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We study the problem of the slow passage through a Hopf bifurcation point for the FitzHugh Nagumo equation (FHN) \[ \begin{aligned} & v_{t}=D v_{x x}-f(v)-w+\phi(x)\left(I_{i}+\varepsilon t\right) \\ & w_{t}=b v-b \gamma w \end{aligned} \] where \(f\) has some properties so that the system has a Hopf bifurcation at \(I=I_{-}\) when \(\varepsilon=0\) and \(I=I_{i}+\varepsilon t\) is regarded as a parameter independent of \(t\). The experimental results of E. Jackobsson and R. Guttman (1981, in "Biophysical Approach to Excitable Systems," Plenum, New York) showed that large amplitude oscillations occurred only after \(I\) reached a value well above \(I_{-}\)when \(\varepsilon\) is positive and small. S. M. Baer, T. Erneux, and J. Rinzel (1989, SIAM Appl. Math. 49, 55-71) studied these phenomena numerically and produced a prediction of the ignition (jumping) time for the system. J. Su (1993, J. Differential Equations 105, 180-215; 1990, "Delayed Oscillation Phenomena in FitzHugh Naguma Equation," Ph.D. thesis) proved the delayed oscillation phenomena when \(\varphi(x) \equiv 1\). In this work, we show that delayed oscillations occur when \(\varepsilon\) is small enough for any \(\phi(

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Author

J.Z. Su

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0022-0396)

Published

1994

Language

EN

Field

Mathematics (Physical Sciences)