数理解析研究所講究録
1118 巻 1999 年 119-128

119

ON DUCKS IN THE MINIMAL SYSTEM’$?_{\mathrm{c}-}^{z\mathscr{M}}$$\wedge\overline{-}\mathrm{f}\mathrm{f}\overline{Z}\#-$KIYOYUKI TCHIZAWA
Dept. of mathematics, Musashi Institute of Technology
A bstrac . In theeedd local model of the generalized BVP system, the winding
number for a duck solution is not defined welL By choosing an adequate regulartransformation, which contains a new parameter, it can be proved that this number
becomes well defined after that and tends to infinity as a regular limit for the newparameter. In case the new one is fixed sufficiently small, the number becomes large
as the absolute value of a coparameter embedded originally becomes small.
$\mathrm{t}$$1\dot{\mathrm{m}}\mathrm{e}\mathrm{a}X\dot{B}$1. INTRODUCTION.
The modified Bonhoeffer-van der Pol(BVP) equations were proposed by
H.Kawakami etin 1999. Their result of the simulation for this system showsthat there exist winding orbits on some projected phase space. Furthermore, thewinding number increases when some parameter contained originalIy in this systemdecreases.
The VP equations are described as follows:$\mathrm{a}1.[5]$$B$$L_{1}di_{1}/dt=E_{1}-R_{1}i_{1}-vJ^{\cdot}$(1.1)$L_{2}di_{2}/dt=E_{2}-R_{22^{-}}iv$,$Cdv/dt=i_{1}+i_{2}+\rho(v)$ ,where
are the currents through the inductors
and the registors,
respectively. Moreover,are the constant voltages, represents at the nonlinear registor $\rho(\rho(v)=v-v^{3}/3)$ and $C$ is a capacitor with very small capacitance.
Let consider the following generized BVP system:$i_{1},$$i_{2}$$L_{1},$$E_{1},$$L_{2}$$E_{2}$$R_{1},$$R_{2}$$v$$dx/dt=-ax-az.$,$dy/dt=-by-bz$ ,(1.2)$\epsilon dz/dt=x+y+z-z^{3}/3$ ,where$C=\epsilon$

is infinitesimally small. In the equation (1.1), put $i_{1}=x,$ $i_{2}=y,$ $v=z$ ,and then assume that $E_{1}=E_{2}=0,$ $R_{1}=R_{2}=1,1/L_{1}=a,$ $1/L_{2}=b$ .
$\epsilon$

1991 Mathematics Subject.
Key words and phrases. modified Bonhoeffer-van der Pol system,constrained system,
singular perturbation, duck solution,winding number.
$\alpha_{aSS\overline{l}fi}cat|O’ k34\mathrm{A}34,34\mathrm{A}47,34\mathrm{C}35.$$\mathrm{q}1^{r}\mathrm{p}\mathrm{a}\mathrm{e}\mathrm{e}\mathrm{t}$by$A_{\lambda\prime}\beta \mathrm{Z}\mathrm{r}\mathrm{x}$120

As the generalized BVP system satisfies the generic conditions;
Section2), some orbit expresses ajumping state with delay (duck solution, or simply
duck) in the system of the niimal polynomials (or in the minimat system). By
giving a relation between two parameters as $a-b=1$ , this system can be reduced tothe problem of one parameter family with incomplete ducks. See Section3. In thispaper, under the above assumptions, we will describe the following two theorems.
$\mathrm{A}1,$$\ldots,\mathrm{A}5(\mathrm{s}\mathrm{a}\mathrm{e}$Theoreml. If the parameter satisfies-l $<b<0$ , then the minimal system has
a duck near the pseudo singular saddle point.
$b$

In the BVP system, the winding number for a duck is not defined well by itself.
By introducing some regular transformation, which contains a parasitic parameter,
the winding number for the duck turns to be well defined as a regular limit in theregularized BVP system. In another point of view, there exists a regular coordinatetransformation containing certain parameter to realize it in the minimal system.
As a result, the following theorem is obtained:

