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FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 43, NO. 6, JUNE 1996

Geometric Structure of Mutually
Coupled Phase-Locked Loops
Hisa-Aki Tanaka, Student Member, IEEE, Shin’ichi Oishi, and Kazuo Horiuchi, Fellow, IEEE

Abstract-Dynamical properties such as lock-in or out-of-lockcondition of mutually coupled phase-locked loops (PLL’s) areproblems of practical interest. The present paper describes a
study of such dynamical properties for mutually coupled PLL’s
incorporating lag filters and triangular phase detectors. The
fourth-order ordinary differential equation (ODE) governing themutually coupled PLL’s is reduced to the equivalent third-order
ODE due to the symmetry, where the system is anafyzed in thecontext of nonlinear dynamical system theory. An understanding
as to how and when lock-in can be obtained or out-of-lockbehavior persists, is provided by the geometric structure of theinvariant manifolds generated in the vector field from the thirdorder ODE. In addition, a connection to the recently developed
theory on chaos and bifurcations from degenerated homoclinic
points is also found to exist. The two-parameter diagrams of
the one-homoclinic orbit are obtained by graphical solution of
a set of nonlinear (finite dimensional) equations. Their graphicalresults useful in determining whether the system undergoes lockin or continues out-of-lock behavior, are verified by numericalsimulations.

I. INTRODUCTION

PLL 1

PLL 2

Fig. 1. Block diagram of considered mutually coupled PLL’sFig. 2. Triangular characteristics of the phase detector.

UTUALLY COUPLED phase-locked loops (PLL’s) arefrequently used in practical communication systems tosynchronize geographically separated timing clocks [ 11, [2].
The dynamical properties of such PLL’s, e.g., their lock-in or
out-of-lock conditions, present a problem of practical interest.
Theoretical studies on lock-in condition have been carriedout near the synchronized state [l], [2], while the dynamics
occurring beyond this state has been recently studied in both
single and mutually coupled PLL’s [3]-[6]. When two PLL’s
are in the critical state where they are almost but not quite
Synchronized, chaotic phenomena can be experimentally andnumerically observed for a wide range of realistic parameters[6]. Endo and Chua called this critical desynchronized statenear the synchronization (lock-in) as the “marginal out-oflock condition” in [6]. In the present paper, we elucidate
the nature of such marginality, thereby establishing the limitof synchronization. To do so, we apply nonlinear dynamicalsystems theory to determine the geometric structure of thesystem-generated invariant manifolds.

Manuscript received May 10, 1994; revised October 11, 1995. This paper
was recommended by Associate Editor M. Ogorzalek.
H.-A. Tanaka and S. Oishi are with the Department of Information andComputer Sciences, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo,
Japan.
K. Horiuchi is with the Department of Electronics and Communication
Engineering, Waseda University. 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, Japan.
Publisher Item Identifier S 1057-7122(96)03901-3.

11. ORDER REDUCTION
Considered here are PLL’s that incorporate a voltagecontrolled oscillator (VCO), a phase detector (PD) having
triangular characteristics (Fig. l), and a loop filter (LF) comprising of a simple RC filter with transfer function F ( S ) =l / ( l + ~ Swhich),
is known as a lag filter. Fig. 2 shows a block
diagram of the system, and the following fourth-order ordinary
differential equation (ODE), the phase model [61, describes the
dynamics of the phases

6,+ 2<1&1+ h(@l CDZ) = 6,
-

where
and
are, respectively, the resultant phase of theoutput of VCOl and VC02 after subtracting Rt. Here, R repsesents the synchronized angular frequency uniquely determinedby the free-running angular frequencies of VCO 1 N C 0 2 andthe system parameters presented in [6]. h is a 27r-periodic
triangular function (Fig. l), while
T I , rz, and S are normalized parameters, respectively, defined as = 1/2=,(2 = 1/2=,
r1 = Kz/Kl, 7-2 = r2/r1, and 6 =(w01 - W O ~ ) / ( KKz),
I
in which Ki, ri, and wgi denotethe total loop gain, loop filter time constant, and free-running
angular frequency of PLL i (i = 1, 2), respectively. Our goalin this section is to reduce (1) to the simplest equivalent ODE.

