SCIENTIFIC ABSTRACT ROYTENBERG, L.YA. - ROYTENBERG, YE.YA.

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• Collection: Scientific Abstracts
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• Release Decision: RIF
• Original Classification: S
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• Document Creation Date: November 2, 2016
• Document Release Date: July 19, 2001
• Sequence Number: 9
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

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S/02 62/142/005/010/022
BI 04YBI 02,
AUTHOR: Roy-t g-ab!E&_j~. Ya
TITLE: Theory of a gyroscopic follow-up system in the case of
random noise
PERIODICAL: Akademiya nauk SSSR. Doklady, v. 142, no. 5, *62, 1050-1053
TEXT: The optimum reproduction of the useful signal of a gyroscopic
follow-up system is studied by means of the least root-mean-square error.
f (D) z (1) e (D) X (1) (D Wilt),
D
D2 + D +
A A
f (D) =11 z
_D + D2
B B
0
e(D) A S x W
0 V)

S/020/62/142/005/010/022
Theory of a gyroscopic B104/BI02
is the equation of motion for such a system. a is the angle of rotation
of the outer gimbal, P is thatof the case, H is the kinetic moment of the
gyroscopet,A is the moment of inertia of the gyroscope with case and outer
gimbal, B is the moment of inertia of the gyroscope and case around the
.case axis, n1al is the-moment of frictional forces in the bearings of the
outer gimbal, miip-xl(t)~ is the moment around the ax is of the outer
gimbal to which a stabilizing motor is fixed, S[a-x2(t)j is the moment
around the case axis of the Syroscdpe with a correcting electromagnet
attached.to it. From (2) it follows that z(t)'= Y(D)x(t),
Y(D) F(D)e(D)/a(D), where F(D) is the adjoint of the matrix f(D) and
6(D) is the determinant of the matrix f(D).
D2 D +
A B (A~ j
(D) A.(D) in, 'H S S
;F~jD + (DI + D)
Still i 2 H.
A (D) D4 + D3 + q2D2 + S +1117" q2D + -g,- q~q
A
holds for the transition matrix. The input signal consi -sts of the useful
siUnal/-'I(t) and random noise n(t): Q(t) m(t) + n(t). Useful signal
Card 2

S,102q62/142/005/010/022
Theory of a gyroscopic BIG4/B102
and noise are not correlated. It is shown that an optimum gyroscopic system
must contain an optimum filter with the transmission function
(D (D) k (D 4- x) (D + p) k 2tWL
XIV UIA~- V, +61), +ell
b/a, where a 2 2 + 2gx + R(x 1 + X2) - "1X211
b 2V3 + OP 3 F-2 )x + 2tLF- 2 + (v 2 + FS. 2 + 2px) (41 + X 2) + xx 1x 2'
4tLV2L 2xAf (V2
SM F(Dz VIP + 41012 1 S" (0)) W2 + k2 +
holds for the spectral densities of the random processes. The output
signal y(b) of the optimum filter is fed into a computer which solves the
integral equation
t 2 1
(22)
W1, (t -r) xj (-r) dx y (t) W2, (I (-r) d-r 0.
Card 3/4

S/02 62/142/005/010/022
Theory of a gyroscopic B104 102
Vf ki are the elements of the weight matrix of the.gyroscope. The solutions
to these equations are the signals which are to be fed into the input of
the gyroscope. The author*thanks A. Yu. Ishlinakiy,for valuable advice.
There are 5 references: 4 Soviet and 1 non-Soviet. The reference to the
English-language publication reads as follows: N. Wiener, Extrapolation,
Interpolation and Smoothing of Stationary Time Series, 11. Y., 1949.
ASSOCIATION: Moskovskiy gosudaratv~lnnyy universitet im. M. V. Lomonosova
(Moscow state University imeni M. V. Lomonosov)
"RESENTED:, June 2, 1961. by A. Yu. Ishlinskiy, Academician
SUBMITTED: May 29, 1961
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28 (2) SOV/115-59-10-25/29
AUTHOR: Roytenberg,.M.N., Head