Theorem2. If the parameter satisfies $-1/2-\sqrt{8}/5/2<b<-1$ , then theminimal systema duck near the pseudo singular node point and the winding
number of the duck tends to infinity as tends to-l.
$b$$\mathrm{h}_{\mathrm{c}\mathrm{t}}\mathrm{s}$$b$2. PRELIMINARIESLet consider a constrained system(2.1):$dx/dt=f(X, y, Z, u)$ ,(2.1)$dy/dt=g(x,y, z,u)$ ,$h(x,y, z,u)=0$ ,where 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 $(A1)-(A5)$ :$(A1)f$ 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=$$\{(x,y, z)\in S\cap T\}$ is a 1-dimensional differentiable manifold.
(A 3) Either the value of or that of is nonzero at any.
Let $(x(t,u),$ $y(t,u),$ $Z(i,u))$ be a solution of (2.1). By differentiating $h(x,y, z,u)$with respect to the time , the following equation holds:$\epsilon$$C^{1}$$g$$h$$C^{2}$

$S$

$f$

$g$

$\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\cdot p\in PL$$t$(2.3) $h_{x}(x, y, Z,u)f(x,y, z, u)+h_{y}(x,y, z,u)g(x,y, z,u)+h_{z}(x,y, z, u)dZ/dt=0$ ,121where $h_{i}(x, y, z, u)=\partial h$ (the following system:$x,y_{\tau}z$, u)/&, $i=x,y,$ . The above system (2.1) becomes$z$$dx/dt=f(x, y, z, u)_{j}$$dy/dt=g(x, y, z, u)$ ,(2.4)$dz/dt=-\{h_{x}(x,y,$ $Z,\mathrm{u}\rangle f(x,y,$ $z,u\rangle+$$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)$dy/dt=-h_{z}(x, y, z, u)g(x, y, z, u)$ ,$dz/dt=h_{x}(x, y, z, u)f(x, y, z, u)+h_{y}(x,y, Z,u)g(x, y, z, u)$.

As the system(2.5) has well posedness at any point of , it has well posedness
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 $(x, y, z)\in S$ , the implicit function theorem holds;$R^{3}$$S\backslash _{\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 by using $y=\varphi(x, z,u)$ or $x=\psi(y, z, u)$ in
the neighborhood of $PL$ . Let $y=\varphi(x, z, u)$ exist, then the projected system, which
restricts the system (2.5) is obtained:$S$$dx \int dt=-h_{z}(x, \varphi(X, Z,u), z,u)f(x, \varphi(x, Z, u), z, u)$

(2.7)

,

$dz/d\mathrm{t}=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,$ $\varphi(X, \varphi(x, Z, u), z,u)$.

AI1 the singuIar points of (2.7) are nondegenerate, theinduced fromthe linearized system of (2.7) 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 singndar points are the singular points of (2.5).
$(A5)$$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{l}\mathrm{X}\sigma$

Definition2.1. Let $p\in PS$ andbe two eigenvalues of the linearized
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$$\mu_{2}(u)$Definition2.2. A
duck, if there exist standard$\mathrm{s}\mathrm{o}\mathrm{I}\mathrm{u}\mathrm{t}\mathrm{i}_{0\mathrm{n}}(x(t, u),y(t,u),$$z(t,u\})$of the system(2.2) is$\mathrm{c}\mathrm{a}\mathrm{J}\mathrm{l}\mathrm{e}\mathrm{d}a$such that
where $*(X)$ denotes the standard part of $X$ ,(2) for $t\in(t_{1},t_{0})$ the segment of the trajectory $(x(t, u),y(t, u),$ $z(t, u))$ is \’infinites\’imally close to the attracting part of the slow curves,(3) for $t\in(t_{0},t_{2})$ , it is infinitesimally close to the repeling part of the slow curves,and(4) the attracting and repelling parts of the trajectory are not infinitesimally small.
If a duck exists as a part of a limmit cycle, it is called a proper duck.
$t_{1}<t_{\mathrm{O}}<t_{2}$$\{1) *(x(t_{0}, u), y(i_{\mathrm{O}}, u), z(t0,u\})\in S,$$\cdot$122