1057-7122/96$05.00 0 1996 IEEEc1, cz,+c1

TANAKA et al.: MUTUALLY COUPLED PLL’S

439

For this purpose, we first normalize (1) to give

Our attention is focused on the following three sets ofchaotic parameters experimentally obtained by Endo and Chua
161:1) Asymmetric high damping [Case l)]4‘1 = p i

42 = p 2 ,P1 = - Plh(% - q 2 ) - alp1
$2

= - Paw%! - 4 1 ) - a2p2

+ 71
+ Y2

(1

(2)

0.614, 52

0.331

f o l = 18 250 Hi<, f o 2 = 25 400 HZ

where q1 = @I, q2 = e2,Q~ = all, a2 = 2
c
2
m
p12
=/
.
T
/
;
.
2
, r1 = 45 500/12 500, 7-2 = 5.027/5.298
1, Pz = TI/TZ, 71 6,and 7 2 = - ( ~ i / ~ 2 ) 6 .
6 Z 2 7 ~f o l - f o 2 / 4 5 500 1 2 500, -0.774 565. (6)It should be noted that symmetry reduces (2) to a third-order
ODE,
by introduciI1g= p1 p 2 , pz = p1 - p 2 ,2) Symletric high damping [Case 2)]
Q i = 41q 2 , and Q2 = Q L - q 2 , (2) becomes

+

+

+

P2 = -

3) Symmetric low damping [Case 3)]++

a2, a - = ail - a2, /3+ = p1
where a+ = a 1
P2,
P- = Pi - P 2 , Y+ = YI YZ,and Y- = 71 - 7 2 .
Note the solution for Pi, P2, and Qz can only be determined by (3b)-(3d). In (3b)-(3d) there are two nonlinear
terms -,L?-h(Q2)and - P + h ( Q 2 ) . To reduce these two into

+

we

One

the transformation

x = Pi/@-- P2/P+, y = J’2/P+, and z = Q 2 , which reduces(3b)-(3d) to the following Ihird-order ODE:

(4)

where A1 = (a-P- - tr+/3+)/20+, A2 = (/3.-//?+ P+/P-)a-/2, Bo = r-/P+ = 6,B1 = -a-P-/2P+, andBz = -(a-,L?- a+p+)/2@+.Although h(x) can be either
sinusoidal or triangular depending on the type of the employedphase detector, we only consider the triangular function here.
When transmission delays are incorporated into either phaseequation (1) or (2), the samie reduction can be carried out by
applying some assumptions (See Appendix I).
111. RECEWISE-LINEARANALYSIS(8)

The following parameter sets are, respectively, obtained forcases11-31:

B1 1-0.019++

r1 = 12 440/12 200, r 2 = 2.16/2.45

6 f 2 ~ f o 1- fo2/12440 + 12200, = -0.186914.

A 1 = -1.242519,

li. = A i z + . 4 q y ,y =Bo + B i z B 2 y - h ( ~ ) ,=P+y,(1 = 0.0914, 52 10.0964
f o i = 22 450H:~,foz = 23 183Hz

A2 = -0.037054

424, B2 = -1.281 369-0.77456!5, p+ = 4.836228
A1 = -1.960438, A2 = -0.502870
B1 = -1.353398, B 2 = -4.167234
Bo 6 = -1.218 127, p+ = 13.686 562

Bo rz 6A1

-0.194297,

A2 = -0.168230

B1 = -0.000891,

B2 = -0.196079

Bo = S = -0.186914, ,8+ = 2.156572.

(9)

(10)

(11)

These parameter sets, respectively, correspond to the real
eigenvalues A,, A,
and A, of the linearized matrix of thevector fields at 0+ and 0 - , which are obtained by solving++

