scholarly journals Subcritical Hopf bifurcations in a car-following model with reaction-time delay

2006 ◽  
Vol 462 (2073) ◽  
pp. 2643-2670 ◽  
Author(s):  
Gábor Orosz ◽  
Gábor Stépán
Keyword(s):  
Hopf Bifurcation ◽  
Reaction Time ◽  
Time Delay ◽  
Hopf Bifurcations ◽  
Highway Traffic ◽  
Subcritical Case ◽  
Car Following ◽  
Car Following Model ◽  
Flow Equilibrium ◽  
The Stability

A nonlinear car-following model of highway traffic is considered, which includes the reaction-time delay of drivers. Linear stability analysis shows that the uniform flow equilibrium of the system loses its stability via Hopf bifurcations and thus oscillations can appear. The stability and amplitudes of the oscillations are determined with the help of normal-form calculations of the Hopf bifurcation that also handles the essential translational symmetry of the system. We show that the subcritical case of the Hopf bifurcation occurs robustly, which indicates the possibility of bistability. We also show how these oscillations lead to spatial wave formation as can be observed in real-world traffic flows.

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.

Analysis of a Car-Following Model with Driver Memory Effect

2015 ◽  
Vol 25 (04) ◽  
pp. 1550057 ◽  
Author(s):  
Zhi Xin ◽  
Jian Xu
Keyword(s):  
Time Delay ◽  
Memory Effect ◽  
Traffic Situation ◽  
System Variable ◽  
Car Following ◽  
State Dependent ◽  
State Dependent Delay ◽  
Car Following Model ◽  
Initial States ◽  
The Stability

A nonlinear car-following model considering memory effect of human drivers is studied by means of theoretical analysis and numerical simulation. We model the memory effect of the response of drivers to the traffic situation for a car-following model by introducing a system variable related to velocity. The memory effect of the drivers is a kind of state-dependent delay. The stability of the car-following model with two kinds of time delay is studied. The hysteresis loop of traffic flow from different initial states is compared. The effect of the maximum time delay on the stability is discussed.

scholarly journals Bifurcations and multiple traffic jams in a car-following model with reaction-time delay

2005 ◽  
Vol 211 (3-4) ◽  
pp. 277-293 ◽  
Author(s):  
Gábor Orosz ◽  
Bernd Krauskopf ◽  
R.Eddie Wilson
Keyword(s):  
Reaction Time ◽  
Time Delay ◽  
Car Following ◽  
Traffic Jams ◽  
Car Following Model

Stabilization strategies of a general nonlinear car-following model with varying reaction-time delay of the drivers

2014 ◽  
Vol 53 (6) ◽  
pp. 1739-1745 ◽  
Author(s):  
Shukai Li ◽  
Lixing Yang ◽  
Ziyou Gao ◽  
Keping Li
Keyword(s):  
Reaction Time ◽  
Time Delay ◽  
Car Following ◽  
Car Following Model

Multi-Anticipative Car-Following Model with Explicit Reaction-Time Delay

2021 ◽  
Vol 13 (6) ◽  
pp. 1109-1115
Author(s):  
V. V. Kurtc ◽  
I. E. Anufriev ◽  
D. O. Trufanov
Keyword(s):  
Reaction Time ◽  
Time Delay ◽  
Car Following ◽  
Car Following Model

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.

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