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Uniqueness and stability of a limit cycle for a third order dynamical system arising in neuron modelling
0362.546X’81/0101~013
Nonlmear Analysis, Theory, Methods & Applications, Vol. 5, No. 1, pp. 13-19 0 Pergamon Press Ltd. 1981. Pnnted ,n Great Bntam
E02.00,0
UNIQUENESS AND STABILITY OF A LIMIT CYCLE FOR A THIRD ORDER DYNAMICAL SYSTEM ARISING IN NEURON MODELLING BINGXI LIP Department
of Mathematics,
University
of California
at Los Angeles,
(Received 21 January
Los Angeles,
California
90024, U.S.A.
1980)
Key words: Third order dynamical system, nonlinear Volterra integral of limit cycle, neuron model, Laplace transform, omega limit set.
equation,
uniqueness
and stability
1. INTRODUCTION
THE FOLLOWING third order dynamical system (I), proposed by Stein, Leung, Mangeron, and Oguztoreli, is an improved neural model for studying neural network [l] : a
X=l+exp{-j-h(q-p)z}-aX’ d = x - PY,
(1)
i = y - qz,
where x(t), 0 < x(t) Q 1, is the normalized axonal impulse frequency at time t, f = f(t) is the input of the nerve cell. The constants a > 0,O < p < q and b are characteristic for the individual neuron or special type of neurons. For details refer to [I]. U. an der Heiden proved (Theorem 1 in [2]): Let f be constant. Then system (1) has at least one non-constant periodic solution if b < -(P + q)(pq + a(~ + P + q))/(q - @x*(1 - x*)>
(2)
where x* is the unique positive root of 1 X* = 1 + exp{ -f
-
(3)
bx*(q - p)/(pq)}’
But the question remains whether the periodic solution is unique and asymptotically stable. In what follows we shall prove by means of nonlinear Volterra integral equations that under conditions:f = const., (2), (3) and p # a # q, (1) does possess a unique and stable limit cycle. 2. LEMMAS
AND
PRELIMINARIES
Consider the n-dimensional (n > 3) dynamical system i-, = M(xn) - cIIX1’ ij =x. t
Permanent
address:
Department
J-1
of Mathematics,
- “,Xj’ Jinan
University, l?
(4)
j = 2,. . . , n, Canton,
the People’s Republic
of China.
B. LI
14
where the C’“- i) map M : R + R satisfies a global Lipschitz condition: IM(r,) - M(r,)l
d
vr,,r2 ER,
Llr, - r21,
andtheconstantsccj>0,j=1,2 ,..., n;x=(x,,x, n}. Moreover, suppose the equation
,...,
xn)ER:={(xl
,...,
xn)ER”lxj>O,
j = 1,2,...,
M(r) - Snr = 0,
Sn = fi
clj,
j=l
possesses a unique positive root x *. In this case, (4) has one and only one singular point (x:3 $9.. ., x,*) E Int(R:), where XT =
Setting yj = xj -
xj*,j
a,x; = a,a,x; = . . . =
x;, M(xjT)
- S,x,* = 0.
= 1,2,. . . , IZ,we can transform (4) into = M(yn + x,*, - S nx*n - a,~,,
jl
~j = Yj_l - ‘jYj,
2,
(5)
where cI. > 0, (y,, . . . , y,) E & = {(y,, . . . , y,) E R”ly, 3 -XT, Vj}. (0,. . . , 0) is the unique singular point o/(5) in Int(Q). It can be seen that the following Lemma 1 holds: LEMMA1. System (5) is equivalent to the nth order nonlinear differential equation
h (9 + cri)y = M(y + x,*) - S,,x;, j=
(6)
1
or (9” + S,W’
+ S,W-*
+ . . . + S,_i9
+ SJy = M(y + x,*) - S,x,*,
where 9 = d/dt; y = yn; S,, . . . , Sn are the elementary symmetric functions of cl,.,j = 1,
s, = f:
aj,
j=l
s, = i
qxj,.