Definition2.3. Let $E$ be a set in . We call a point is a micro–galaxy of $E$
when the distance from to $E$ is less than $exp(-n/\delta)$ , where is some positive( is indnitesimally small).
integer andDefinition2.4. Let is an angle of the polar coordinate after changing the coordinates in the ”local model” such as the orbit passing through the pseudo singularpoint becomes the -axis itself as the below. See $[3],[4]$ . Then, the winding number$N(\psi)$ of a duckis defined as follows:$R^{3}$$\delta-$

$p$

$n$

$p$

$\delta=\epsilon/\alpha^{2}$$\alpha$$\theta$$z$$\psi$(2.8)$N(\psi\rangle=(1/2\pi\rangle I_{\psi}d\theta$

where$\psi$

is contained partially in the$\delta- \mathrm{m}\mathrm{i}_{\mathrm{C}}\mathrm{r}\mathrm{o}_{-}$,

galaxy of$\gamma_{\mu}$.

. In the system(2.1), if the following two conditions at a$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}2.1(\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{t})$pseudo singular saddle or node point;(1) $f(O,u)\simeq h(O,u)\simeq h_{y}(O, u)\simeq h_{z}(O, u)\simeq 0$ ,,, where(2)of theapproximationin the firstare satisfied, the explicit duck solutions
local model can be constructed:$O=(\mathrm{O}, 0, \mathrm{o})\in PS$$g(O,u)\not\simeq \mathrm{O},$ $h_{x}(O, u)\not\simeq \mathrm{O},$ $h_{zz}(o, u)\not\simeq \mathrm{O}$$\gamma_{\mu.(\tau \mathrm{A})}$(2.9)$\gamma_{\mu_{i}}(u\rangle(t)=(-\mu i(u)^{2}\# 2-\delta\mu i(u),t,\mu i(u)t)(i=1,2)$

wehere$\delta$,

is an infinitesimally small constant.

The above Definition2.3 is based on the following fact. If is fixed arbitrariIy
is within $exp(-n/\delta)$ in some neighborhood of theis a duck nearand.
pseudo singular point.
In the system(2.2), under the conditions (1) and (2) in the Theorem2.1, making
the following coordinate transformations (2.10) and (2.11) successively;$\epsilon$$\gamma(t)$$\gamma_{\mu(u)}(t)$$\mathrm{S}\mathrm{e}\mathrm{e}[10]$$=,$(2.10)$(\alpha\simeq 0,\epsilon/\alpha^{2}\simeq \mathrm{o})$(2.11)$=(^{h_{x}(\mathrm{o},u})hzz(0,u)_{\tilde{X}/+}2(h_{y,\tilde{y}}(0,u)h_{zz}(\mathrm{o},u)-hz(y0,u)^{2})\tilde{y}^{2}/41-h(yz0,u)\tilde{y}/2-y/g(\mathrm{o},uh(zz0,u))\tilde{z}/2$

the following local model (2.12) is obtained:$dX/dt=p\mathrm{Y}+qZ+\xi(X, \mathrm{Y}, Z,u)$(2.12)$d\mathrm{Y}/dt=1+\eta(X, \mathrm{Y}, z,u)$,,$\delta dZ/dt=-(Z^{2}+X)+\zeta(X, \mathrm{Y}, Z, u)$

,

where
$p=g(\mathrm{O}, u)hx(\mathrm{o}, u)(f_{y}(\mathrm{o}, u)hzz(0, u)-f_{z}(\mathrm{o}, u)h_{yz}(0, u))/2$$+g(\mathrm{O},u)^{2}(h_{yy}(\mathrm{o}, u)h(zz0, u)-h_{y}(z\mathrm{o}, u)^{2})/2$(2.13)$q=-h_{x}(\mathrm{o},u)f_{z}(\mathrm{o},u)$$\delta=\epsilon/\alpha^{2}$

.