TITLE: About the Organization and Activities of Testing
tD
Laboratories in Plants
PERIODICAL: Izmeritel1naya tekhnika, 1959, Nr 10, PP 59-60 (USSR)
ABSTRACT: The article contains comments on the article by K.N.
Katsman entitled "About Some Problems of Organization
and Activities of Measuring Laboratories:in Plants"
which appeared.in "Izmeritel'naya tekhnika",, 195.9,~
Nr 2. The author'is of.the opinion that Katsman is not
taking the problem seriously, since he suggeststhat:,all
plant laboratories be united into one central laborato-
ry. The activities of plant laboratories are varied:and
such a merger:would only deteriorate their work. Let
the laboratories do their work as they do today, only
the quality of work should be improved and the labo,-.
ratories.should keel) pace with the modern technical

ss. The author agrees with the ug at.
progre s gestion th
Card 1/2. all laboratories should be responsible to the chief

7"
ALITSHULKR, Z.Ye., inzh.; BASTUNSKLY, R.A., inzh.; BERSTEL', Y.N., inzh.;
BIRXIIB~MG, I.E., inzh.; BOGOPOLSKIY, B.Kh., inzh.; BUKHARIN, S.I.,
inzh.; GNRbIiTFYII# B.G., Inzh.; GRIUSHPUU, L.Y., inzh.; DRETYER, G.I.,
inzh.; D11MRSHWII, A.G., inzh.-, ZIATOPOLISKIY, D.S., iznh.; KIANM,
A.V., inzh.; KOZIN, Yu.V., inzh.; MVITIN. 1.F., tnzb.; HFJLINIKOVI
L.F., inzh.; MELIKUMOV, L.G., inzh.; "HADEL', M.B., Jnzh.: PAVJOV,
N.A., inzh.; PASLYelf. D.A., inzh.; PESIN, B.Ya., inzh.; PYATKOVSKIT,
P.I., inzh.; RAZI40SCHIKOV, D.V., inzh.; ROZENOYER, G-Ya.. iazh.;
ROZY.11BERG, R.L.. inzh.; 1!qjA;,W inzh.; RYABI14SKIY, T&J.,
inzh.; SYMENKO, I.I., inzh.; TARACHNIKOV. L.D., inzh.; FELIDMAII.
A.S., inzh.; SHIRAITHAN. G.Ya., Inzh.; SHTMIIGAS, N.S., in2h.;
LHVITIN, I.P., otvetstvennyy red.; STHLIMAKH, A.11., red.izd-va;
BEMR, O.G., tekbri.red.
[overall mechanization and automatization of production processes in
the coal industry] Komp3eksnaia mekbanizatsiis i avtomatizatsiia
proizvod,stvennVkh protsessov v ugollnoi prouqshlennosti. Pod red,
IU.V.Kozina i dr. Moskva, Ugletekhizdat, 1957. 82 p. (MIRA 11:3),
1. Gosudarstvannyy proyektno-konstruktorskiy institut. 2. Institut
Giprougleavtomatizateiya i Tekhnicheskogo Upra,vleniya Ministerstva
ugolltioy promyshlennosti (foi- all except: Levitin, Stellmakh,
Bekker)
(Automatic control) (Coal mining Mchinery)

