数理解析研究所講究録
1072 巻 1998 年 28-38

28

ON AN

$\omega$-INCOMPLETE DUCK AND ITS APPLICATIONKIYOYUKI TCHIZAWA$k\nearrow \mathcal{K}=$$\simeq\dot{\mathrm{F}}-\neg\neq=-$

Dept. of mathematics, Musashi Institute of Technology. Introducing a parameter newly to the singular perturbation problem,
an -incomplete duck solution; a singular -limit of the duck solution for the parameter will be discussed. It will become clear that the winding number for one of
the solutions goes to infinity at that limit. Furthermore, the problem :when cer? will betain coefficients have different infinitesimals, what happens in thesolved.

A bst$\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}$$\omega$$\omega$$\Re \mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$1. INTRODUCTION.
The explicit duck solutions (or simply ducks) were constructed bybut
not include a parameter in the differential equations. He showed, in his paper,that if the difference of each the winding numbers associated with the ducks ismore than 3/2, there exists a duck which is not
is a smoothness class innonstandard analysis). It is important to remark that this solution could not be
constructed explicitly yet. The explicit -incomplete solution in the local modelof the FitzHugh-Nagumo(FHN) equation will be discussed, however, this solution
is the exact solution in the first approximation of this model. The solutions very
near by this exact solution are winding so many times; the winding number goes toinfinity as the parameter tends to infinity. In Section2 and Section3, introducing
a parameter in the differential equations, the -incomplete ducks will be discussed
as a singular limit of the ducks for the parameter. In Section4 and Section5, thispaper will treat of the FitzHugh-Nagumo equation and, in Section6, analyze its-incomplete duck in the first approximation of the ”local model” ([4], [18]).
$\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{t}[4]$$S^{1}(S^{1}$$\omega$$\omega$$\omega$

2. PRELIMINARIES.
Let consider a constrained system(2.1):$dx/dt=f(x, y, z, u)$ ,(2.1)$dy/dt=g(X, y, z, u)$ ,$h(x, y, z,u)=0$ ,1991 Mathematics Subject Classification.
.
Key words and phrases. constrained system, singular perturbation, duck solution, winding
number, FitzHugh-Nagumo equation..
$34\mathrm{A}34,34\mathrm{A}47,34\mathrm{C}35.$

Typeset by$A_{\lambda 4}\mathrm{s}\tau \mathrm{E}\mathrm{x}$29where is a parameter (any fixed) and $f,g,h$ are defined in. Furthermore,
let consider the singular perturbation problem of the system (2.1):$R^{3}\cross R^{1}$$u$$dx/dt=f(x, y, z, u)$ ,$dy/dt=g(x, y, z, u)$ ,

(2.2)

,

$\epsilon dz/dt=h(x, y, z, u)$where is infinitesimally small.
We assume that the system (2.1) satisfies the following conditions (A ) $-(A5)$ :$(A1)f$ and are of classand is of class.
$(A2)$ The set $S=\{(x, y, z)\in R^{3} : h(x, y, z, u)=0\}$ is a 2-dimensional differentiable manifold and the set intersects the set$T=\{(x, y, z)\in R^{3} : \partial h(x, y, Z, u)/\partial z=0\}$ transversely so that the set $PL=$$\{(x, y, z)\in S\cap T\}$ is a 1-dimensional differentiable.
$(A3)$ Either the value ofor that of is nonzero at any point $p\in PL$ .
Let $(x(t, u),$ $y(t, u),$ $z(t, u))$ be a solution of (2.1). By differentiating $h(x, y, z, u)$with respect to the time , the following equation holds:$\epsilon$$1$$C^{1}$$g$$C^{2}$$h$$S$$\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\sim \mathrm{l}\mathrm{d}$

$f$

$g$

$t$

(2.3)