,

,

’

123

Here
and
are infinitesimal when $X,$ and $Z$
are limited. Note that the solutions (2.9) are in the first approximation system of
the system(2.12).
By applying the following transformations ofthe coordinates as mentioned above,in Definition2.4, successively;$\xi(X, \mathrm{Y}, Z,u),\eta(X, \mathrm{Y}, z,u)$$\zeta(X, \mathrm{Y}, z, u)$$u=X+z^{2}+\delta\mu$(2.14)$v=\mathrm{Y}-Z/\mu$$\mathrm{Y}$,,$z=Z$ ,,
$v=rsin\theta$ ,$u=rcos\theta$(2.15)

the Hermite equation (2.16) is obtained. This equation associated with$1,2)$ is the following:(2.16)$\delta\ddot{z}-\tau\dot{z}+K_{i}z=0,\dot{z}=dz/d\tau,$ $t=\tau/\alpha,$$\gamma_{\mu_{i}(u)}(i=$$(i=1,2)$ ,whereis a positive integer and $K_{1}=1+\mu_{2}(u)/\mu 1(u),$ $K_{2}=1+\mu_{1}(u)/\mu_{2}(u)$ .
See [3].
It is said that a duckhas a jump if the shadow of it contains a verticalsegment and thatis long if it is in an infinitesimally small neighborhood at thepseudo singular point. It can be proved that if is not long, the standard part of
the winding number
associated withis an integer. If the pseudo singularpoint 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.
. If the duck . which is not long has 2 jumps,$N(\psi_{1})\approx-[K1/2]$ , and if the duckhas 2 jumps,.
$K_{i}$$\psi(t)$$\psi(t)$$\psi$$N(\psi_{i})$$\mu_{i}$$K_{i}$$N(\psi_{i})$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{n}2.2(\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{t})$$\psi_{1_{}}$$N(\psi_{2})\approx 0$$\psi_{2}$3. INCOMPLETE DUCKS.
Definition3.1. In the system(2.12), 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 $\omega-in\omega mptebe$ duck.
$u$

$u$

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

at a,

$b$

(3.1)

$\frac{\psi(_{X},u)-\psi(y,u)}{x-y}\approx b$

for any $x,y(x\approx a,y\approx a)$ .
A duck is called an duck if it isof the pseudo singular point.
$S^{1}$$S^{1}$,

in some neighborhood

124. In the first approximation of the system $(2.12\}$ ,$(>3)$ but no an integer, then all theifpositiveisducks are exponentially close to one of the two explicit ducks and there exists non
du&s.
$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{n}3.1(\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{t})$$\mu_{1}(u)/\mu_{2}(u)$$S^{1}$$S^{1}$In the system(2.12), we assume that(3.2)

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

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
only the coefficient can take an unlimited number:$u$,is negligible, that is,

$q$

(3.3)

$q=c_{1}u+o(1),$$C_{1}\not\simeq 0$.

Then, blowing up only the variable $Z$ again;(3.4)$Z=(1/u)\tilde{Z}$,the first approximation of the system(2.12) becomes the following:$dX/dt=p\mathrm{Y}+c_{1}\tilde{Z}$(3.5)$d\mathrm{Y}/dt=1$,,$(\delta/u)d\tilde{Z}/dt=-(\tilde{Z}^{2}/u^{2}+X)$

where is limited (does not contain ) andsystem$(3,5)$ are$u$$c_{1}$(3.6)$\delta/u\simeq \mathrm{O}$,. The explicit solutions in the$\gamma_{\mu_{i}}(u)(t)=(-\mu i(u)2t-2\delta\mu i(u),t,u\mu i(u)t)(i=1,2)$,$\mu_{2(u)}(\mu_{1}(u)>\mu_{2}(u))$ are the solutions of the characteristic equation
whereof the system(3.5) in case.
The above system satifies the conditions $(A1)-(A5)$ and the solutions(3.6)
the condition (1) and satisfies the condition (2) whenin Definition3.1.
In fact, if, then the existence of such a duck is ensured. We choose$\epsilon=1/n(n=2,3, \ldots)(u=1/n^{1/3})$ , then $1/n^{1/3}\gg exp(-n/\delta)$ for any $n(n\geq 2)$ .
In the system for each any fixed , let $J=[AB]$ be a connected segment in,
where the solution which starts at
belongs to the family of the duck(or). It can be proved that if any solution starting at $p\in J$ is not long, thenit has the same winding number. Rom Theorem3.1, a duck passing through thepseudo singular node point belongs to one of two families of the above ducks. On
the other hand, there exists a segment $[CD]\subset J$ such that any solution startingat $p\in[CD]$ is not long and the solutions passing through $C$ or $D$ are ducks. This
fact ensures the existence of a nonduck. Note thatis positive but
no an integer. If it is a positive integer , it indicates the fact that the slow vector
field has two
trajectories but only one of them is. Then, it is not possible
to have an asymptotic expansion in powers of with the coefficients analytic in$\mu_{1}(u),$$\delta/u\simeq \mathrm{O}$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}6\Gamma$$uarrow\omega$$q=\epsilon^{-1/3}$$R^{3}$$n$$A(\mathrm{o}\mathrm{r}B)$$\gamma_{\mu_{1}}$$\gamma_{\mu_{2}}$$S^{1}$$\mu_{1}(u)/\mu_{2}(u)$$k$$C^{1}$$C^{k}$$\epsilon$

