• Title/Summary/Keyword: Hopf-bifurcation

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A GLOBALITY OF A HOPF BIFURCATION IN A FREE BOUNDARY PROBLEM

  • Ham, Yoon-Mee
    • Journal of the Korean Mathematical Society
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    • v.34 no.2
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    • pp.395-405
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    • 1997
  • A globality of the Hopf bifurcation in a free boundary problem for a parabolic partial differential equation is investigated in this paper. We shall examine the global behavior of the Hopf critical eigenvalues and and apply the center-index theory to show the globality.

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A HOPF BIFURCATION IN AN ATTRACTION-ATTRACTION CHEMOTAXIS SYSTEM WITH GLOBAL COUPLING

  • YoonMee Ham
    • Korean Journal of Mathematics
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    • v.31 no.2
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    • pp.203-216
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    • 2023
  • We consider a bistable attraction-attraction chemotaxis system with global coupling term. The study in this paper asserts that conditions for chemotactic coefficients for attraction and attraction and the global coupling constant to show existence of stationary solutions and Hopf bifurcation in the interfacial problem as the bifurcation parameters vary are obtained analytically.

Global Periodic Solutions in a Delayed Predator-Prey System with Holling II Functional Response

  • Jiang, Zhichao;Wang, Hongtao;Wang, Hongmei
    • Kyungpook Mathematical Journal
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    • v.50 no.2
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    • pp.255-266
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    • 2010
  • We consider a delayed predator-prey system with Holling II functional response. Firstly, the paper considers the stability and local Hopf bifurcation for a delayed prey-predator model using the basic theorem on zeros of general transcendental function, which was established by Cook etc.. Secondly, special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are given.

HOPF BIFURCATION IN NUMERICAL APPROXIMATION OF THE SUNFLOWER EQUATION

  • Zhang Chunrui;Zheng Baodong
    • Journal of applied mathematics & informatics
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    • v.22 no.1_2
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    • pp.113-124
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    • 2006
  • In this paper we consider the numerical solution of the sunflower equation. We prove that if the sunflower equation has a Hopf bifurcation point at a = ao, then the numerical solution with the Euler-method of the equation has a Hopf bifurcation point at ah = ao + O(h).

HOPF BIFURCATION IN NUMERICAL APPROXIMATION FOR DELAY DIFFERENTIAL EQUATIONS

  • Zhang, Chunrui;Liu, Mingzhu;Zheng, Baodong
    • Journal of applied mathematics & informatics
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    • v.14 no.1_2
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    • pp.319-328
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    • 2004
  • In this paper we investigate the qualitative behaviour of numerical approximation to a class delay differential equation. We consider the numerical solution of the delay differential equations undergoing a Hopf bifurcation. We prove the numerical approximation of delay differential equation had a Hopf bifurcation point if the true solution does.

STABILITY AND BIFURCATION ANALYSIS OF A LOTKA-VOLTERRA MODEL WITH TIME DELAYS

  • Xu, Changjin;Liao, Maoxin
    • Journal of applied mathematics & informatics
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    • v.29 no.1_2
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    • pp.1-22
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    • 2011
  • In this paper, a Lotka-Volterra model with time delays is considered. A set of sufficient conditions for the existence of Hopf bifurcation are obtained via analyzing the associated characteristic transcendental equation. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by applying the normal form method and center manifold theory. Finally, the main results are illustrated by some numerical simulations.

BIFURCATION ANALYSIS OF A DELAYED EPIDEMIC MODEL WITH DIFFUSION

  • Xu, Changjin;Liao, Maoxin
    • Communications of the Korean Mathematical Society
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    • v.26 no.2
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    • pp.321-338
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    • 2011
  • In this paper, a class of delayed epidemic model with diffusion is investigated. By analyzing the associated characteristic transcendental equation, its linear stability is investigated and Hopf bifurcation is demonstrated. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulation are also carried out to support our analytical findings. Finally, biological explanations and main conclusions are given.