$h_{x}(x, y, z, u)f(x, y, z, u)+h_{y}(x, y, z, u)g(x, y, z, u)+h_{z}(x, y, z, u)dZ/dt=0$ ,where $h_{i}(x, y, z, u)=\partial h(x, y, z, u)/\partial i,$ $i=x,$ $y,$ . The above system (2.1) becomesthe following system:$z$$dx/dt=f(x, y, z, u)$ ,$dy/dt=g(X, y, Z, u)$ ,(2.4)$dz/dt=-\{h_{x}(x, y, z,u)f(x,y, z, u)+$$h_{y}(x, y, z, u)g(x, y, z, u)\}/h_{z}(x, y, Z, u)$,where $(x, y, z)\in S\backslash PL$ . The system (2.1) coincides with the system (2.4) at any
point $p\in S\backslash PL$ . In order to study the system (2.4), let consider the following
system:$dx/dt=-h_{z}(x, y, z, u)f(x, y, z,u)$ ,(2.5)$dy/dt=-h_{z}(x, y, z, u)g(X, y_{j}z, u)$ ,$dz/dt=h_{x}(x, y, z, u)f(x, y, z, u)+h_{y}(x, y, z, u)g(x, y, z, u)$ .

As the system(2.5) is well defined at any point of , it is well defined indeed at
any point of $PL$ . The solutions of (2.4) coincide with those of (2.1) on
except the velocity when they start from the same initial points.
$(A4)$ For any $(x, y, z)\in S$ , either of the following holds;$R^{3}$$S\backslash PL$(2.6)$h_{y}(x, y, Z, u)\neq 0,$ $h_{x}(x, y, z, u)\neq 0$,that is, the surface can be expressed as $y=\varphi(x, z, u)$ or $x=\psi(y, z, u)$ in theneighborhood of $PL$ . Let $y=\varphi(x, z, u)$ exist, then the projected system, which
restricts the system (2.5) is obtained:$S$$dx/dt=-h_{z}(x, \varphi(X, Z, u), z,u)f(x, \varphi(x, Z, u), z, u)$(2.7)$dz/dt=h_{x}(x, \varphi(X, z, u), z, u)f(X, \varphi(x, z, u), z, u)+$$h_{y}(x, \varphi(X, Z, u), z, u)g(x,$ $\varphi(X,\cdot\varphi(x, Z, u), z, u)$

.

,

30
All the singular points of (2.7) are nondegenerate, the matrix induced
the linearized system of (2.6) at a singular point has two nonzero eigenvalues. Notethat all the points contained in $PS=\{(x, y, z)\in PL:dz/dt=0\}$ , which is called
pseudo singular points are the singular points of (2.7).
$(A5)$$\mathrm{h}\mathrm{o}\mathrm{m}$Definition2.1. Let $p\in PS$ andbe two eigenvalues of the linealized
system of (2.7), then the point is called pseudo singular saddle if $\mu_{1}(u)<0<$
and called pseudo singular node if $\mu_{1}(u)<\mu_{2}(u)<0$ or $\mu_{1}(u)>\mu_{2}(u)>0$ .
$\mu_{1}(u),$$\mu_{2}(u)$$p$$\mu_{2}(u)$Definition2.2. A solution $(x(t, u),$ $y(t, u),$ $z(t, u))$ of the system(2.2) is called
duck, if there exist standard $t_{1}<t_{0}<t_{2}$ such that
(1) $*(x(b_{0}, u),$ $y(t0, u),$ $Z(t_{0}, u))\in S,$ where $*(X)$ denotes the standard part of $X$ ,(2) for $t\in(t_{1}, t_{0})$ the segment of the trajectory $(x(t, u),$ $y(t, u)Y^{\cdot}Z(b, u))$ is infinitesimally close to the attracting part of the slow curves,(3) for $t\in(t_{0}, t_{2})$ , it is infinitesimally close to the repelling part of the slow curves,and(4) the attracting and repelling parts of the trajectory are not infinitesimally small.

