Gierer-Meinhard Model吸引域,极限环和Hopf分岔现象分析
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摘要
在这篇文章中,研究著名的Gierer-Meinhard模型的动力学性态.首先讨论了平衡点稳定和不稳定的条件.在获得平衡点稳定与不稳定条件后,在稳定的情形,通过构造李雅普诺夫函数,对吸引域的范围进行了估计,在不稳定的情形,通过内外界周线的控制来确定极限环的大致位置.此外,我们还断言本系统在临界情形时会发生霍普夫分岔现象,并借助一些图像来直观地解释我们的理论结果.
In this paper,we are concerned with the dynamic behavior for the well-known Gierer-Meinhard Model. We first discuss the conditions on the stability and instability of the equilibrium point. Then,in the case of stability,we seek for the attract domain; While in the case of instability,we will prove the existence of limit cycle. We claim that the occurrence of Hopf bifurcation in the critical case. Graphs are also used to illustrate our theoretic results.
In this paper,we are concerned with the dynamic behavior for the well-known Gierer-Meinhard Model. We first discuss the conditions on the stability and instability of the equilibrium point. Then,in the case of stability,we seek for the attract domain; While in the case of instability,we will prove the existence of limit cycle. We claim that the occurrence of Hopf bifurcation in the critical case. Graphs are also used to illustrate our theoretic results.
引文
[1]Zou H.On global existence for the Gierer-Meinhardt system[J].Discrete Contin.Dyn.Syst,2015(35):583-591.
[2]Gierer A,Meinhardt H.A theory of biological pattern formation[J].Kybernetik,1972,(12):30-39.
[3]Karali G,Suzuki T,Yamada Y.Global-in-time behavior of the solution to a Gierer-Meinhardt system[J].Discrete Contin.Dyn.Syst,2013(33):2885-2900.
[4]Kolokolonikov T,Ren X.Smoke-ring solutions of GiererMeinhardt system in[J].SIAM J.Appl.Dyn.Syst,2011(10):251-277.
[5]Kolokolnikov T,Wei J,Yang W.On large ring solutions for Gierer-Meinhardt system in[J].Journal Differential Equations,2013,(255):1408-1436.
[6]Li F,Yip N.Finite time blow-up of parabolic systems with nonlocal terms[J].Indiana Univ.Math.J,2014,(63):783-829.
[7]Li F,Peng Rui,Song X F.Global existence and finite time blow-up of solutions of a Gierer-Meinhardt system[J].Journal Differential Equations,2017(262):559-589.
[8]Ni W M,Suzuki K,Takagi I.The dynamics of a kynetics activator-inhibitor system[J].Journal Differential Equations,2006(229):426-465.
[9]Ni W M,Wei J.On positive solutions concentrating on spheres for the Gierer-Meinhardt system[J].Journal Differential Equations,2006(221):158-189.
[10]Turing A.The chemical basis of morphogenesis[J].Philos.Trans.R.Soc.B,1952(237):37-72.
[11]Zou H.Finite-time blow-up and blow-up rates for the Gierer-Meinhardt system[J].Appl.Anal,2015(94):2110-2132.
[12]陆启韶,彭临平,杨卓琴.常微分方程与动力系统[M].北京:北京航空航天大学出版社,2010:51-61.
[13]张家忠.非线性动力系统的运动稳定性、分岔理论及其应用[M].西安:西安交通大学大学出版社,2010:21-30.
[2]Gierer A,Meinhardt H.A theory of biological pattern formation[J].Kybernetik,1972,(12):30-39.
[3]Karali G,Suzuki T,Yamada Y.Global-in-time behavior of the solution to a Gierer-Meinhardt system[J].Discrete Contin.Dyn.Syst,2013(33):2885-2900.
[4]Kolokolonikov T,Ren X.Smoke-ring solutions of GiererMeinhardt system in[J].SIAM J.Appl.Dyn.Syst,2011(10):251-277.
[5]Kolokolnikov T,Wei J,Yang W.On large ring solutions for Gierer-Meinhardt system in[J].Journal Differential Equations,2013,(255):1408-1436.
[6]Li F,Yip N.Finite time blow-up of parabolic systems with nonlocal terms[J].Indiana Univ.Math.J,2014,(63):783-829.
[7]Li F,Peng Rui,Song X F.Global existence and finite time blow-up of solutions of a Gierer-Meinhardt system[J].Journal Differential Equations,2017(262):559-589.
[8]Ni W M,Suzuki K,Takagi I.The dynamics of a kynetics activator-inhibitor system[J].Journal Differential Equations,2006(229):426-465.
[9]Ni W M,Wei J.On positive solutions concentrating on spheres for the Gierer-Meinhardt system[J].Journal Differential Equations,2006(221):158-189.
[10]Turing A.The chemical basis of morphogenesis[J].Philos.Trans.R.Soc.B,1952(237):37-72.
[11]Zou H.Finite-time blow-up and blow-up rates for the Gierer-Meinhardt system[J].Appl.Anal,2015(94):2110-2132.
[12]陆启韶,彭临平,杨卓琴.常微分方程与动力系统[M].北京:北京航空航天大学出版社,2010:51-61.
[13]张家忠.非线性动力系统的运动稳定性、分岔理论及其应用[M].西安:西安交通大学大学出版社,2010:21-30.