• 제목, 요약, 키워드: 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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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.

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.

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.

PUSHCHINO DYNAMICS OF INTERNAL LAYER

  • Yum, Sang Sup
    • Korean Journal of Mathematics
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    • v.12 no.1
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    • pp.7-14
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    • 2004
  • The existence of solutions and the occurence of a Hopf bifurcation for the free boundary problem with Pushchino dynamics was shown in [3]. In this paper we shall show a Hopf bifurcation occurs for the free boundary which is given by (1).

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BIFURCATION OF A PREDATOR-PREY SYSTEM WITH GENERATION DELAY AND HABITAT COMPLEXITY

  • Ma, Zhihui;Tang, Haopeng;Wang, Shufan;Wang, Tingting
    • Journal of the Korean Mathematical Society
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    • v.55 no.1
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    • pp.43-58
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    • 2018
  • In this paper, we study a delayed predator-prey system with Holling type IV functional response incorporating the effect of habitat complexity. The results show that there exist stability switches and Hopf bifurcation occurs while the delay crosses a set of critical values. The explicit formulas which determine the direction and stability of Hopf bifurcation are obtained by the normal form theory and the center manifold theorem.

Bifurcations of non-semi-simple eigenvalues and the zero-order approximations of responses at critical points of Hopf bifurcation in nonlinear systems

  • Chen, Yu Dong;Pei, Chun Yan;Chen, Su Huan
    • Structural Engineering and Mechanics
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    • v.40 no.3
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    • pp.335-346
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    • 2011
  • This paper deals with the bifurcations of non-semi-simple eigenvalues at critical point of Hopf bifurcation to understand the dynamic behavior of the system. By using the Puiseux expansion, the expression of the bifurcation of non-semi-simple eigenvalues and the corresponding topological structure in the parameter space are obtained. The zero-order approximate solutions in the vicinity of the critical points at which the multiple Hopf bifurcation may occur are developed. A numerical example, the flutter problem of an airfoil in simplified model, is given to illustrate the application of the proposed method.