. .,s,
= fi
(6’) , n, i.e.
aj
j=l
i,j=l i#j
LEMMA2. If a, # clj, i # j, then the solution y(t) of (6) with initial conditions: y(O),y”‘(O), . . . , y”‘- l’(O),satisfies the nonlinear Volterra integral equation
y(t) +
x,*= *A(t s 0
- T)M[~(T) + x,*-J dr +
h(t),
t>O
(7)
Uniqueness
and stability
of a limit cycle for a third order dynamical
system arising
in neuron
modelling
15
where
“ilCk(-~j)“-k-l
Snx:] {exp(-ajt)>{pj(-“j)>-l,
-
J
k=O
(8)
{exp( -NY)} {Pi(--~j)}-~,
.4(s) = i j=l
Pj(0) = (a + cxj)-l fi (0 + cl,), 1=1 Ipclj)
= (a1 - “j)...(cfj_l
- cxj)(mj+l - ctj)...(ccn - aj),
and C,, k = 0,...,n - 1 are the linear combinations
co= Y(O), c, = Y”‘(O)
+
c2 = y’yo)
+ S,y”‘(O)
...
...
ck = .. Cn-l
S,Y(O), ...
+ S,y(O),
... ...
(9)
+ s1yCk-“(0) + . . . + s,Y(o)T
y'k'(o)
... =
...
of y(O),y”‘(O), . . . , y’“-“(O):
...
y(“-l’(o)
...
...
...
+ sly’“-2’(o)
+ . . . + Lyly(0).
Proof: Applying the Laplace Transform [3] to (6), we obtain
where Y(a) = T{y(t)}. From this
On taking the inverse Laplace Transform 2-l y(t) +
x,*=
fr
A(t - r)M[y(r)
J 0
[3], we get +
x;] dr
+ h(t),
t 2
0.
16
B.LI
LEMMA 3.The mapping h :R + -+ R is bounded and uniformly continuous, moreover the following relations hold : t u(t) =
A(t - z)M(u(z)) dr + h(t), s0
u”‘(t)
’A”‘(t
=
- r)M(u(r))d~ +
k+(t))
+
P’(t),
s0
(10)
’kk’(t
dk)(t) =
- r)M(u(z)) dz + i
s0 k=l
i
I=1
j=l
(+
M(k-r)(u(r)) + h’k’(r),
‘jCAaj)
, . . . , n - 1; where u(t) = y(t) + x,*
Proof. Since h: R+ -+ R is continuous and lim h(t) = 0, it is uniformly continuous, and the *+OO boundedness of h is clear from its definition. The relations (10) can be obtained by differentiation of (7) and setting y(t) + x,* = u(t). From (5) and (lo), it follows that y,(t) Y_
=
u(t) -
xn*,
,@I= YJt).+ any,(t),
... ... ... ... y,(t) = y;(t) + a,y,(t); or y,(t) y,_,(t)
=
u(t) -
xn*,
= u”‘(t) f anu(t) - anxn*,
Yn_ ,(t) = uC2’(t)+ (cl”_ * + a,W”(t) ....
....
....
y,(t) = u’“-“(t)
- a,_la,xn*,
(11)
....
+ (S1 - al)ff’“-2)(t)
+ (S, - a,@,
- a1))u(“-3)(t)
+ (S, - a,@,
- a,(S,
+ Snu(t) - S”X,*,
where u(t) satisfies equations (10).
+ an-lan4t)
- al)))u(“-4)(t) + . . .
17
Uniqueness and stability of a limit cycle for a third order dynamical system arising in neuron modelling 3. UNIQUENESS
AND
STABILITY
In order to establish the uniqueness and stability theorem of Miller [4,5]. Consider the nonlinear Volterra integral equation u(t) = -
OF LIMIT
of the limit cycle for system (5), we need a on a Hilbert
space H: t > 0,
’ F(t - T)&(T)) dr + h(t), s0
where F(t) is a family of bounded, linear self-adjoint operators mapping, g :H + H. If lim h(t) = ho, then we call f+m % u,(t) = F(+&(t - 5)) dr + ho, s0 the limit equation
of(E). The following
theorem
CYCLE
(E)
on H, g is a bounded,
t
>
0,
nonlinear
@,I
is due to Miller.