$a$. We call a point is a
Definition2.3. Let $E$ be a set in$E$
is less than $exp(-n/\delta)$ , where
when the distance from to( is infinitesimally small).
integer and$R^{3}$$p$$p$$\delta=\epsilon/\alpha^{2}$-galaxy of $E$
is some positive$\delta- \mathrm{m}\mathrm{i}\mathrm{C}\mathrm{r}\mathrm{o}$$n$$\alpha$Theorem2.1(Benoit). In the system(2.1), if the following two conditions at a
pseudo singular saddle or node point;(1) $f(O,u)\simeq h(O, u)\simeq h_{y}(O, u)\simeq h_{z}(O, u)\simeq 0$ ,(2)
, where $O=(0,0, \mathrm{o})\in PS$ ,are satisfied, the first approximation of the ”local model”$g(O, u)\not\simeq \mathrm{O},$ $h_{x}(O, u)\not\simeq \mathrm{O},$ $h_{zz}(O, u)\not\simeq \mathrm{O}$$dX/dt=p\mathrm{Y}+qZ$(2.8)$d\mathrm{Y}/dt=1$,,$\delta dZ/dt=-(Z^{2}+X)$ ,is obtained, where
are constants and. Then, the explicit duck solutionsin the first approximation of the system(2.2) can be constructed:$\delta\simeq 0$$p,$ $q$$\gamma_{\mu_{i}(u)}$(2.9)$\gamma_{\mu i}(u)(b)=(-\mu_{i}(u)2t2-\delta\mu i(u), t, \mu i(u)t)(i=1,2)$

Proceeding the following coordinate transformations,$u=X+z^{2}+\delta\mu$(2.10)$v=\mathrm{Y}-Z/\mu$,$z=Z$ ,,
$v=rsin\theta$ ,$u=rcos\theta$(2.11),.

31
is transformed to $(0,0, \mu t)$ . Moreover, proceeding the coordinate

the solution
transformation$\gamma_{\mu}(t)$(2.12)$\rho=\delta\log r$,

the system ( $(2.13)$ is obtained:,$\delta d\theta/dt=2z\cos\theta\sin\theta+\cos^{2}\theta/\mu-\delta p\sin^{2}\theta$(2.13)$d\rho/dt=-2z\cos^{2}\theta+(1/\mu+\delta p)\cos\theta\sin\theta$$dz/dt=-\exp(\rho/\delta)\cos\theta+\delta\mu$,,

and proceeding(2.14)$w=-\cot\theta$,

the first equation in the equations (2.13) becomes the following Riccati equation,(2.15)$\delta dw/dt=w(w-ct)/\mu-\delta p,$ ( $c$: some constant)

is obtained. Fhrthermore, applying time scaling and applying the certain cordinatetransformation to $W$ , the Hermite equations associated withareobtained as the following:$w$(2.16)$\gamma_{\mu_{\mathrm{t}}(u)}(i=1,2)$$\delta\ddot{W}-\tau\dot{W}+K_{i}z=0,t=\tau/\alpha,$where
See [4].

$K_{i}$( : any constant)(i $=1,2$ ),$\alpha$is a positive integer and $K_{1}=1+\mu_{2}(u)/\mu_{1}(u),$ $K_{2}--1+\mu_{1}(u)/\mu_{2}(u)$ .

Definition2.4. The winding number $N(\psi)$ of a duck
(2.17)

where$N( \psi)=(1/2\pi)\int_{\psi}d\theta$$\psi$$\psi$

is defined as follows:,is contained partially in the -micr-galaxy of$\delta$$\gamma_{\mu}$.

The above Definition2.3 is based on the following fact. If is fixed arbitrarily
andis a duck near
, then the distance ffom
tois within$exp(-n/\delta)$ in some neighborhood of the pseudo singular point. See [19].
It is said that a duckhas a jump if the shadow of it contains a vertical segment
and thatis long if it is in an infinitesimally small neighborhood at the pseudo
singular point. It can be proved that if is not long, the standard part of thewinding number
is an integer. If the pseudo singularassociated withpoint is node, it is positive. If the point is saddle, it needs some conditions such as
is poitive. The relation between
and $K_{i}(i=1,2)$ is as follows.
$\epsilon$$\gamma(t)$$\gamma(t)$$\gamma_{\mu(u)()}t$$\gamma_{\mu(u)()}t$$\psi(t)$$\psi(t)$$\psi$$N(\psi_{i})$$\mu_{i}$$N(\psi_{i})$$K_{i}$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\Gamma \mathrm{e}\mathrm{m}2.2(\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{t})$$N(\psi_{1})\approx-[K1/2]$. If the duck

, and if the duck

, which is not long has 2 jumps,
has 2 jumps,.