Since (4) is 27r-periodic with respect to z , let us concentrate-A(A - A1)(A - B,) P+(A1 - A) AaBlA = 0 (12)on the region D = ((2, y, ,?)I - 7r - 6 I
z 5 7r - S} in orderA), = (-1.242336, -2.931 525, 1.649973)
to examine the geometric structure of the defined vector field. giving (A,. A,
for
(9),
(A,,
A
,
A), = (-1.405973, - 6.432011, 2.210312)
Region D can be divided into three segments: D+, DO,and= (-0.194283, - 1.569894,
D - , respectively, defined by D+ = {(x, y, z)l7~/2< z 5 for (lo), and (A,, A,, A),
7T - S}, Do = ( ( 5 , y, .)I121 I../a}> and D- = {(z, Y, .)I - 1.373 800) for (11). In segment DO,the real eigenvalue A 1 andthe pair of complex-conjugate eigenvalues A2, 3 are obtained
7r - 6 < z < -7r/2}, where the vector field in each segment
is linear. Further, denote thle planes {(x,y, z ) ( z= 3:7r/2} as by
E+, and E-, respectively. From (4) and the form of h ( z ) ,it-X(A - A 1 ) ( A - B2) +/?+(A - A i ) AzBlA = 0 (13)follows that unique equilibrium points at 0+,00,0- exists
in a+,Do, D- , with their positions being, respectively,
giving ( A i , X2,3) =(1.242706, -0.640591 f 2.103602i) for(9), ( A i , & , 3 ) = (-1.571786, -2.027943
2.933330i)(0, 0, f~ - S), 0 0 = (0, 0, 6).
0k, ( 5 ) for (lo), and (AI, X 2 . 3 ) = (-0.194310, -0.098033
It should be noted that GI+ and 0- can be identified if we f 1.465 200i) for (11). The corresponding eigenvectors
consider the vector field is {definedon the cylinderR2 x S1 = e,, e,, e,,, e l , and e 2 , 3 to the above eigenvalues determine
the invariant manifolds W s ( O - ) , W;(Oo), W,S(Oo), and((2,Y, Z)l(X, Y) E R 2 , ‘F S’}.

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W'(O-)

FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 43, NO. 6, JUNE 1996

(Fig. 4) or 3) p+ is mapped to the right of Ws(O-) forS < 6, (Fig. 5 ) , where p+ denotes the intersection of theplane C+ and the unstable manifold W u ( O + ) .Situation 1)
leads to a set of two nonlinear (finite dimensional) equations
derived as shown in Appendix I1 where the values of fo2
and T are unknown. By using the parameter values off o l , (1 2 , and r1,2 in (9)-(ll), the critical parameter 6, (thecntical free-running angular frequency 6 2 ) for situation 1)
can be obtained by solving this set of nonlinear equations,
respectively, giving

T = 0.787078
and&z = 25 390.248,
Y

Fig. 3.

Zi.e.,

6, = -0.773 508

Saddle connection in piecewise-linear vector field. Generically,W " ( O + ) becomes tangent to the weakly stable direction (eS in this case)after extended onto W " ( 0 - ) .Vt;B(Oo) and LV,"(Oo), respectively, denotes
the eigenspaces of the real eigenvalue and the complex conjugate eigenvaluesfor (9),T = 0.614 754

at 0 0 .

and

f o =~ 27811.590
i.e.,

6, = -1.217405
for (lo), andT = 1.075 136,
and
= 23 181.330

i.e.,

S, = -0.186 488(16)
for (1 I), where T is the time interval p+ being mapped fromE+ to C-. When 6 = 6,, the point mapped from p+ to theYZintersection of W s ( O - ) and C- asymptotically goes to 0because A, < A, < 0; being a situation that indicates a saddle
Fig. 4. Lock-in dynamics for 15 > 6,. W " ( O + ) is extended eventually
to Oo (the lock-in state). Lock-in can be obtained from almost all initialconnection (a homoclinic orbit in this case) exists between O+
conditions.
and 0-as shown in Fig. 3. When S is slightly larger than S,,
the orbits from the neighborhood of p+ go to the sink 00,
W'(O*), as well as the extended unstable manifold i.e., the two PLL's undergo lock-in from their broad initial
Weu(O+),where W'''(O+) is tangent at O+ to the linear states (Fig. 4). However, when 6 is slightly less than 6,,being
space spanned by e" and e' (see [7] which proves the existence the case if the set of parameters are per (9)-(ll), then theof such an invariant manifold). W" and W " , respectively, point mapped from p+ on C- is on the other side of 00
represent the stable manifold and the unstable manifold. Fig. 3 with respect to W s ( O - ) .Such a situation can be consideredschematically depicts these invariant manifolds. From ( 5 ) , to lead to the loss of the global lock-in because W e u ( O + )(12), and (13), it is evident that the positions of O+, 00,and on C - (Fig. 5 ) prevents most initial points from reaching the0- depend only on the parameter 6,and that the eigenvalues sink 00.Adding to this lock-in mechanism, the homoclinic(vectors) are not dependent on 6. Hence, if we only change orbit at (14) in Case 1) is shown to appear near a degenerated
6,i.e., the free-running angular frequency f o 2 of PLL 2, then homoclinic point-orbit-flip homoclinic point [8]-[ 101-at
W s ( O - ) , WG(OO),W,"(Oo), Wu(O*), and W""(O+) do which the homoclinic orbit is doubly asymptotic to the strongly
not change their normal vectors or directions, though they are stable direction ess and the unstable direction e" at the saddle
shifted in the z-direction by the position changes of 0- , 0+, [ll]. Kokubu and Oka [12] demonstrated the existence of theand 00.
Therefore, a critical parameter 6, is expected to Smale horseshoe and the invariant foliation in an unfolding
exist under the following situation: 1) p+ is mapped onto the of the orbit-flip homoclinic point, while the resultant chaotic
intersection of W s ( O - ) and C - by the linear flow in Do phenomena near the orbit-flip homoclinic point is reported
for S = S, (Fig. 3). If S # S,, the following situations are in [ 111. Homburg, Kokubu, and Krupa also demonstrated the
expected to occur depending on whether S > S, or S < 6,: Smale horseshoe generation in another degenarated homoclinic2) p+ is mapped to the left of W s ( O - ) on C- for 6 > 6, point-inclination-flip homoclinic point [ 101-at which the44 1

