Hopf Bifurcation Analysis of KdV–Burgers–Kuramoto Chaotic System with Distributed Delay Feedback

2019 ◽  
Vol 29 (01) ◽  
pp. 1950011 ◽  
Author(s):  
Jie Liu ◽  
Junbiao Guan ◽  
Zhaosheng Feng
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic System ◽  
Equilibrium Points ◽  
Distributed Delay ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
The Mean ◽  
Mean Time ◽  
Theoretical Results

In this paper, the KdV–Burgers–Kuramoto chaotic system with distributed delay feedback is studied. The local stability of equilibrium points of this system is analyzed and the conditions under which Hopf bifurcation occurs are obtained by choosing the mean time delay as a bifurcation parameter. The direction and stability of bifurcating periodic solutions are derived by means of the normal form theory and the center manifold theorem. Numerical simulations are also illustrated which are in agreement with our theoretical results.

Hopf Bifurcation of KdV–Burgers–Kuramoto System with Delay Feedback

2020 ◽  
Vol 30 (14) ◽  
pp. 2050213
Author(s):  
Junbiao Guan ◽  
Jie Liu ◽  
Zhaosheng Feng
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic System ◽  
Center Manifold ◽  
Evolution Equations ◽  
Three Dimensional ◽  
Nonlinear Evolution Equations ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Theoretical Results

Chaotic phenomena may exist in nonlinear evolution equations. In many cases, they are undesirable but can be controlled. In this study, we deal with the chaos control of a three-dimensional chaotic system, reduced from a KdV–Burgers–Kuramoto equation. By adding a single delay feedback term into the chaotic system, we investigate the local stability and occurrence of Hopf bifurcation near the equilibrium point. Some dynamical properties including the direction and stability of bifurcated periodic solutions are presented by using the normal form theory and the center manifold theorem. Numerical simulations are illustrated which agree well with the theoretical results.

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.

scholarly journals Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Tao Dong ◽  
Xiaofeng Liao ◽  
Huaqing Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Computer Virus ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Form Theory ◽  
Virus Model ◽  
Theoretical Results

By considering that people may immunize their computers with countermeasures in susceptible state, exposed state and using anti-virus software may take a period of time, a computer virus model with time delay based on an SEIR model is proposed. We regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when time delay passes through a sequence of critical value. The linerized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.

scholarly journals Control of Hopf Bifurcation in Autonomous System Based on Washout Filter

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Wenju Du ◽  
Yandong Chu ◽  
Jiangang Zhang ◽  
Yingxiang Chang ◽  
Jianning Yu ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Equilibrium Points ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Bifurcation Phenomenon ◽  
Controlled Systems ◽  
Washout Filter ◽  
Form Theory ◽  
The Stability

In order to further understand a Lorenz-like system, we study the stability of the equilibrium points and the existence of Hopf bifurcation by center manifold theorem and normal form theory. More precisely, we designed a washout controller such that the equilibriumE0undergoes a controllable Hopf bifurcation, and by adjusting the controller parameters, we delayed Hopf bifurcation phenomenon of the equilibriumE+. Besides, numerical simulation is given to illustrate the theoretical analysis. Finally, two possible electronic circuits are given to realize the uncontrolled and the controlled systems.

scholarly journals Bifurcation Analysis for an SEIRS-V Model with Delays on the Transmission of Worms in a Wireless Sensor Network

2017 ◽  
Vol 2017 ◽  
pp. 1-15
Author(s):  
Zizhen Zhang ◽  
Yougang Wang
Keyword(s):  
Wireless Sensor Network ◽  
Hopf Bifurcation ◽  
Sensor Network ◽  
Center Manifold ◽  
Wireless Sensor ◽  
Bifurcation Parameter ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Theoretical Results

Hopf bifurcation for an SEIRS-V model with delays on the transmission of worms in a wireless sensor network is investigated. We focus on existence of the Hopf bifurcation by regarding the diverse delay as a bifurcation parameter. The results show that propagation of worms in the wireless sensor network can be controlled when the delay is suitably small under some certain conditions. Then, we study properties of the Hopf bifurcation by using the normal form theory and center manifold theorem. Finally, we give a numerical example to support the theoretical results.

scholarly journals Effects of delayed immune-activation in the dynamics of tumor-immune interactions