$\psi_{1}$$\psi_{2}$$N(\psi_{2})\approx 0$

323. DEFINITION OF AN$\omega$-INCOMPLETE DUCK.

In the system(2.2), making the following coordinate transformations (3.1) and(3.2) successively;

$=,$

(3.1)

(3.2)

$(\alpha\simeq 0,$$\epsilon/\alpha^{2}\simeq^{\mathrm{o})}$$=(^{h_{x}()}\mathrm{o},$$uh_{zz}(0, u)\tilde{x}/2+(h_{yy,\tilde{y}}(\mathrm{o},,zz\mathrm{o},u)-h_{yz}(0, u)^{2})\tilde{y}^{2}/4)-h_{y}z(\mathrm{o},u)\tilde{y}/2/g(\mathrm{o}^{u)h(}u)-hzz(0,u)_{\tilde{Z}/2}$’

the following system(3.3) is obtained:$dX/dt=p\mathrm{Y}+qZ+\xi(x, \mathrm{Y}, z, u)$(3.3)$d\mathrm{Y}/dt=1+\eta(X, \mathrm{Y}, z, u)$,,$\delta dZ/dt=-(Z^{2}+X)+\zeta(X, \mathrm{Y}, z, u)$

,

where
$p=g(0, u)hx(\mathrm{o}, u)(f_{y}(\mathrm{o}, u)h(zz0, u)-fz(\mathrm{o}, u)h_{yz}(0, u))/2$$+g(\mathrm{O}, u)^{2}(h_{yy}(0, u)h(zz\mathrm{o}, u)-h_{y}(z\mathrm{o}, u)^{2})/2$$q=-h_{x}(0, u)fz(\mathrm{o}, u)$$\delta=\epsilon/\alpha^{2}$

,

,

.

Here
are limited.

and$\xi(X, \mathrm{Y}, Z, u),\eta(X, \mathrm{Y}, Z, u)$$\zeta(X, \mathrm{Y}, z, u)$

are infinitesimal when $X,$

Definition3.1. In the system(3.3), if the followings (1) and (2):(1) for any limited parameter ,it satisfies the conditions $(A1)_{-}(A5)$ and has a duck,(2) when the parameter tends to infinity, one of the winding numbers
tends to infinity and the other tends to zero,and the system does not have a duck as a singular limit,
are established, this solution is called an -incomplete duck.
$u$

$u$

$\omega$

Definition3.2. A solution $\psi(x, u)$ is called
if there exists a real number such that$S^{1}$

at a,

$b$

(3.4)

$\frac{\psi(x,u)-^{\psi()}y,u}{x-y}\overline{\sim}b$

for any $y(x\approx a, y\approx a)$ .
A duck is called an duck if it isof the pseudo singular point.

,

$x,$

$S^{1}$$S^{1}$

in some neighborhood$\mathrm{Y}$

and

$Z$

33
Theorem3.1 (Benoit). In the first approximation of the system(3.3),
if $\mu_{1}(u)/\mu_{2}(u)$ is positive $(>3)$ but no an integer, then all theducks are exponentially close to one of the two explicit ducks and there exists non
ducks.
$S^{1}$$S^{1}$In the system(3.3), we assume that
(3.5)

,

$f_{y}(\mathrm{o}, u)=g_{u}(0, u)=h_{yz}(\mathrm{o}, u)=h_{yyu}(\mathrm{o}, u)=h_{zzu}(0, u)=0$

and that the following (1) or (2):(1) $h_{x}(\mathrm{o}, u)=O(u)$ and $f_{z}(0,u)=O(1)$ ,(2) $f_{z}(0, u)=O(u)$ and $h_{x}(\mathrm{o}, u)=O(1)$ ,where all the coefficients of higher order (more than 2) for is negligible, that is,
only the coefficient can take an unlimited number ( $q=c_{1}u+o(1)$ , a constant. Then, blowing up only the variable $Z$ again;
$u$