125

$t$

. Furthermore, one of the solutions(3.6) may tend tangent to theas$\mu_{2}(u)arrow-\omega/2$$uarrow\omega$

following(3.7)-axis, since$X$. In fact, for the first component of the solutions, the,$\frac{-(\omega/2)^{2}(2/\omega)^{2}+(\omega/2)^{2}(1/\omega)^{2}}{2/\omega-1/\omega}arrow-3\omega/4$, the windin number
establishes. In thisassociated withtends
to infinity and the otherassociated withtends to infinitesimal.
$uarrow\omega=\epsilon^{-1/3}$, the eigen space of the linear part of the slow vector field foris(for). The ducks are almost
tangent to thespaces and therefore the -hmit of the duck with respect tothe parameter ( -incomplete duck) is not .
Let $v=1/u$ , thenholds and then the following conditions are$f(x,y,z,u)=\tilde{f}(x,y_{J}.z,v)\in
C^{3},$ $g(x,y, z.u)=\tilde{g}(x,y,z,v)\in C^{1}$ 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.
$8\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}$$N(\psi_{2})$$g$$N(\psi_{1})$$\mu_{2}\simeq-\epsilon^{-1/3}$$z\simeq\epsilon^{1/3}y$$z\simeq-\epsilon^{1/3}y$$\mathrm{e}\mathrm{i}_{\mathrm{c})}\sigma \mathrm{e}\mathrm{n}$$u$$\mu_{2}$$\mathrm{V}^{\gamma}\mathrm{h}\mathrm{e}\mathrm{n}$$\mu_{1}$$\mu_{1}\simeq-\epsilon^{1/3}$$\omega$$S^{1}$$\omega$$\partial_{8\mathrm{A}}=-v^{2}\partial_{v}$$v$Theorem3.2. In the first approximation of the system(2.12), if
ispositive but no integer under the condition (3.2) and ifwheneither the condition (1) or (2) ;
(1)
and
,
(2)
, and
,
where all the coefficients of higher order (more than 2) for is negligible are satisfied, then this system has an -incomplete duck.
$\mu_{1}(u)/\mu_{2}(u)$$\tilde{h}_{xv}(0,v)\tilde{f}\mathrm{z}v(0,v)=0$$\tilde{f}_{z}(0,v)=0,$$\tilde{h}_{x}(\mathrm{o}, v)\tilde{f}zvv(0,v)=0$$\tilde{h}_{x}(\mathrm{o},v)=0$$\tilde{h}_{xvv}(\mathrm{o},v)\tilde{f}_{z}(0,v)=0$$u$$\omega$

Corollary3.3. In the system(2.12), if the coefficient satisfies $q=c_{1}u+O(1)$ , thatis, $q=c_{1}u+c_{2}$ where
and $p>0$ or $0>p\geq-1/32$ , then there exists a
finite value
which makes the winding number infinite when tends tothecorresponding duck is called incomplete , simply.
$q$$c_{1},$ $c_{2}\not\simeq 0$$u$$u_{0}$$u_{0;}$

Remark. In this situation, the singular perturbation problem is equivalent to thefollowing system of two parameters family with and :$\epsilon_{1}$$\epsilon_{2}$$\epsilon_{1}dXfdt=\epsilon_{1}p\mathrm{Y}+qZ+\xi(X, \mathrm{Y}, z,\epsilon_{1},\epsilon_{2})$(3.8)$d\mathrm{Y}/dt=1+\eta(X, \mathrm{Y}, Z, \epsilon_{1,2}\epsilon)$,,$\epsilon_{2}dZ/dt=-(Z^{2}+X)+\zeta(X,\mathrm{Y}, Z,\epsilon_{1},\epsilon 2)$

where$\epsilon_{1}$and$\epsilon_{2}$are infinitesimally small.
4. THE REGULARIZED BVP SYSTEM