TANAKA et al.: MUTUALLY COUPLED PLUS

by solving the nonlinear equations defining a homoclinic orbit.
Numerical simulations show a good agreement to the expected
critical change of dynamics for various parameters in theneighborhood of the one-homoclinic branches.
APPENIDIX 1
TRANSMISSION
DELAYSY

Z

Fig. 5. Out-oflock dynamics for 6 < 6,. Most initial conditions in thepositive y region or in the neighborhood of 00eventually go to 0 0 ,while
solutions starting from initial conditions away from 00 in the negative yregion exhibit out-of-lock persistenltly.

invariant manifolds We" and W" become tangent along thehomoclinic orbit [131. The inclination-flip homoclinic point
and the resultant dynamics in (4) will be reported in the future.

This Appendix describes the reduction of the phase model(1) or (2) which incorporates transmission delays. We consider a system of two mutually coupled PLL's incorporatingtransmission delays of the signals between two PLL's and thecorresponding estimates of these delays-a system employing
the so-called delay line compensation technique [2]. The
governing equation of the phase differences for such a systemthat incorporates a loop filter whose transfer function F ( s ) =1/(1 7s)can be written as follows:+&&) + 2<1&(t) + h[@l(t- 712) - @ 2 ( t - ?12)] = 6&,(t) + 2 ( 2 f i & 2 ( i )+(2)

h[@2(t- 721) - @l(t- +21)] = -(17)

Iv. BRANCHOF ,4 HOMOCLINICORBITAs mentioned in the previous section, a saddle connection,in this case a one-homoclinic orbit, appears at 6,; beyond
which global lock-in of the two PLL's is lost. Such onehomoclinic orbit forms a submanifold of codimension one
in the parameter space (see Theorem 3.5.1 in [lo], p. 399),
namely, a one dimensional homoclinic branch in a suitable
two-parameter space. Here, we consider a set of two practicalparameters f 0 2 and K2, for which a one-homoclinic branch canbe computed by solving the set of two nonlinear equations
contained in Appendix I1 for various values of Kz. Fig. 6
shows the computed one-homoclinic branch (dotted by 0)
in the ( 1 0 2 , K2)-parameter space for Case 1). The ones forCases 2) and 3), respectively, exhibit a similar characteristics
between K2 and f 0 2 as in Fig. 6. Namely, the critical f02
value is almost linearly proportional to K2 in the practicalrange of K2. These linear characteristics observed in Fig. 6
seem to coincide with the experimental facts observed in apractical range of parameter Kz values. Employing the fourthorder Runge-Kutta integration scheme with the time step 0.01
and setting certain (wide range of) initial states away from thesynchronized states in (4), the critical change of dynamics isverified at the various sets of parameters dotted by
and 0
in Fig. 6. The insets of Fig. 6 show the typical solutions in
the corresponding dynamics projected to the (y, 2)-plane.

where ~ i denotesjthe delay of signals traveling from PLL j
to PLL i, while; i , j = 1, 2; i # j . ? ~ jdenotes the estimateof q j for delay compensation; i , j = 1, 2; i # j . It is notedthat (17) becomes (1) in case 712 = 0, ? l ~= 0, 7-21 = 0, and?2j21 = 0. If ~ i =j ? i j hold, Pi = 6
1 + 62,Pz = 61 - 62,
Q1 = @1 @ 2 , and QZ = Q1 -- @ 2 , transform (17) to+

p1 = -

for which ithe solution for F1, P2, and Q 2 can only be
determined by (18b)-(18d). In addition, when rij = ~ j =i T
holds, the transformation 5 = P l / / L - E$//?+, y = P2//3+mand z = 1 2 2 , reduces (18) 'to the following differentialdifference equation:+