THEOREM (MILLER). Suppose H = R”, F E L, (0, co); h: R+ + R” is bounded and uniformly continuous, g is continuous, and equation (E) possesses a bounded solution u: R+ + R”. Then there exists a solution a0 of equation (E,) and a sequence of real numbers {t,}, lim t, = co, N-CC such that the relation lim u(t + tN) = u,(t) v- 7 is valid. Now we are in a position
to state and prove the uniqueness
and stability
result:
THEOREM 1. If system (5) possesses
positive periodic as
a nonconstant periodic orbit y c IT:where T c Q is a bounded invariant set of (5), free from singular points, and cli # mj, i # j, then y is the unique orbit (nonconstant) of (5) in the whole phase space g. The equation of y can be expressed (Qt),
tER+,
. . . 9q(O),
Qt),
where
q(t) = u,(t) -
xn*,
cQ1(t)
= ub’)(t) + ctnuo(t) - c(nx;,
q-&)
= q(t) + (cl”_
o,(t)
l
= q’(t)
+
Co&w + yp,uo(t)
+ (S1 - cc,)ub”-2)(t) +
+ (s, - cgs, - cqs, and u,(t) is the nonconstant
solution
u,(t) =
- U”_,cg,
(12)
(S, - Lqs, - q)ub”-3)(t)
tq))Ub”-4)(t) + . . . + Snuo(t)-
s,x;,
of
sm
A(z)M(u,(t
0
- T))
dr,
teR+.
(13)
18
B. LI
Moreover,
if @(r; t) c T is any orbit of (5), then its omega limit set Q, = y.
trajectory of (5), with the initial conditions Proof: If (y,(t), Y (t), . . . 2Y,(O) is an arbitrary Y(O), Y”‘(C%. . . 3Ycn-‘)(0), then the y .(t)‘s are given by (11). Hence the behavior of this trajectory depends on that of u(t), u(‘)(t), . . . , zhel) (t). If this trajectory lies in T, then it is bounded, so are the y,(t)‘s, in particular u(t), t E Rf, is bounded. Applying the Theorem of Miller, we have a real sequence (tN>, lim t, = co, such that N+a,
lim y(t -t tN) = u,(t) - x,*. N-tm
It can be proved that this u,(t) is not constant, Moreover, it can be shown that lim uCk)(t+ t N )
=
and that (13) has a unique
dk’(t) 0
k = l,...,n
’
-
nonconstant
solution.
1.
N-tCC
In other words, there exists a sequence
{tN}, lim t, = GO,such that
of real numbers
N+‘X
lim N+CC
(y,(t + t,), y,(t + t,), . . . , y,(t + t,)) = (al(t), Qt),
Next, let us consider
the nonconstant I?,;
where r. is a point on y. According {tN}, lim t, = co, such that
periodic
. . . , o,(t)).
orbit y c T. Denote
it by
t) = (Y;(t), Y;(t), . . . 9Y,“(N> to the above argument
there exists a certain
real sequence
N+CG
lim T(r,;
t +
tN) =
(ml(t), o,(t), . . ., o,(t)).
N-+02
Noticing
lim
y,"(t +
tN) =
u,(t) -
xX, y”(t) is periodic
(with period
8 > 0, say) and applying
the TheorEIhmof Miller, we get u,(t + fi) = lim
y,"(t+
fl +
yf(t +
tN)
tN) +
x,*
N-CC
= lim
+
x,*
N-+‘W
=
u,@),
VteR+.
That is to say, u,(t) is periodic with period 8 > 0, and hence o,(t), j = 1,2, . . . , IZare also periodic with the same period 8. The relation .lir”, T(r,; t + tN) = (ml(t), o,(t), . . . , o,(t)) implies the closed curve {(ml(t), o,(t), . . . , o,(t))) t E R+} c Rio, and since R10 is a connected ((W,(Q, QG
. . . >con(t))lt E R+} = Silo = I-(r,; I),
set, we have
I = (+co,co).
To prove the uniqueness of the limit cycle for system (5), let us suppose the contrary, and let f(r,; t) be a nonconstant closed orbit of (5) other than I+-,; t), hence T‘(r,; I) n T(r,; I) = 8. But from the above argument we have I?,,;
I) = al,, = {(Qt),
which is a contradiction,
w,(t), . . . , q$))~~ E R+} = Qrl = fk,
and the proof is complete.