Let consider the following equations (regularized BVP):,$L_{2}di_{2}/dt=E_{\mathrm{z}-}R_{2}i2-R(i_{1}+i_{2})-v$ ,$L_{1}di_{1}/dt=E_{1}-R_{11^{-}}iR(i_{1}+i_{2})-v$(4.1)$Cdv/dt=i_{1}+i_{2}+v-v^{3}/3$ ,

,

126

where $R$ is a linear reistor and $C$ has a very small capacitance. The equation (4.1)
is composed of a parasitic resistor (as a parameter) to the BVP system. When
tends to zero, the system (4.1) tends to quite the same one (1.1) as a regular
limit. In the linearized local model of (4.1), the winding numbers for the ducks
are well defined since the matrix associated with the model at the pseudo singular. The parameter
point are nondegenerate; it is called regularized
duck fromincomplete
holds the relation (3.3) as $u=R$. Thus, this system has antends to zero again, only one of the eigenvaJues tends toCorollary3.3. Whenzero, that is, the winding number for the duck tends to infinity.
The linearized local model of the system(1.2) could not make one of the winding
numbers defined well because the first equation does not contain the variable .
This fact causes that the value of in the equation (2.13) takes zero, therefore oneof the eigenvalues takes zero. There exists a problem how to avoid this trouble.
The mathematical model which it may concern is obtained as follows.
$R$

$R$

$R$

$BVP_{\mathit{8}}ystem$

$R$

$y$

$p$

Lemma4.1. In the system(1.2), there exists a regular coordina
such that the winding numbers become well defined.

$te$transformation(proof)$\mathrm{C}\mathrm{h}\mathrm{o}\mathrm{o}8\mathrm{i}\mathrm{n}\mathrm{g}$(4.2)

the following transformation:$=$,

where $u(\neq 1)$ is a parasitic parameter, the system(1.2) becomes$dX/dt=-aX+u(a-b)\mathrm{Y}-(a+bu)Z$ ,(4.3)$d\mathrm{Y}/dt=-b\mathrm{Y}-bZ$,$\epsilon dZ/dt=X+(1-u)\mathrm{Y}+Z-Z^{3}/3$.

Considering the coxstrained system of the equation(4.3) and the relation $a-b=1$ ,the following(4.4)$dX/dt=(-(b+1)X+u\mathrm{Y}-(bu+b+1)Z)(1-z^{2})$ ,$dZ/dt=(b+1)X-(bu+u-b)\mathrm{Y}+(2b+1\rangle Z$ ,establishes, where $b,u$ are parameters. Substituting the following(4.5)$\mathrm{Y}=(X+Z-Z^{3}/3)/(u-1)$ ,for the above, the equation(4.4) becomes$dX/dt=(-(b+1-u/(u-1))X-(bu+b+1-u/(u-1))z$(4.6)$-uZ^{3}/3(u-1))(1-Z^{2})$ ,$dZ/dt=(1-u/(u-1))\mathrm{x}+(b+1-u/(u-1))z$$+(b+u/(u-1))z^{3}/3$ .

127

The pseudo singular points are$PS_{-1}=(X0-, \mathrm{Y}_{0}-, Z0-)$,$X_{0-}=(-(4b+1)u+4b+3)/3$ ,(4.7)$\mathrm{Y}_{\mathrm{O}-}=(-(4b+1)u+4b+1)/3(u-1)$,$z_{0-=}-1$ ,and$PS_{1}=(x0+, \mathrm{Y}0+, z_{0+})$,$X_{0+}=((4b+1)u-4b-3)/3$ ,(4.8)$Y_{0+}=((4b+1)u-4b-\iota)/3(u-\iota)$ ,$Z_{0+}=1$.