$q$

$c_{1}\not\simeq 0)$(3.6)$Z=(1/u)\overline{Z}$,the first approximation of the system(3.3) becomes the following:$dX/dt=p\mathrm{Y}+c_{1}\tilde{Z}$(3.7)$d\mathrm{Y}/dt=1$,,$(\delta/u)d\tilde{Z}/dt=-(\tilde{Z}^{2}/u^{2}+X)$

where is limited (does not contain ) andsystem(3.7) are$u$$c_{1}$(3.8)$\delta/u\simeq \mathrm{O}$,. The explicit solutions in the$\gamma_{\mu_{i}}(u)(t)=(-\mu i(u)2t-2\delta\mu i(u), t, u\mu i(u)t)(i=1,2)$,$\mu_{2}(u)(\mu_{1}(u)>\mu_{2}(u))$ are the solutions of the characteristic equation
where.
of the system(3.7) in case
The above system satifies the conditions (A ) $-(A5)$ and the solutions(3.8) satisfy
in Definition3.1.
the condition (1) and satisfies the condition (2) when
In fact, if $q=\epsilon^{-1/3}$ , then the existence of such a duck is ensured. We choose$\epsilon=1/n(n=2,3, \ldots)(u=1/n^{1/3})$ , then $1/n^{1/3}\gg exp(-n/\delta)$ for any $n(n\geq 2)$ .
,In the system for each any fixed , let $J=[AB]$ be a connected segment inbelongs to the family of the duckwhere the solution which starts at). It can be proved that if any solution starting at $p\in J$ is not long, then(orit has the same winding number. FYom Theorem3.1, a duck passing through thepseudo singular node point belongs to one of two families of the above ducks. On
the other hand, there exists a segment $[CD]\subset J$ such that any solution startingat $p\in[CD]$ is not long and the solutions passing through $C$ or $D$ are ducks. This
duck. Note that $\mu_{1}(u)/\mu_{2}(u)$ is positive but
fact ensures the existence of a nonno an integer. If it is a positive integer , it indicates the fact that the slow vector. Then, it is not possible
trajectories but only one of them isfield has twoto have an asymptotic expansion in powers of with the coefficients analytic in. Furthermore, one of the solutions(3.8) may tend tangent to the $X$ -axis, since$\mu_{2}(u)arrow-\omega/2$ as
and for the first component of (3.8), the following$\mu_{1}(u),$$\delta/u\simeq \mathrm{O}$$1$$uarrow\omega$$R^{3}$$n$$A(\mathrm{o}\mathrm{r}B)$$\gamma_{\mu_{1}}$$\gamma_{\mu_{2}}$$S^{1}$$k$$C^{k}$$C^{1}$$\epsilon$$t$$uarrow\omega$(3.9)$\frac{-(\omega/2)^{2}(2/\omega)^{2}+(\omega/2)^{2}(1/\omega)^{2}}{2/\omega-1/\omega}arrow-3\omega/4$,

34
establishes. In this state, the winding number
associated withtends
to infinity and the otherassociated withtends to infinitesimal. When$uarrow\omega=\epsilon^{-1/3}$, the eigen space of the linear part of the slow vector field for(is). The ducks are almost
fortangent to the eigen spaces and therefore the -limit of the duck with respect tothe parameter ( -incomplete duck) is not .
holds and then the following conditions are
Let $v=1/u$ , then$f(x,y,z,u)=\tilde{f}(x,y,z,v)\inC^{3},$ $g(x, y, z, u)=\tilde{g}(x, y, z,v)\in C^{1}$ andassumed;$h(x, y, z, u)=\tilde{h}(x, y, z, v)\in C^{3}$ at almost every where but $v=v_{0}=0$ .
Rom
the assumptions, the relation $q=-h_{x}(0, u)f_{z}(0, u)=c_{1}u$ holds. Differentiating
the both side of this equation by the parameter , we can lead to the following
theorem.
$N(\psi_{2})$$N(\psi_{1})$$z\simeq-\epsilon^{1/3}y$$z\simeq\epsilon^{1/3}y$$\mu_{2}\simeq-\epsilon^{-1/3}$$\mu_{2}$$\mu_{1}$$\mu_{1}\simeq-\epsilon^{1/3}$$\omega$$u$$S^{1}$$\omega$$\partial_{u}=-v^{2}\partial_{v}$$v$Theorem3.2. In the first approximation of the system(3.3), if
is pos$\tilde{h}_{xv}(0,v)\tilde{f}zv(0,v)=0$
itive but no integer under the condition (3.5) and ifwheneither the condition (1) or (2) ;
, and
,
(1)
(2)
, and
,
where all the coefficients of higher order (more han 2) for is negligible are satisfied, then this system has an -incomplete duck.
$\mu_{1}(u)/\mu_{2}(u)$$\tilde{f}_{z}(0, v)=0$$\tilde{h}_{x}(0, v)\tilde{f}_{z}v\underline{v}(0, v)=0$$\tilde{h}_{x}(0, v)=0$$\tilde{h}_{xvv}(0,v)fz(\mathrm{o},v)=0$$\dot{\mathrm{t}}$$u$$\omega$

