Rank One Chaos in Periodically-Kicked Time-Delayed Chen System

2015 ◽  
Vol 25 (08) ◽  
pp. 1550097 ◽  
Author(s):  
Yunxian Dai ◽  
Yiping Lin ◽  
Wenjie Yang ◽  
Huitao Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Chen System ◽  
Periodic Force ◽  
Center Manifold Theorem ◽  
Delayed Differential Equations ◽  
Rank One ◽  
Delayed System ◽  
Normal Form Method ◽  
System With Time Delay

In this paper, we study the existence of rank one strange attractor in time-delayed system. First, we try to develop rank one theory for delayed differential equations. Then, we consider Chen system with time-delay, the conditions under which a supercritical Hopf bifurcation occurs are given by using the normal form method and center manifold theorem. Then, we add an external periodic force as an input and observe rank one strange attractors. Finally, several numerical simulations supporting the theoretical analysis are also given.

Rank One Strange Attractors in Periodically Kicked Predator–Prey System with Time-Delay

2016 ◽  
Vol 26 (07) ◽  
pp. 1640114 ◽  
Author(s):  
Wenjie Yang ◽  
Yiping Lin ◽  
Yunxian Dai ◽  
Huitao Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Strange Attractors ◽  
Periodic Force ◽  
Delayed Systems ◽  
Predator Prey ◽  
Prey System ◽  
Rank One ◽  
Delayed System ◽  
System With Time Delay

This paper is devoted to the study of the problem of rank one strange attractor in a periodically kicked predator–prey system with time-delay. Our discussion is based on the theory of rank one maps formulated by Wang and Young. Firstly, we develop the rank one chaotic theory to delayed systems. It is shown that strange attractors occur when the delayed system undergoes a Hopf bifurcation and encounters an external periodic force. Then we use the theory to the periodically kicked predator–prey system with delay, deriving the conditions for Hopf bifurcation and rank one chaos along with the results of numerical simulations.

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 Hopf Bifurcation and Chaos of a Delayed Finance System

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Xuebing Zhang ◽  
Honglan Zhu
Keyword(s):  
Numerical Simulation ◽  
Hopf Bifurcation ◽  
Center Manifold ◽  
Chaotic State ◽  
Bifurcation And Chaos ◽  
Finance System ◽  
Center Manifold Theorem ◽  
Normal Form Method ◽  
Simulation Results ◽  
The Stability

In this paper, a finance system with delay is considered. By analyzing the corresponding characteristic equations, the local stability of equilibrium is established. The existence of Hopf bifurcations at the equilibrium is also discussed. Furthermore, 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. Numerical simulation results show that delay can lead a stable system into a chaotic state.

scholarly journals Hopf Bifurcation and Global Periodic Solutions in a Predator-Prey System with Michaelis-Menten Type Functional Response and Two Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Yunxian Dai ◽  
Yiping Lin ◽  
Huitao Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Center Manifold ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Prey System ◽  
Model Study ◽  
Two Delays ◽  
Normal Form Method

We consider a predator-prey system with Michaelis-Menten type functional response and two delays. We focus on the case with two unequal and non-zero delays present in the model, study the local stability of the equilibria and the existence of Hopf bifurcation, and then obtain explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation when the delaysτ1≠τ2.

scholarly journals Hopf Bifurcation Analysis on General Gause-Type Predator-Prey Models with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Shuang Guo ◽  
Weihua Jiang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Normal Form Method ◽  
Predator Prey Models ◽  
The Stability

A class of three-dimensional Gause-type predator-prey model with delay is considered. Firstly, a group of sufficient conditions for the existence of Hopf bifurcation is obtained via employing the polynomial theorem by analyzing the distribution of the roots of the associated characteristic equation. Secondly, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions are determined by applying the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the obtained results.

STABILITY AND BIFURCATION IN A LOGISTIC EQUATION WITH PIECEWISE CONSTANT ARGUMENTS

2009 ◽  
Vol 19 (04) ◽  
pp. 1373-1379 ◽  
Author(s):  
CHUNRUI ZHANG ◽  
BAODONG ZHENG ◽  
YAZHUO ZHANG
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Critical Value ◽  
Logistic Equation ◽  
Piecewise Constant Arguments ◽  
Piecewise Constant ◽  
Center Manifold Theorem ◽  
Normal Form Method ◽  
The Stability ◽  
Bifurcating Periodic Solution

A logistic equation with piecewise constant arguments is investigated. Firstly, the linear stability of the model is studied. It is found that there exists a Hopf bifurcation when the parameter passes a critical value. Then the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solution is derived by using the normal form method and center manifold theorem. Finally, computer simulations are performed to illustrate the analytical results found.

scholarly journals Synchronized Hopf Bifurcation Analysis in a Delay-Coupled Semiconductor Lasers System

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Gang Zhu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Semiconductor Lasers ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Stability Switches ◽  
Center Manifold Theorem ◽  
Coupling Parameters ◽  
Normal Form Method ◽  
The Stability

The dynamics of a system of two semiconductor lasers, which are delay coupled via a passive relay within the synchronization manifold, are investigated. Depending on the coupling parameters, the system exhibits synchronized Hopf bifurcation and the stability switches as the delay varies. Employing the center manifold theorem and normal form method, an algorithm is derived for determining the Hopf bifurcation properties. Some numerical simulations are carried out to illustrate the analysis results.

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 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.

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.

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