibrium point by selecting refuge as the bifurcation parameter. Moreover, we also conclude that the stability of all populations occurs faster when the harvesting rate increases.  In the end, several numerical solutions are presented to check the analytical results.

Hopf bifurcation analysis in a turbidostat model with Beddington–DeAngelis functional response and discrete delay

2017 ◽  
Vol 10 (05) ◽  
pp. 1750061
Author(s):  
Yong Yao ◽  
Zuxiong Li ◽  
Huili Xiang ◽  
Hailing Wang ◽  
Zhijun Liu
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Response ◽  
Center Manifold ◽  
Critical Value ◽  
Hopf Bifurcations ◽  
Bifurcation Parameter ◽  
Discrete Delay ◽  
Center Manifold Theorem ◽  
The Stability

In this paper, regarding the time delay as a bifurcation parameter, the stability and Hopf bifurcation of the model of competition between two species in a turbidostat with Beddington–DeAngelis functional response and discrete delay are studied. The Hopf bifurcations can be shown when the delay crosses the critical value. Furthermore, based on the normal form and the center manifold theorem, the type, stability and other properties of the bifurcating periodic solutions are determined. Finally, some numerical simulations are given to illustrate the results.

scholarly journals Hopf Bifurcation of a Differential-Algebraic Bioeconomic Model with Time Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Xiaojian Zhou ◽  
Xin Chen ◽  
Yongzhong Song
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Time Delays ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Bifurcation Parameter ◽  
Bioeconomic Model ◽  
Numerical Tests ◽  
The Stability

We investigate the dynamics of a differential-algebraic bioeconomic model with two time delays. Regarding time delay as a bifurcation parameter, we show that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Using the theories of normal form and center manifold, we also give the explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Numerical tests are provided to verify our theoretical analysis.

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.

Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System

2018 ◽  
Vol 28 (11) ◽  
pp. 1850136 ◽  
Author(s):  
Ben Niu ◽  
Yuxiao Guo ◽  
Yanfei Du
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Immune Cell ◽  
Tumor Treatment ◽  
Delay Effect ◽  
Stable Periodic ◽  
The Stability ◽  
Bifurcation And Stability

Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay [Formula: see text] of immune system where the immune cell has a probability [Formula: see text] in killing tumor cells. We find increasing time delay [Formula: see text] destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability [Formula: see text] on Hopf bifurcation values is also discussed.

DYNAMICS AND DOUBLE HOPF BIFURCATIONS OF THE ROSE–HINDMARSH MODEL WITH TIME DELAY

2009 ◽  
Vol 19 (11) ◽  
pp. 3733-3751 ◽  
Author(s):  
SUQI MA ◽  
ZHAOSHENG FENG ◽  
QISHAI LU
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Value Of Time ◽  
Asymptotically Stable ◽  
Universal Unfolding ◽  
Double Hopf Bifurcation ◽  
Bifurcation Points ◽  
The Rose

In this paper, we are concerned with the Rose–Hindmarsh model with time delay. By applying the generalized Sturm criterion, a number of imaginary roots of the characteristic equation are classified. The absolutely stable regions for any value of time delay are detected. By the continuous software DDE-Biftool, both the Hopf bifurcation curves and double Hopf bifurcation points are illustrated in parametric spaces. The normal form and universal unfolding at double Hopf bifurcation points are considered by the center manifold method. Some examples also indicate that the corresponding unique attractor near each double Hopf point is asymptotically stable.

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.

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