S/020/62/146/006/064/016
B172/B186
AUTHORS: Roytberg, Ya. A., Sheftell, Z. G.
TITLE: On eauations of the elliptical type with non-continuouB
coefiicient
PERIODICALt Akademiya,nauk SSSR. Doklady, v. 146, no. 6, 1962,,1275-1278-
TEXT: Results which hold also for equations-of higher order (uniqueness
and existence-of generalized solutions of the boundary value problems
formulated below) are given by the author for an equation of second order.
These results take the form tu f
n m
b
pj(x)D;li + b(.
tu D (b (x)D U) + 6x. b k k
j ik k i j,
The operator 8? is considered on determined Sobolev functional spaces. The
~omplex-valued coefficients b Pip b are defined in a domain G of the
jk'
n-dimensional space. The boundary of G is piecewise smooth. G is de-
composed'into two domains G and G by a (n-l)-dimensional continuously
1 2
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S/020/62/146/006/004/016
On equations of the elliptical ... B172/B186
differentiable areat which is spherically homomorphic and has no common
points with f, so that the coefficients of X are elements from CO(G
) and
,
) respectively, if considered as functions in G
(i 1,2). The
C (G
i
i
boundary conditions take the -form
m u + Tu + -u 0
r
Cu
a I a tul 0
2
Here T designates A linear combination of the tangential derivative with
real coefficient from Cl(r); Q a limited linear operator in L2 m a
constant being equal to 0 for T 0, 1, and otherwise being equal to 1;
2 i
ive furiction from C (Gi); 15 Z:b' ~ D U. are
a. (i 1,2) a posit
(v
,
k
d'-i jk k
the components of the normal to the surface r. pointing away from G
),
i
LU) =. u - u1
, The question of smoothness of the generalized
j
,
r _0
+ 0
,
solutions is treated in a manner similar to that described in a study by
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S/020/62/146/006/004/016
On equations of the elliptical ... B172/B186
Nirenberg (Comm. Pure and Appl. Math., 8, no- 4 (1955))- -The results ob-
tained hold also for decompositions into more than two'domains G
Further, the corresponding eigenvalue problem is treated.
ASSOCIATIONt Stanislavskiy pedagogicheskiy institut (Stanislav Pedagogics.
Institute). Drogobyc'n'skiy pedagogiciLeskiy institut-
(Drogobych Pedagogical Institute.)
PRESENTEDi May 21, 1962, by S. L. Sobolev, Academician
SUBMITTEDs May 4, 1962
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&6iiditi6n 'for 'stabil
C64`

-Axii
is
J%Card 0
-, 1.- -71

tit .11 01
-.11 err. 7

-, ~~VCoV7rf~N (3 ER C-i~~ - [- I-N---
--- -- w-

, YA. M.

AUTHOR: Roytenberg, Ya.N. (Moscovv) 40-22-2-3/21
TITLE: On a Method for the Construction of a Lyapunov Function for
Linear Systems With Variable Coefficients (0b odnom metode
.postroyeniya funktsiy lyapunova dlya lineynykh sistem s pere-
mennymi koeffitsiyentami)
PERIODICAL: Prikladnaya matematiks, i mekhanika,1958,Vol 22,Nr 2,
pp 167-172 (USSR)
ABSTRACT: For the construction of Lyapunor functions.for systems of
linear differential equations with constant coefficients
Chetaev elaborated effective methods. The author now transfers
these methods to systems of differential equations with variable
coefficients. By consideration of the system of differential
equations which is obtained from the initial system by setting,-
the coefficients constantg it.is possible to transform the
initial system into a form so that a Lyapunoy function can be
given. The stability conditions can be found in well-known
manner from the condition for the definiteness of the Lyapunor
function. The Lyapunov function itself is set up in the usual
way in the form
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On a.Method for the Construction of &- Lyapunov 40-22-2-3/21
Function for Linear Systems 'Nith Variable Coefficients
NT+N'?
V 1 2 + 2 2
2 % h + -?h
9 Z-) (9
h=Nl+l
whereby thag I represent the variables of state of.the system.
-:ned stabilityconditions-are suf aient,,but no b v,n 6,
The obta.L
cessary.
The method is illustrated by an example..
Thare are et referenaes.
Sovi
SUBMITTED: Septembe*r 16, 1957
2- t ab li f--.Y- ~T heory 2. Equations of statp-Theory
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...
16(1) ................ ..
AUTHOR: Roytenberg, Ya.N. (Moscow) SOV/40-22-4--14/26
TITLE: On the Accumulation of,Perturbations in Nonsteady Linear Im-
pulse Systems (0 nakoplenii vozmushcheniy v nestatsionarnykh
lineynykh impul'snykh sistemakh)
PERIODICAL: Prikladnaya matematika i mekhanika,1958,Vol 22iNr 49
PP 534 - 536 (USSR)
ABSTRACT; In addition to investigations which have been carried out during.~
the last years by different authors on disturbances in impulse
systems the author considers in the present paper the problem
of accumulation of disturbances in nonsteady linear impulse
systems which are under the influence.of external forces. The
maximum amplitudes of the disturbing forces are to.be bounded.
The,author considers an impulse system, the equations of which
can be written as difference equations in the following form I
n
Yk (t +T-) + ak1MY1 (t) xk(t) (k=l,.-.,n)
0)
1
The system of these equations is equivalent to a matrix
Card 1/3 equation of the form