Remark. In the system(3.3), if the coefficient satisfies $q=c_{1}u+O(1)$ , that is,
and $p>0$ or $0>p\geq-1/32$ , then there exists a finite
where
which makes the winding number infinite when tends to .
value
$q$

$q=c_{1}u+c_{2}$$c_{1},$$c_{2}\not\simeq 0$$u$$u_{0}$$u_{0}$

4. THE FHN EQUATION.
We consider the space-clamped FitzHugh-Nagumo(FHN) equation (4.1) withslowly varing in the time dependent parameter I;$dv/dt=-p(v)-w+I$(4.1)$dw/dt=b(v-\gamma w),$ $\rho(v)=v(v-1)(v-a)$where $0<a<1/2,$ and are positive constants. Each variable is denoted asfollows;$v(t)$ : the potential difference at the time
across the membrane of the axon,$w(t)$ : a recovery current which is often taken to be the sum of all ion flows,: an injected electric current treated as a control or bifurcation parameter depending on the time on the membrane such that$b$$\gamma$$\mathrm{t}$$I$$\mathrm{t}$(4.2)$I=I_{0}+\epsilon 1t$,

is an initially given constant. The first of
is infinitesimally small andwhere(4.1) expresses Kirchhoff’s law applied to the membrane. The second relates thebiophysicalrecovery current with the potential. The constant is restrictedconsiderations so that$\epsilon_{1}$$I_{0}$$\mathrm{h}\mathrm{o}\mathrm{m}$$\gamma$(4.3)$0<\gamma<3/(1-a+a^{2})$35standard form of a slow-fast system with two slow variables and one fast variable,that is,(4.4)$b=c\epsilon_{2}$,and is any constant.
where
Though we first start at this condition, in Section6, we will refer to the case.
gave a sufficient condition for the existence of the ducks: a
pseudo singular point is a saddle point. Under the assumptions (4.2) and (4.4), thesystem (4.1) has a pseudo singular saddle point [18], that is, this system has ducks.
After using several coordinate transformations, the system (4.1) is transformed to
the local model. This model has a pseudo singular node point under a certain
condition. Then two duck solutions $\gamma_{\mu_{i}}(i=1,2)$ at the pseudo singular point
are obtained where $\mu_{i}(i=1,2)$ are the eigenvalues associated with the linearized, the winding
system. Furthermore, in a sufficiently small neighborhood of
number of the duck for , which depends on the Hermite equation derived by theseveral transformations becomes infinity when the constant in the system (4.1)
is limited and greater
tends to zero [11]. The winding number of the duck forhas already pointed out
than 1 in this state. As mentioned in Sectionl,that if the difference of each the ratio of the eigenvalues, which relates the winding
number directly is greater than 3, there exists a duck without . Therefore, thisfact tells us the first approximation system of the local model has a duck without
but this is not an explicit form.
In this back ground, there is the delay problem $[1],[2],[14]$ . When $k\leq 1$ under$\epsilon=O(b^{k})$ , it had not been solved yet because of the difficulty: the uniformity ofthe solution is not ensured in the asymptotic expansion. Some authors $([6],[9],[12])$
directly in their papers and obtained various results, however, they did
put
not touch ducks.
$\epsilon_{1}=\epsilon 2=\epsilon$$\epsilon_{1}\neq\epsilon_{2}$$c$$\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{t}[3]$$\gamma_{\mu_{i}}$$\mu_{1}$$\gamma$$\mu_{\mathit{2}}$$\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{t}[4]$$S^{1}$$S^{1}$$b=\epsilon$