; 0,
Uniqueness
and stability
of a limit cycle for a third order dynamical
system arising in neuron
modelling
19
An application of Theorem 1 to the system (1) yields the following result. THEOREM
2.
Iff = constant, 0 < a, 0 < p < b
<
_
q, p #
a #
q,
and
(P+ 4NP4 + a@ + P + 4)) (q -
@x*(1 - X*)
’
where x* is the unique positive root of 1 ‘* = 1 + exp{ -f - bx*(q - p)/(pq)}’ then system (1) possesses a unique limit cycle, which is stable. Moreover, the equation of this limit cycle can be expressed as (u’,O’(t)+ (p + q)ub’)(t) + pqu,(t)
- pqx*
solution of a, A(++& u,(t) = s0
+ x*, u;‘(t)
+ quo(t) -
qx* + x*, u,(t)),
(14)
were u,(t) is the nonconstant
- 3) d7,
teR+,
and A(s) = M(r) =
exp ( - as)
exp ( - Ps)
exp (- 44
CP - a)(q - 4 + (a - PN4 - P) + (a - 4) CP- 4)’
u/{1 +
exp [I-f - Wq - pP1).
Acknowledgment-The author would like to express his gratitude to the Department of Mathematics, UCLA, for offering him the opportunity of visiting UCLA as a visiting scholar. He would also like to thank Professor Earl Coddington for his helpful suggestions.
REFERENCES 1. SPIN R. B., LEUNG K. V., MANGERON D. & O&JZTBRELI M. N., Improved neuronal models for studying neural networks, Kybernetik 15, l-9 (1974). 2. AN DER HEIDEN U., Existence of periodic solutions of a nerve equation, Biol. Cybern. 21, 37-39 (1976). New York (1973). 3. OBERHETTINGERF. & BADII LARRY, Tables of Luplace Transforms. Springer-Verlag, 4. MILLER R. K., Nonlinear Volterra Integral Equations. Benjamin, New York (1971). BUN. Am. math. Sot. M(4), 5. MACCAMY R. C. & SMITH R. L., Limits of solutions of Volterra integral equations, 739-742 (1975).
Nonlmear Analysis, Theory, Methods & Applications, Vol. 5, No. 1, pp. 13-19 0 Pergamon Press Ltd. 1981. Pnnted ,n Great Bntam
E02.00,0
UNIQUENESS AND STABILITY OF A LIMIT CYCLE FOR A THIRD ORDER DYNAMICAL SYSTEM ARISING IN NEURON MODELLING BINGXI LIP Department
of Mathematics,
University
of California
at Los Angeles,
(Received 21 January
Los Angeles,
California
90024, U.S.A.
1980)
Key words: Third order dynamical system, nonlinear Volterra integral of limit cycle, neuron model, Laplace transform, omega limit set.
equation,
uniqueness
and stability
1. INTRODUCTION
THE FOLLOWING third order dynamical system (I), proposed by Stein, Leung, Mangeron, and Oguztoreli, is an improved neural model for studying neural network [l] : a
X=l+exp{-j-h(q-p)z}-aX’ d = x - PY,
(1)
i = y - qz,
where x(t), 0 < x(t) Q 1, is the normalized axonal impulse frequency at time t, f = f(t) is the input of the nerve cell. The constants a > 0,O < p < q and b are characteristic for the individual neuron or special type of neurons. For details refer to [I]. U. an der Heiden proved (Theorem 1 in [2]): Let f be constant. Then system (1) has at least one non-constant periodic solution if b < -(P + q)(pq + a(~ + P + q))/(q - @x*(1 - x*)>
(2)
where x* is the unique positive root of 1 X* = 1 + exp{ -f
-
(3)
bx*(q - p)/(pq)}’
But the question remains whether the periodic solution is unique and asymptotically stable. In what follows we shall prove by means of nonlinear Volterra integral equations that under conditions:f = const., (2), (3) and p # a # q, (1) does possess a unique and stable limit cycle. 2. LEMMAS
AND
PRELIMINARIES
Consider the n-dimensional (n > 3) dynamical system i-, = M(xn) - cIIX1’ ij =x. t
Permanent
address:
Department
J-1
of Mathematics,
- “,Xj’ Jinan
University, l?
