scholarly journals Bifurcation of a Cohen-Grossberg Neural Network with Discrete Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Qiming Liu ◽  
Wang Zheng
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Sufficient Conditions ◽  
Hopf Bifurcations ◽  
Bifurcation Parameter ◽  
Explicit Formulas ◽  
Global Hopf Bifurcation ◽  
Discrete Delays ◽  
Bifurcation And Stability

A simple Cohen-Grossberg neural network with discrete delays is investigated in this paper. The existence of local Hopf bifurcations is first considered by choosing the appropriate bifurcation parameter, and then explicit formulas are given to determine the direction of Hopf bifurcation and stability of the periodic solutions. Moreover, a set of sufficient conditions are given to guarantee the global Hopf bifurcation. Numerical simulations are given to illustrate the obtained results.

scholarly journals Stability and Hopf Bifurcation of ann-Neuron Cohen-Grossberg Neural Network with Time Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Qiming Liu ◽  
Sumin Yang
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Discrete Delays ◽  
Form Theory ◽  
Distribution Of Roots

A Cohen-Grossberg neural network with discrete delays is investigated in this paper. Sufficient conditions for the existence of local Hopf bifurcation are obtained by analyzing the distribution of roots of characteristic equation. Moreover, the direction and stability of Hopf bifurcation are obtained by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the obtained results.

Heteroclinic Trajectory and Hopf Bifurcation in an Extended Lorenz System

2018 ◽  
Vol 28 (09) ◽  
pp. 1850111
Author(s):  
Xianyi Li ◽  
Haijun Wang
Keyword(s):  
Hopf Bifurcation ◽  
Lyapunov Function ◽  
Numerical Simulations ◽  
Lorenz System ◽  
The Other ◽  
Hopf Bifurcations ◽  
Limit Sets ◽  
Initial Values ◽  
Heteroclinic Trajectory ◽  
The One

This note revisits an extended Lorenz system, which was presented in the paper entitled “Hopf bifurcations in an extended Lorenz system” by Zhou et al. [2017]. On the one hand, one points out and corrects some wrong results in that paper on the Hopf bifurcation at the symmetric equilibria [Formula: see text] and [Formula: see text]. On the other hand, combining Lyapunov function and the concepts of [Formula: see text]- and [Formula: see text]-limit sets, it is rigorously proved that there exists two and only two heteroclinic trajectories but no homoclinic trajectories under some certain conditions of its parameters and initial values. In addition, numerical simulations illustrate the consistence with the theoretical conclusions. The results together not only improve and complement the known ones, but also provide support in some future applications.

scholarly journals Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge

2019 ◽  
Vol 17 (1) ◽  
pp. 141-159 ◽  
Author(s):  
Zaowang Xiao ◽  
Zhong Li ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Bifurcation And Stability

Abstract In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

scholarly journals Hopf Bifurcation of a Mathematical Model for Growth of Tumors with an Action of Inhibitor and Two Time Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Bao Shi ◽  
Fangwei Zhang ◽  
Shihe Xu
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Time Delays ◽  
Cell Loss ◽  
Bifurcation Parameter ◽  
Discrete Delays

A mathematical model for growth of tumors with two discrete delays is studied. The delays, respectively, represent the time taken for cells to undergo mitosis and the time taken for the cell to modify the rate of cell loss due to apoptosis and kill of cells by the inhibitor. We show the influence of time delays on the Hopf bifurcation when one of delays is used as a bifurcation parameter.

QUALITATIVE ANALYSIS FOR THE MERKIN ENZYME REACTION SYSTEM

2002 ◽  
Vol 10 (02) ◽  
pp. 167-182
Author(s):  
YUQUAN WANG ◽  
ZUORUI SHEN
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Local Stability ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Reaction System ◽  
Enzyme Reaction ◽  
Qualitative Theory ◽  
Positive Equilibrium ◽  
Hopf Bifurcations

Applying qualitative theory and Hopf bifurcation theory, we detailedly discuss the Merkin enzyme reaction system, and the sufficient conditions derived for the global stability of the unique positive equilibrium, the local stability of three equilibria and the existence of limit cycles. Meanwhile, we show that the Hopf bifurcations may occur. Using MATLAB software, we present three examples to simulate these conclusions in this paper.

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.

Hopf Bifurcation of a Generalized Delay-Induced Predator–Prey System with Habitat Complexity

2020 ◽  
Vol 30 (06) ◽  
pp. 2050082
Author(s):  
Zhihui Ma
Keyword(s):  
Hopf Bifurcation ◽  
Habitat Complexity ◽  
Sufficient Conditions ◽  
Explicit Formulas ◽  
Predator Prey ◽  
Dynamical Behaviors ◽  
Prey System ◽  
The Stability ◽  
Theoretical Results ◽  
Positivity And Boundedness

A delay-induced nonautonomous predator–prey system with variable habitat complexity is proposed based on mathematical and ecological issues, and this system is more realistic than the published models. Firstly, the permanence of the nonautonomous predation system is studied and some sufficient conditions are obtained. Secondly, the dynamical behaviors of the corresponding autonomous predation system are investigated, including the positivity and boundedness, and local and global stabilities. Thirdly, the properties of Hopf bifurcation of the autonomous predation system without/with delay are investigated and the explicit formulas which determine the stability and the direction of periodic solutions are obtained. Finally, a numerical example is given to test our theoretical results.

Period Motions and Limit Cycle in a Periodically Forced, Plunged Galloping Oscillator

2017 ◽  
Author(s):  
Bo Yu ◽  
Albert C. J. Luo
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Limit Cycle ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Hopf Bifurcations ◽  
Periodic Motions ◽  
Stability And Bifurcations ◽  
Generalized Harmonic Balance Method

In this paper, periodic motions of a periodically forced, plunged galloping oscillator are investigated. The analytical solutions of stable and unstable periodic motions are obtained by the generalized harmonic balance method. Stability and bifurcations of the periodic motions are discussed through the eigenvalue analysis. The saddle-node and Hopf bifurcations of periodic motions are presented through frequency-amplitude curves. The Hopf bifurcation generates the quasiperiodic motions. Numerical simulations of stable and unstable periodic motions are illustrated.

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