5. THE DUCKS IN THE FHN.
Using the assumptions (4.2) and (4.4), the system(4.1) becomes the system(5.1):(5.1)$\epsilon dv/dI=-\rho(v)-w+I$ ,$dw/dI=C(v-\gamma w)$ .

By changing the coordinates, $w=X,$
the system(5.2):$I=\mathrm{Y}$and $v=Z$ , the system(5.1) becomes$dX/dI=c(Z-\gamma X)$ ,(5.2)$d\mathrm{Y}/dI=1$,$\epsilon dZ/dI=(-p(Z)-X+\mathrm{Y})$.

,Note that the system(5.2) satisfies the conditions $(A1)-(A5)$ in Section2 when
the conditons (1) and (2) in the Theorem2.1.
but that this system does not
Using the following coordinate transformation (5.3):$\epsilon=0$$=$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{S}\mathfrak{h}\Gamma$

(5.3)

,

36
the newly revised system on which the above conditions (i) and (ii) establish is
obtained [18]:$dx/dt=-\eta(2x+y)+cz$ ,$dy/dt=c\gamma(2x+y)-cz+1$ ,(5.4)$\epsilon dz/dt=-x-p(Z+Z_{0})+\rho(Z_{0})$

,

where
is one of the two pseudo singular points of the system(5.2)
in case. By the ”local model theory”, the explicit ducks of the system(5.5)
which is the local model of the system(5.4) are obtained as follows.
$P_{0}=(X_{0}, \mathrm{Y}_{0}, Z_{0})$$\epsilon=0$$dX/dt=p\mathrm{Y}+cZ$(5.5)$d\mathrm{Y}/dt=1$,,$\delta dZ/dt=-(Z^{2}+X)$ ,where $p=(-1)^{i_{C}}\gamma\sqrt{(a^{2}-a+1)}(i=1,2)$ . The system (5.5) is ,the exact local
model which effects (5.4) globally. In fact, the first and second equations are firstorder for the variables
and . Furthermore, the third equation does not containthe variable and co–variables are third order at most. So, it may be admissible
for us to use the system (3.3) globally. Restricting the system(5.5) on the surface$-(Z^{2}+X)=0$ , the linearized system(5.6) is obtained:$x,$ $y$$z$$y$$d\mathrm{Y}/dt=Z$

(5.6)

,

$dZ/dt=-(p\mathrm{Y}+cZ)/2$ .

The characteristic equation for (5.6) is(5.7)$2\mu^{2}+c\mu+c\gamma\sqrt{a^{2}-a+1}=0$.

If we choose $p=c\gamma\sqrt{a^{2}-a+1}$ and $c>8\gamma\sqrt{a^{2}-a+1}$ so that the system (5.5)
has a pseudo singular node point, then the explicit ducks are(5.8)$\gamma_{\mu_{i}}(t)=(-\mu_{i}^{2}t^{2}-\delta\mu i, t, \mu_{i}t)(i=1,2)$,

where $\mu_{i}(i=1,2)$ are the solutions of (5.7).