(4)
j = 2,. . . , n, Canton,
the People’s Republic
of China.
B. LI
14
where the C’“- i) map M : R + R satisfies a global Lipschitz condition: IM(r,) - M(r,)l
d
vr,,r2 ER,
Llr, - r21,
andtheconstantsccj>0,j=1,2 ,..., n;x=(x,,x, n}. Moreover, suppose the equation
,...,
xn)ER:={(xl
,...,
xn)ER”lxj>O,
j = 1,2,...,
M(r) - Snr = 0,
Sn = fi
clj,
j=l
possesses a unique positive root x *. In this case, (4) has one and only one singular point (x:3 $9.. ., x,*) E Int(R:), where XT =
Setting yj = xj -
xj*,j
a,x; = a,a,x; = . . . =
x;, M(xjT)
- S,x,* = 0.
= 1,2,. . . , IZ,we can transform (4) into = M(yn + x,*, - S nx*n - a,~,,
jl
~j = Yj_l - ‘jYj,
2,
(5)
where cI. > 0, (y,, . . . , y,) E & = {(y,, . . . , y,) E R”ly, 3 -XT, Vj}. (0,. . . , 0) is the unique singular point o/(5) in Int(Q). It can be seen that the following Lemma 1 holds: LEMMA1. System (5) is equivalent to the nth order nonlinear differential equation
h (9 + cri)y = M(y + x,*) - S,,x;, j=
(6)
1
or (9” + S,W’
+ S,W-*
+ . . . + S,_i9
+ SJy = M(y + x,*) - S,x,*,
where 9 = d/dt; y = yn; S,, . . . , Sn are the elementary symmetric functions of cl,.,j = 1,
s, = f:
aj,
j=l
s, = i
qxj,.
. .,s,
= fi
(6’) , n, i.e.
aj
j=l
i,j=l i#j
LEMMA2. If a, # clj, i # j, then the solution y(t) of (6) with initial conditions: y(O),y”‘(O), . . . , y”‘- l’(O),satisfies the nonlinear Volterra integral equation
y(t) +
x,*= *A(t s 0
- T)M[~(T) + x,*-J dr +
h(t),
t>O
(7)
Uniqueness
and stability
of a limit cycle for a third order dynamical
system arising
in neuron
modelling
15
where
“ilCk(-~j)“-k-l
Snx:] {exp(-ajt)>{pj(-“j)>-l,
-
J
k=O
(8)
{exp( -NY)} {Pi(--~j)}-~,
.4(s) = i j=l
Pj(0) = (a + cxj)-l fi (0 + cl,), 1=1 Ipclj)
= (a1 - “j)...(cfj_l
- cxj)(mj+l - ctj)...(ccn - aj),
and C,, k = 0,...,n - 1 are the linear combinations
co= Y(O), c, = Y”‘(O)
+
c2 = y’yo)
+ S,y”‘(O)
...
...
ck = .. Cn-l
S,Y(O), ...
+ S,y(O),
... ...
(9)
+ s1yCk-“(0) + . . . + s,Y(o)T
y'k'(o)
... =
...
of y(O),y”‘(O), . . . , y’“-“(O):
...
y(“-l’(o)
...
...
...
+ sly’“-2’(o)
+ . . . + Lyly(0).
Proof: Applying the Laplace Transform [3] to (6), we obtain
where Y(a) = T{y(t)}. From this
On taking the inverse Laplace Transform 2-l y(t) +
x,*=
fr
A(t - r)M[y(r)
J 0
[3], we get +
x;] dr
+ h(t),
t 2
0.