V. CONCLUSION
We presented theoretical and numerical results that explain
experimental results of mutually coupled PLL's in the criticalstate between lock-in and out-of lock dynamics. Such criticalstate, namely, the limit of captureflock dynamics is now considered to be the onset of the homoclinic orbit. For three typical, realisic parameter sets, the one dimensional homoclinic
bifurcation set in the ( f 0 2 , Ka)-parameter space is obtainedin which the time delay r appears only in h.
APPENDIX I1NONLINEAR EQUATIONFOR A HOMOCLINIC
ORBITThis Appendix describes the derivation of the finite dimensional nonlinear equations that determine T and f o 2 of thehomoclinic orbit for given parameters. A saddle connection-a
homoclinic orbit-exists when p+ is mapped by the linear flow

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442

FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 43, NO. 6, JUNE 1996

't

-

-e-

+

: One-Homoclinic Branch: Out-Of-Lock: Lock-in

Lock-in

9Y

25000

I

I

I

I

I

I

I

I

1

1

K 2

40500

45500

Fig. 6. One-homoclinic bifurcation set for Case (1) in the ( f 0 2 , I<Z)-parameter space. Two insets show the lock-in (observed at U) and the out-of-lockdynamics (at +) projected to the (y, 2)-plane, respectively. This bifurcation set also presents the limit of synchronization between the two PLL's.

in DO onto the line defined by W s ( O - )n E-,as in Fig. 2.
Such a situation leads the following two conditions on T andtransformations

fo2:

1) 9 ( p + ) is located on W " ( O - ) ,where 9 is a flow-defined
map from E+ to E-. (Cond. 1);
2) From the definition of 9, the z element of 9 ( p + ) equals
to -7r/2. (Cond. 2).

where Q is given as

Since the flow in D+ is linear, W u ( O + ) becomes a line.
This leads the exact position of p+, i.e., the intersection ofW"(O+) and C+
in which f1 and f 2 are given by

where a, w denotes the real and imaginary parts of X2,
respectively. Equations (22) and (23) lead to the following
expression of 9
Similary, W s ( O - )becomes a plane spanned by e, and ess at
0-,satisfying the following equation:ab+) = QJQ-lpY + ( 0 , 0 , 6IT(26)

where p; = p+ - ( 0 , 0, 6 ) T , and J is given as

To obtain 9,we transform the original coordinate ( x , y, Z )in DO to a canonical coordinate ( X , Y , 2)by the following

Combining (20), (21), (24), (26), and (27), Cond. 1, and Cond.
2 lead to a set of two nonlinear equations in the unknowns T

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TANAKA et al.: MUTUALLY COUPLED PLL‘S

and f o 2 , which can be solved via Newton’s method. Suchcomputation for a saddle connection has been previously
carried out in [4] for the second-order piecewise-linear ODE.

ACKNOWLEDGMENT
The authors would like to thank Professor T. Endo of Meiji
University for providing us ihe experimental data. One of theauthors (H. Tanaka) would like to thank Professor T. Endo,
Meiji University, T. Matsuda, NTT Transmission Systems
Laboratories, and Dr. S. Watanabe, Niels Bohr Institute forconstructive comments and discussions. Also, he acknowledges Assistant Professor R.. Tokunaga, Tsukuba University
for providing him a useful reference [ 101.