6. AN$\omega$-INCOMPLETE DUCK IN THE FHN EQUATION.

In this section, we will describe how to construct an -incomplete duck in thefirst approximation of the FHN local model. By the elementary calculations, thefollowings are obtained. See [11].
$\omega$Lemma6.1. If tends to zero, the winding number$(\mu_{2}<\mu_{1}<0)$ tends to infinity.
.
$\gamma$$N(\psi_{1})$asssociated with$\mu_{1}$37
Lemma6.2. If $\psi_{i}(i=1,2)$ are not long, then in the first approximation of theFHN local model the lower boundis 1 and of
is infinitesimal.
$\mathrm{o}\mathrm{f}-N(\psi_{1})$$N(\psi_{\mathit{2}})$

As mentioned before, if $K_{2}-K_{1}>3$ , then there exists a duck which is notin
the first approximation of the local model. So, these facts lead to the conclusion:
there exists a duck which is notin the approximation model.
In this framework, we would try to consider the solution as a singular limit forthe parameter of the duck. In the system(5.5), we choose the parameter such as
up the variable again$S^{1}$$S^{1}$$v$$Z$$c=\mathrm{l}\mathrm{S}\mathrm{O}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}/v\mathrm{a}\mathrm{n}\mathrm{d}\gamma=v.\mathrm{L}\mathrm{e}\mathrm{t}v=\beta(z=\beta\tilde{Z},\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}(5.5\beta \mathrm{i}\mathrm{s}_{\mathrm{b}\mathrm{m}}\mathrm{a}\mathrm{n}_{0}\mathrm{y}\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}),\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{b}1_{0}\mathrm{w})\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{S}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{l}]\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{i},\cdot \mathrm{n}\mathrm{g}$$dX/dt=(-1)^{i}\sqrt{a^{2}-a+1}\mathrm{Y}+\tilde{Z}$(6.1)$d\mathrm{Y}/dt=1$,,$(\delta\beta)d\tilde{Z}/dt=-(\beta^{2}\tilde{Z}^{2}+X)$,. Under the above assumptions, the system(6.1) satisfies the condiwheretions $(A1)-(A5)$ in Section2. Therefore, the explicit duck solutions) of thesystem become$\delta\beta\approx 0$$\gamma_{\mu:}\langle\beta$

(6.2)

where

$\gamma_{\mu_{i}(\beta)}(t)=(-\mu_{i}^{2}t^{2}-\delta\mu i, t, \mu_{i}t/\beta)(i=1,2)$$\mu_{1}(\beta),$$\mu_{2}(\beta)(\mu_{1}(\beta)>\mu_{2}(\beta))$

(6.3)

,

are the solutions of the following equation:$2\mu^{2}+\mu/\beta+\sqrt{a^{2}-a+1}=0$.

Whentends to zero, the eigenvalue
tends to zero and the other eigenvalue
infinity.
may tend to the solution whichtends to
Then the first element of
is not. However, the conditions $(A1)-(A5)$ are satisfied. In this state, we canobtain an -incomplete duck:$\gamma=\beta$$\mu_{1}$$\mu_{2}$$\gamma_{\mu_{2}}(\beta)$$S^{1}$$\gamma_{\omega}^{\sim}(t)$$\omega$(6.4)$\gamma_{\omega}^{\sim}(t)\overline{\sim}(-t^{2}/\beta^{2}+\delta/\beta, t, -t/\beta^{2})$since this solution does not$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\infty$,

the definition of the ducks when$\beta$tends to .
$0$

Theorem6.1. In the first approximation system(5.5) of the FHN local model, if
the constant tends to infinity when the constant tends to zero, there exists an-incomplete duck.
$c$$\gamma$$\omega$

Corollary6.2. In the singular limit, when $\gamma=1/\omega=\epsilon$ and $c=1/\epsilon^{1/n}(n\geq 2$ ;integer), satisfies. If we put
again, thenthe FHN equation still preserve the assumptions (4.2) and (4.4). This fact ensuresthat even in the case ofandthe first approximation of
the local model has an -incomplete duck at the singular limit of the parameter.
$b$$b=\epsilon(1/\epsilon^{1/n})=\epsilon 1-1/n$$b=c\epsilon_{1}$$I=I_{\mathrm{O}}+\epsilon^{1-1/n}t$$I=I_{0}+\epsilon_{2}t(\epsilon_{1}\neq\epsilon_{2})$$\omega$

ACKNOWLEDGMENT
We are grateful to Prof.E.Benoit for useful suggestions and to Prof.N.Kikuchi,
Prof.S.Shimomura for many stimulating discussions and useful comments.

38
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