16
B.LI
LEMMA 3.The mapping h :R + -+ R is bounded and uniformly continuous, moreover the following relations hold : t u(t) =
A(t - z)M(u(z)) dr + h(t), s0
u”‘(t)
’A”‘(t
=
- r)M(u(r))d~ +
k+(t))
+
P’(t),
s0
(10)
’kk’(t
dk)(t) =
- r)M(u(z)) dz + i
s0 k=l
i
I=1
j=l
(+
M(k-r)(u(r)) + h’k’(r),
‘jCAaj)
, . . . , n - 1; where u(t) = y(t) + x,*
Proof. Since h: R+ -+ R is continuous and lim h(t) = 0, it is uniformly continuous, and the *+OO boundedness of h is clear from its definition. The relations (10) can be obtained by differentiation of (7) and setting y(t) + x,* = u(t). From (5) and (lo), it follows that y,(t) Y_
=
u(t) -
xn*,
,@I= YJt).+ any,(t),
... ... ... ... y,(t) = y;(t) + a,y,(t); or y,(t) y,_,(t)
=
u(t) -
xn*,
= u”‘(t) f anu(t) - anxn*,
Yn_ ,(t) = uC2’(t)+ (cl”_ * + a,W”(t) ....
....
....
y,(t) = u’“-“(t)
- a,_la,xn*,
(11)
....
+ (S1 - al)ff’“-2)(t)
+ (S, - a,@,
- a1))u(“-3)(t)
+ (S, - a,@,
- a,(S,
+ Snu(t) - S”X,*,
where u(t) satisfies equations (10).
+ an-lan4t)
- al)))u(“-4)(t) + . . .
17
Uniqueness and stability of a limit cycle for a third order dynamical system arising in neuron modelling 3. UNIQUENESS
AND
STABILITY
In order to establish the uniqueness and stability theorem of Miller [4,5]. Consider the nonlinear Volterra integral equation u(t) = -
OF LIMIT
of the limit cycle for system (5), we need a on a Hilbert
space H: t > 0,
’ F(t - T)&(T)) dr + h(t), s0
where F(t) is a family of bounded, linear self-adjoint operators mapping, g :H + H. If lim h(t) = ho, then we call f+m % u,(t) = F(+&(t - 5)) dr + ho, s0 the limit equation
of(E). The following
theorem
CYCLE
(E)
on H, g is a bounded,
t
>
0,
nonlinear
@,I
is due to Miller.
THEOREM (MILLER). Suppose H = R”, F E L, (0, co); h: R+ + R” is bounded and uniformly continuous, g is continuous, and equation (E) possesses a bounded solution u: R+ + R”. Then there exists a solution a0 of equation (E,) and a sequence of real numbers {t,}, lim t, = co, N-CC such that the relation lim u(t + tN) = u,(t) v- 7 is valid. Now we are in a position
to state and prove the uniqueness
and stability
result:
THEOREM 1. If system (5) possesses
positive periodic as
a nonconstant periodic orbit y c IT:where T c Q is a bounded invariant set of (5), free from singular points, and cli # mj, i # j, then y is the unique orbit (nonconstant) of (5) in the whole phase space g. The equation of y can be expressed (Qt),
tER+,
. . . 9q(O),
Qt),
where
q(t) = u,(t) -
xn*,
cQ1(t)
= ub’)(t) + ctnuo(t) - c(nx;,
q-&)
= q(t) + (cl”_
o,(t)
l
= q’(t)
+
Co&w + yp,uo(t)
+ (S1 - cc,)ub”-2)(t) +
+ (s, - cgs, - cqs, and u,(t) is the nonconstant
solution
u,(t) =
- U”_,cg,
(12)
(S, - Lqs, - q)ub”-3)(t)
tq))Ub”-4)(t) + . . . + Snuo(t)-
s,x;,
of
sm
A(z)M(u,(t
0
- T))
dr,
teR+.
(13)
18
B. LI
Moreover,
if @(r; t) c T is any orbit of (5), then its omega limit set Q, = y.
trajectory of (5), with the initial conditions Proof: If (y,(t), Y (t), . . . 2Y,(O) is an arbitrary Y(O), Y”‘(C%. . . 3Ycn-‘)(0), then the y .(t)‘s are given by (11). Hence the behavior of this trajectory depends on that of u(t), u(‘)(t), . . . , zhel) (t). If this trajectory lies in T, then it is bounded, so are the y,(t)‘s, in particular u(t), t E Rf, is bounded. Applying the Theorem of Miller, we have a real sequence (tN>, lim t, = co, such that N+a,
lim y(t -t tN) = u,(t) - x,*. N-tm
It can be proved that this u,(t) is not constant, Moreover, it can be shown that lim uCk)(t+ t N )
=
and that (13) has a unique
dk’(t) 0
k = l,...,n
’
-
nonconstant
solution.
