Dynamics in a Van Der Pol Model with Delay

2012 ◽  
Vol 204-208 ◽  
pp. 4586-4589
Author(s):  
Chang Jin Xu ◽  
Pei Luan Li ◽  
Ling Yun Yao
Keyword(s):  
Asymptotic Behavior ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Parameter ◽  
Bifurcation Problem ◽  
Van Der Pol ◽  
Van Der Pol Model ◽  
The Stability

In this paper, the dynamics of a van der pol model with delay are considered. It is shown that the asymptotic behavior depends crucially on the time delay parameter. By regarded the delay as a bifurcation parameter, we are particularly interested in the study of the Hopf bifurcation problem. The length of delay which preserves the stability of the equilibrium is calculated. Some numerical simulations for justifying the analytical findings are included.

scholarly journals Chaos and Hopf Bifurcation Analysis of the Delayed Local Lengyel-Epstein System

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao ◽  
Jinde Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Critical Value ◽  
Hopf Bifurcations ◽  
The Stability

The local reaction-diffusion Lengyel-Epstein system with delay is investigated. By choosingτas bifurcating parameter, we show that Hopf bifurcations occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.

scholarly journals Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response

2006 ◽  
Vol 2006 ◽  
pp. 1-9 ◽  
Author(s):  
Radouane Yafia
Keyword(s):  
Immune Response ◽  
Asymptotic Behavior ◽  
Steady State ◽  
Immune System ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Limit Cycle ◽  
Numerical Study ◽  
Bifurcation Problem ◽  
Delayed Model

The dynamics of the model for tumor-immune system competition with negative immune response and with one delay investigated. We show that the asymptotic behavior depends crucially on the time delay parameter. We are particularly interested in the study of the Hopf bifurcation problem to predict the occurrence of a limit cycle bifurcating from the nontrivial steady state, by using the delay as a parameter of bifurcation. The obtained results provide the oscillations given by the numerical study in M. Gałach (2003), which are observed in reality by Kirschner and Panetta (1998).

scholarly journals Dynamics of a Discrete Internet Congestion Control Model

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Yingguo Li
Keyword(s):  
Time Delay ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Control Model ◽  
Bifurcation Parameter ◽  
Single Source ◽  
Internet Congestion Control ◽  
Single Link ◽  
The Stability

We consider a discrete Internet model with a single link accessed by a single source, which responds to congestion signals from the network. Firstly, the stability of the equilibria of the system is investigated by analyzing the characteristic equation. By choosing the time delay as a bifurcation parameter, we prove that Neimark-Sacker bifurcations occur when the delay passes a sequence of critical values. Then, the explicit algorithm for determining the direction of the Neimark-Sacker bifurcations and the stability of the bifurcating periodic solutions is derived. Finally, some numerical simulations are given to verify the theoretical analysis.

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.

Bifurcation Analysis in a Lotka-Volterra Model with Delay

2012 ◽  
Vol 594-597 ◽  
pp. 2693-2696
Author(s):  
Chang Jin Xu
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Volterra Model ◽  
The Stability ◽  
Theoretical Results

In this paper, a Lotka-Volterra model with time delay is considered. The stability of the equilibrium of the model is investigated and the existence of Hopf bifurcation is proved. Numerical simulations are performed to justify the theoretical results. Finally, main conclusions are included.

ECO-EPIDEMIOLOGICAL MODELS OF SALTON SEA WITH INFECTED PREY

2013 ◽  
Vol 21 (01) ◽  
pp. 1350003 ◽  
Author(s):  
Q. J. A. KHAN ◽  
E. BALAKRISHNAN ◽  
AZZA HAMOOD AL HARTHI
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Disease Transmission ◽  
Salton Sea ◽  
Bifurcation Parameter ◽  
Epidemiological Models ◽  
Analytical Results ◽  
Infected Prey

Two models for the interaction of susceptible and infected Tilapia population with Pelican population are studied. Here, we considered that Pelican interact with both susceptible and infected Tilapia in proportion to their abundance. Stability near nonzero equilibria is presented. In the second model, time delay is incorporated in the disease transmission term and Hopf bifurcation is analyzed by taking time delay as a bifurcation parameter. Numerical simulations are performed to support the analytical results.

scholarly journals Stability and Hopf Bifurcation Analysis of a Fractional-Order Epidemic Model with Time Delay

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Zhen Wang ◽  
Xinhe Wang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Equilibrium Point ◽  
Fractional Order ◽  
Bifurcation Analysis ◽  
Epidemic Model ◽  
Bifurcation Parameter ◽  
Theoretical Results ◽  
Disease Free

A fractional-order epidemic model with time delay is considered. Firstly, stability of the disease-free equilibrium point and endemic equilibrium point is studied. Then, by choosing the time delay as a bifurcation parameter, the existence of Hopf bifurcation is studied. Finally, numerical simulations are given to illustrate the effectiveness and feasibility of theoretical 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.

HOPF BIFURCATION OF A DELAYED DIFFERENTIAL EQUATION

2007 ◽  
Vol 17 (04) ◽  
pp. 1367-1374 ◽  
Author(s):  
QIAN GUO ◽  
CHANGPIN LI
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Nonlinear Differential Equation ◽  
Trivial Solution ◽  
The Stability ◽  
Analytical Expressions ◽  
Theoretical Results

In this paper, we study Hopf bifurcation of a second-order nonlinear differential equation with time delay by using the Lyapunov–Schmidt reduction. The approximate analytical expressions of the periodic solutions bifurcated from the trivial solution are given. We also discuss the stability of these periodic solutions. The numerical simulations line up with the theoretical results.

scholarly journals Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2444
Author(s):  
Yani Chen ◽  
Youhua Qian
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Degree Of Freedom ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Double Hopf Bifurcation ◽  
Central Manifold ◽  
The Stability ◽  
Two Degree Of Freedom

In this paper, the normal form and central manifold theories are used to discuss the influence of two-degree-of-freedom coupled van der Pol oscillators with time delay feedback. Compared with the single-degree-of-freedom time delay van der Pol oscillator, the system studied in this paper has richer dynamical behavior. The results obtained include: the change of time delay causing the stability switching of the system, and the greater the time delay, the more complicated the stability switching. Near the double Hopf bifurcation point, the system is simplified by using the normal form and central manifold theories. The system is divided into six regions with different dynamical properties. With the above results, for practical engineering problems, we can perform time delay feedback adjustment to make the system show amplitude death, limit loop, and so on. It is worth noting that because of the existence of unstable limit cycles in the system, the limit cycle cannot be obtained by numerical solution. Therefore, we derive the approximate analytical solution of the system and simulate the time history of the interaction between two frequencies in Region IV.

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