REFERENCES
111 H. hose, H. Fujis&, and T. Saito, “Theory of mutually synchronizedsystems,” Electron. Commun. Japan., vol. 49, no. 4, pp. 755-763, Apr.
1966.
121 K. Dessouky and W. C. Lindsey, “Phase and frequency transfer between
mutuallv svnchronized oscillators,” IEEE Trans. Commun., vol. 32, up.
110-115, Feb. 1984.
[31 T. Endo and L. 0. Chua, “Chaos from phase-locked loops,” IEEE Trans
Circuits Syst., vol. 35, no. 8, pp. 98711003, Aug. 1988.
“Piecewise-linear analysis of high-damping chaotic phase[41 ~,locked loops using Melnikov’s method,” IEEE Trans. Circuit Syst., vol.
40, pp. 801-807, Nov. 1993.
E. Bradley, “Using chaos to broaden the capture range of a phase-locked
loop,” IEEE Trans. Circuits Syst., vol. 40, pp. 808-818, Nov. 1993.
T. Endo and L. 0. Chua, “Chaos from two-coupled phase-locked loops,”
IEEE Trans. Circuits Syst., vol. 37, pp. 1183-1187, Sept. 1990.
M. Hirsch, C. Pugh, and IM. Shub, Invariant Manifolds. Berlin:
Springer-Verlag, 1977, LNM583.
K. Ion, E. Yanagida, and T. Matsumoto, “N-homoclinic bifurcations ofpiecewise-linear vector fields,” in Structure and Bifurcations ojDynumical Systems, S. Ushiki, Ed. Singapore: World Scientific, (Advanced
Series in Dynamical Systems), 1993, vol. 11, pp. 82-97.
M. Kisaka. H. Kokubu. and H. Oka. “Bifurcations to N-homoclinic
orbits and .I-periodic orbits in vector fields,” J. Dynam. DZ@ Eq., vol.
5, pp. 305-357, 1993.
[IO] T. Matsumoto, M. Komuro, H.Kokubu, and R. Tokunaga, Btjkrcarions.
Tokyo: Springer-Verlag, 1993.
1111 H. Tanaka, “Chaos from orhit-flip homoclinic orbits generated in apractical circuit,” Phys. Rev. Lett., vol. 70, pp. 1339-1342, Feb. 1995.
1121 H. Kokubu and H. Oka, (in preparation).
[13] A. J. Homburg, H. Kokubu, and M. Krupa, “The cusp horseshoe andits bifurcations in the unfolding of an inclination-flip homoclinic orbit,”.
I.
Ergod. Th., vol. 14, no. 4, pp. 667-693, 1994.

__

Hisa-Aki Tanaka (S’94) received the B E degree
in electrical engineering, the M E. and D.E. degrees
in electronics and communication engineering, fromWaseda IJniversity, Tokyo, Japan, respectively, in
1990, 1992, and 1995.
Since 1995, he has been a Visiting Research
Fellow at the University California, Berkeley His
current research interests are in the interdiciplinary
area of communication systems, biology, and nonlinear physics
Dr Tanaka has received the 1995-1996 Research
Fellowships of the Japan Society for the Promotion of Science for Young
Scientists He IS a member of the Institute of Elcctronics, Information, andCommunication Engineer3 (IEICE) and Physical Society of Japan.

Shin’ichi Oishi was born in Japan on May 9, 1953.
He recieved the B.E., M.E., and D.E. degrees, allin electronics and communication engineering, fromWaseda University Tokyo, Japan, in 1976, 1978, and1981, respectively.
In 1980, he joined the faculty of the School of
Science and Engineering, Waseda University. Since
then he has engaged in education and research in thefields of nonlinear science, applied mathematics, andnumerical analysis. He is now a Professor with the
Department of Information and Computer Sciences.
ON FUNDAMENTALSDr. Oishi wds the Editor bf the IEICE TRANSACTIONS
during 1993-1994. He received Best F’aper Awards in 1992 and 1995 fromIEICE.

Kazuo Horinchi (SM’83-F’90) was born in Japanon January 7, 1929. He received the B.E., M.E.,
and D.E. degrees, all in electrical communicationengineering, from Waseda University Tokyo, Japan,
in 1952, 1954, and 1958, respectively.
In 1957, he joined the faculty of the Schoolof Science and Engineering, Waseda University,and since then, as a faculty member, he has been
engaged in education and research in the fields
of electromagnetic wave theory, information andcommunication theorv.
,. circuit and svstem theorv.
applied mathematics, and others. He is now a Professor with the Department
of Electronics and Communication Engineering. Recently, he is particularlyinterested in studying the functional analysis of nonlinear systems, including
nonlinear wave: propagation. He is also an author of the books: Mathematics
for Electrical Engineering, Electromag,aetic Theory, Applied Analysis, and A
Course of Applied Mathematics.
Dr. Horiuchi is the President of IEICE, the Chairperson of the Excutive
Committee of the Symposia on Nonlinear Theory and Its Applications(NOLTA), and a Member of the Science Council of Japan. He was a Presidentof the Society of Information Theoq and Its Applications, and a VisePresident, a Director, the first Chairperson of Engineering Sciences Groupand the first Editor-in-Chief of the IEICE. He received the DistinguishedServices Award in 1990, the Achievem,ent Award in 1981 and the Best Paper
Award in 1994 from IEICE.