1.
N-tCC
In other words, there exists a sequence
{tN}, lim t, = GO,such that
of real numbers
N+‘X
lim N+CC
(y,(t + t,), y,(t + t,), . . . , y,(t + t,)) = (al(t), Qt),
Next, let us consider
the nonconstant I?,;
where r. is a point on y. According {tN}, lim t, = co, such that
periodic
. . . , o,(t)).
orbit y c T. Denote
it by
t) = (Y;(t), Y;(t), . . . 9Y,“(N> to the above argument
there exists a certain
real sequence
N+CG
lim T(r,;
t +
tN) =
(ml(t), o,(t), . . ., o,(t)).
N-+02
Noticing
lim
y,"(t +
tN) =
u,(t) -
xX, y”(t) is periodic
(with period
8 > 0, say) and applying
the TheorEIhmof Miller, we get u,(t + fi) = lim
y,"(t+
fl +
yf(t +
tN)
tN) +
x,*
N-CC
= lim
+
x,*
N-+‘W
=
u,@),
VteR+.
That is to say, u,(t) is periodic with period 8 > 0, and hence o,(t), j = 1,2, . . . , IZare also periodic with the same period 8. The relation .lir”, T(r,; t + tN) = (ml(t), o,(t), . . . , o,(t)) implies the closed curve {(ml(t), o,(t), . . . , o,(t))) t E R+} c Rio, and since R10 is a connected ((W,(Q, QG
. . . >con(t))lt E R+} = Silo = I-(r,; I),
set, we have
I = (+co,co).
To prove the uniqueness of the limit cycle for system (5), let us suppose the contrary, and let f(r,; t) be a nonconstant closed orbit of (5) other than I+-,; t), hence T‘(r,; I) n T(r,; I) = 8. But from the above argument we have I?,,;
I) = al,, = {(Qt),
which is a contradiction,
w,(t), . . . , q$))~~ E R+} = Qrl = fk,
and the proof is complete.
; 0,
Uniqueness
and stability
of a limit cycle for a third order dynamical
system arising in neuron
modelling
19
An application of Theorem 1 to the system (1) yields the following result. THEOREM
2.
Iff = constant, 0 < a, 0 < p < b
<
_
q, p #
a #
q,
and
(P+ 4NP4 + a@ + P + 4)) (q -
@x*(1 - X*)
’
where x* is the unique positive root of 1 ‘* = 1 + exp{ -f - bx*(q - p)/(pq)}’ then system (1) possesses a unique limit cycle, which is stable. Moreover, the equation of this limit cycle can be expressed as (u’,O’(t)+ (p + q)ub’)(t) + pqu,(t)
- pqx*
solution of a, A(++& u,(t) = s0
+ x*, u;‘(t)
+ quo(t) -
qx* + x*, u,(t)),
(14)
were u,(t) is the nonconstant
- 3) d7,
teR+,
and A(s) = M(r) =
exp ( - as)
exp ( - Ps)
exp (- 44
CP - a)(q - 4 + (a - PN4 - P) + (a - 4) CP- 4)’
u/{1 +
exp [I-f - Wq - pP1).
Acknowledgment-The author would like to express his gratitude to the Department of Mathematics, UCLA, for offering him the opportunity of visiting UCLA as a visiting scholar. He would also like to thank Professor Earl Coddington for his helpful suggestions.
REFERENCES 1. SPIN R. B., LEUNG K. V., MANGERON D. & O&JZTBRELI M. N., Improved neuronal models for studying neural networks, Kybernetik 15, l-9 (1974). 2. AN DER HEIDEN U., Existence of periodic solutions of a nerve equation, Biol. Cybern. 21, 37-39 (1976). New York (1973). 3. OBERHETTINGERF. & BADII LARRY, Tables of Luplace Transforms. Springer-Verlag, 4. MILLER R. K., Nonlinear Volterra Integral Equations. Benjamin, New York (1971). BUN. Am. math. Sot. M(4), 5. MACCAMY R. C. & SMITH R. L., Limits of solutions of Volterra integral equations, 739-742 (1975).