2020 ◽  
Vol 15 ◽  
pp. 45 ◽  
Author(s):  
Parthasakha Das ◽  
Pritha Das ◽  
Samhita Das
Keyword(s):  
Hopf Bifurcation ◽  
Immune Activation ◽  
Distributed Delay ◽  
Normal Form Theory ◽  
Effector Cells ◽  
Center Manifold Theorem ◽  
Discrete Delays ◽  
Form Theory ◽  
The Stability ◽  
The Impact

This article presents the impact of distributed and discrete delays that emerge in the formulation of a mathematical model of the human immunological system describing the interactions of effector cells (ECs), tumor cells (TCs) and helper T-cells (HTCs). We investigate the stability of equilibria and the commencement of sustained oscillations after Hopf-bifurcation. Moreover, based on the center manifold theorem and normal form theory, the expression for direction and stability of Hopf-bifurcation occurring at tumor presence equilibrium point of the system has been derived explicitly. The effect of distributed delay involved in immune-activation on the system dynamics of the tumor is demonstrated. Numerical simulations are also illustrated for elucidating the change of dynamic behavior by varying system parameters.

scholarly journals Bifurcation of a Delayed SEIS Epidemic Model with a Changing Delitescence and Nonlinear Incidence Rate

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Juan Liu
Keyword(s):  
Hopf Bifurcation ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Center Manifold ◽  
Nonlinear Incidence Rate ◽  
Normal Form Theory ◽  
Nonlinear Incidence ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Theoretical Results

This paper is concerned with a delayed SEIS (Susceptible-Exposed-Infectious-Susceptible) epidemic model with a changing delitescence and nonlinear incidence rate. First of all, local stability of the endemic equilibrium and the existence of a Hopf bifurcation are studied by choosing the time delay as the bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are determined based on the normal form theory and the center manifold theorem. At last, numerical simulations are carried out to illustrate the obtained theoretical results.

scholarly journals Hopf Bifurcation Analysis of a Two-Delay HIV-1 Virus Model with Delay-Dependent Parameters

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Yu Xiao ◽  
Yunxian Dai ◽  
Jinde Cao
Keyword(s):  
Hopf Bifurcation ◽  
Positive Equilibrium ◽  
Reproduction Rate ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Virus Model ◽  
Delay Dependent ◽  
Theoretical Results ◽  
Hiv 1

In this paper, a two-delay HIV-1 virus model with delay-dependent parameters is considered. The model includes both virus-to-cell and cell-to-cell transmissions. Firstly, immune-inactivated reproduction rate R 0 and immune-activated reproduction rate R 1 are deduced. When R 1 > 1 , the system has the unique positive equilibrium E ∗ . The local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the characteristic equation at the positive equilibrium with the time delay as the bifurcation parameter and four different cases. Besides, we obtain the direction and stability of the Hopf bifurcation by using the center manifold theorem and the normal form theory. Finally, the theoretical results are validated by numerical simulation.

HOPF BIFURCATION AND STABILITY ANALYSIS FOR A STAGE-STRUCTURED SYSTEM

2010 ◽  
Vol 03 (01) ◽  
pp. 21-41 ◽  
Author(s):  
JUNLI LIU ◽  
TAILEI ZHANG
Keyword(s):  
Hopf Bifurcation ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Structured System ◽  
Explicit Formulae ◽  
Bifurcation And Stability ◽  
Theoretical Results

In this paper, we considered a time-delay predator–prey system, in which the prey has two life stages, juvenile and mature. Delay was regarded as the bifurcation parameter, we analyzed the characteristic equation of the system at the positive equilibrium, stability of the positive equilibrium and existence of Hopf bifurcation with delay τ in the term of degree are investigated. The explicit formulae which determine the direction of the bifurcations, stability, and other properties of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. To verify our theoretical results, a numerical example is also included.

scholarly journals Hopf bifurcation of a delayed worm model with two latent periods

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Juan Liu ◽  
Zizhen Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Sensor Network ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Worm Model ◽  
Distribution Of Roots ◽  
Influential Parameters

Abstract We investigate a delayed epidemic model for the propagation of worm in wireless sensor network with two latent periods. We derive sufficient conditions for local stability of the worm-induced equilibrium of the system and the existence of Hopf bifurcation by regarding different combination of two latent time delays as the bifurcation parameter and analyzing the distribution of roots of the associated characteristic equation. In particular, we investigate the direction and stability of the Hopf bifurcation by means of the normal form theory and center manifold theorem. To verify analytical results, we present numerical simulations. Also, the effect of some influential parameters of sensor network is properly executed so that the oscillations can be reduced and removed from the network.

Sign in / Sign up

Export Citation Format

Share Document