Delayed Oscillation Phenomena in the FitzHugh Nagumo Equation
Delayed Oscillation Phenomena in the FitzHugh Nagumo Equation
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DOI：
10.1006/jdeq.1993.1087
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发表时间：
1993-09
影响因子：
2.4
通讯作者：
Jianzhong Su
中科院分区：
文献类型：
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作者：
Jianzhong Su
研究了FitzHugh Nagumo方程（FHN）vt = Dvxx − <$（v）− w + I 0 +<$t，（0.1a）wt = bv − Bγw，（0.1b）的Hopf分支点的慢通过问题，其中<$具有一些性质，使得当<$= 0时，系统在I = I−处有一个Hopf分支，且I = I 0 +<$t被认为是一个与t无关的参数. Jakobsson和Guttman的实验结果表明，只有当I达到远高于I−的值时，当I为正值且很小时，才会发生大振幅振荡。S. M. Baer，T. Erneux和J. Rinzel（Siam J.Appl.Math.49，1989，55-71）用数值方法研究了这些现象，并预测了系统的点火（跳跃）时间。在这项工作中，我们提供了一个严格的证明的结果由贝尔，Erneux，和Rinzel（参考上文）。我们证明，如果我们从框架解附近的任何点开始求解（0.1），即（0.1）右侧的零点，在任何Ii < I-处，那么解保持在框架解附近，直到I达到某个Iq < I-。此外，对于Ii接近I−的情况，我们证明了当I移动到某个Iq < I−以上时，解从标架解移动到大振幅解。
We study the problem of the slow passage through a Hopf bifurcation point for the FitzHugh Nagumo equation (FHN) vt = Dvxx − ƒ(v) − w + I0 + ϵt, (0.1a)wt = bv − bγw, (0.1b) where ƒ has some properties so that the system has a Hopf bifurcation at I = I− when ϵ = 0 and I = I0 + ϵt is regarded as a parameter independent of t. The experimental results of Jakobsson and Guttman showed that large amplitude oscillations occurred only after I reaches a value well above I− when ϵ is positive and small. The paper of S. M. Baer, T. Erneux, and J. Rinzel (Siam J. Appl. Math.49, 1989, 55-71) studied these phenomena numerically, and produced a prediction of the ignition (jumping) time for the system. In this work, we provide a rigorous proof of the results conjectured by Baer, Erneux, and Rinzel (referenced above). We show that if we start the solution of (0.1) at any point near the frame solution, which is the zero of the right-hand side of (0.1), at any Ii < I−, then the solution stays near the frame solution until I reaches some Iq < I−. Furthermore, for those cases in which Ii is close to I−, we show that the solution moves from the frame solution to become a large amplitude solution after I moves above some Iq < I−.