The values of $p,q$ in the equation $(2.13)$ at these points are$p_{-}=uz_{0_{-}}(b\mathrm{Y}_{0-}-b)$,$q_{-}=b(u+1)+1$ ,(4.9)$p_{+}=u(-b\mathrm{Y}_{0}++b)$$q+=q_{-}$,.

Since the value of is not zero, it becomes clear that the eigenvalu es at thesepoints are nondegenerate, so the winding number is well defined. This completes$p$

the proof.
Lemma4.2. If a duck of the system $(4.3)$ is proper, that is, there exists a limit
cycle, which has the duck as a part of the solu tion, the right hand of thishas minimal degree of the polynonials (minimal system) for the proper duck.

$s\mathrm{y}\mathrm{s}t\mathrm{e}\mathrm{m}$(proof)If the degree of the polynomials are smaller than 3, there is not ajumping orbit toreturn to an initial point. Therefore, there is not a limit cycle, which contains a
duck. This completes the proof.
In this paper, the condition for the existence of the proper duck does not betreated.

5.

$\mathrm{T}\mathrm{H}\mathrm{E}\mathrm{p}\mathrm{R}\mathrm{O}\mathrm{o}\mathrm{F}\mathrm{S}$OF THEOREMI AND THEOREM2

In the generalized BVP system(1.2), there are three pseudo singular points$(z=-1,0,1);PS_{-1}=(1+4b/3, -1/3-4b/3, -1),$ $PS_{0}=(0,0,\mathrm{o}),$ $PS_{1}=(1+$$4b/3,$ $-5/3-4b/3,1)$ , since we assumed $a-b=1$ . In fact, the constrained system
for the system(4.3) becomes the following; corresponding to the equation(2.7):$PS_{z}$$dx/dt=-(1+b)(x+z)(1-Z^{2})$ ,(5.1)$dz/dt=x+(1+b)z+bz^{3}/3$ ,is a parameter. However, the point
does notthe genericcondions, especialy the
in Section2. So, let consider other two points$PS_{-1}$ and.

where only

$PS_{0}$$b$$\infty \mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}(A\mathrm{s})$$PS_{1}$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\Psi$128

In both cases, the characteristic equations associated with the linearized systemof the equation(5.1) are quite the same as follows:(5.2)$\lambda^{2}-(1+2b)\lambda+8b(1+b)/3=0$.

If satisfies-l $<b<0,$ $PS_{-1}$ andare the pseudo singular saddle points, thatis, there exist the ducks. This completes the proof of Theoreml.
In the regularized system(4.3), if the parameter$-1$ , the pseudo singular points $PS_{-1},$
are the node type when coparameter$u=R$ tends to zero. The characteristic equation of the linearized local model under
the condition (2.13) is$b$$PS_{1}$$b\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{S}\mathrm{f}\mathrm{i}\mathrm{a}\mathrm{e}-1/2-\sqrt{8/5}/2<b<$$PS_{1}$(5.3)$2\lambda^{2}-q\lambda+p=0$,

where $p,q$ are in the equation(4.9). Ron Lemma4.1, the winding number is welldefined. Furthermore, the constantthe equation(4.9) ensures therelation(3.3). Therefore, this regularized system has an incomplete duck. Then,
one of the eigenrues tends to zero as the parameter tends to zero. At that time,
the corresponding index in the equation(2.16) tends to infinity, so the winding
number associated with the duck tends to infinity by Theorem2.2. Let the parameter $u(\neq 0)$ is fixed sufficiently, then the winding number tends to infinity as
the coparameter tends to-l. In fact, it should be available if holds $u=O(\epsilon^{1/})2$
whenever satisfies $b+1=\epsilon$ . This completes the proof of Theorem2.
$p\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{S}\Phi \mathrm{i}\mathrm{n}\mathrm{g}$$u$$\mathrm{K}$$\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{I}\mathrm{l}$

$b$

$u$

$b$

Remark. When tends to-l, the absolute value of becomes small as possible
in the above restriction. This fact coinsides with the reults of the simulation [5]:whenbecomes smaller, the winding number becomes larger.
$b$

$b$

$1/L_{1}$ACKNOWLEDGMENT
I am grateful to Professors E.Benoit, A.FYuchard and G.Wallet for stimulating
meetings in which many discussions and useful suggestions were done in La Rochelle.

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