On the Accumulation of Perturbations in Nonsteady SOV/40-22--i4-14/26
Linear Impulse Systems
(2) y(t+T) + a(,;.-)y(t) x(t)
Here y is the coordinate by which the behavior of the impulse
system is described, x are the external disturbing forces.
With a series set up for the coordinates y now a general? but
very nontransparent expression is obtained which can be ex- q
pressed as a double sum. Under the assumption that the forces
are bounded in their absolute value
xk(t)l '2
2 ) 2 m 2
q a~ Tq, 2
P
2
Here A A + I
A moment of
inertia
of
the frame + object + gyros-
.
cope with respect to the collimating axis. I moment of inertia of the un-
,
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-----------
84760
On the Motion of Gyroscopic Apparata Under the S/040/60/024/003/0, /0 XX
Influence-of-Random Forces G1II/C222
loading motor, j tranrmission gear ratio-from the motor~shaft to the
collimating axiB, a = j (j n + Kc friction coefficient
2
for frame supportings, K = ik, c j k2 magnetic flux caused
2
K 2 H
by the exciter coils of the unlo adirrg motor), M r,r 2 ? q TBT
kinetic moment of the gyroscope, B - equatorial moment of inertia of
thegyroscope.
For the gyroscope pendulum the dispersion of the stabilizing angle is also
determined-Iffith the aid of-the obtained formulas it is stated that for a
motion of-the sea the Oscillations around-the-axis of the box are very large
and oscillations around the axis of suspension are very small.
Under restriction to the precession motion the intercardinal deviation
x al sin 2
4
is determined for the gyrocompass; it vanishes for cardinal azimuths of
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8476o
On.the.11otion of Gyroscopic Apparata. Under the S/040/60/024/003/021/021 X)r
Influence of Random Forces C111/C222
0 0 0
course 0 goo 180 270 and reaches its maximum for the inter,
0
cardinal angles 45 , 1350 , 2250 , 3150 . Explicit expressions are
given for I and a . BX a numerical example it is shown that the inter-
4
cardinal deviation is sufficiently small if the gyrocompass e.g. is of two
rotoric type.
The author mentions A.k. Sveshnikov and S.S. Rivkin. 7
There are I table and 5 references: 3 Soviet, 1 English and 1 American.
SUBMITTED: Pebruary 29, 1960
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S/040/60/024/04/22/023
333
AUTHOR: Ro,-tenberg, Ya, N ("Toscow)
TITLE: On t -a-;-7L7h-e-o-r-y--o-f -Mor Dix e c t Gyroscope Stabilizers
PERIODICALt Prikladnaya,,,materiatika i mckhanika, 1960,, Vol, 24, No. 4,
pp,
TEXT! Under the assumption that the undulations of a ship be a stationary
random process, the author estimates tile accuracy of active and passive
roscopic stabilizers. For the mean quadratic deviation of the
Gy
stabilization angle the author gives a-pproximation formulas. A comparison
of these formulas shows that the activa gyroscopic stabilizers guarantee
a higher stabilization velocity:
There are 4 references: 2 Soviet, lfterican-and 1 German.
SUBMITTED: 11i'laY 3, 1960
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S/020/60/133/005/003/019
B019/BO54
AUTHOR: Roytenberg, Ya. N.
TITLE: On the Motions of a Gyroscopic Compass Under the Action of
Random Forces
PERIODICLL: Doklady Akademii nauk SSSR, 1960, Vol. 133, No. 5,
pp. 1045 - 1048
TEXT: The author investigates the motion of a gyroscopic compass on the
premise that the rolling of the ship is a steady random process which has
a "fractional-rational" spectral density. Proceeding from the equation of
motion (1) for a gyroscopic compass in a ballistic mercury container during
the rolling of the ship, the author obtains - after substituting (3) in
(1) - the system of differential equations (4) on the premise of a
straight-lined steered course. The equations obtained from (4) in first
approximationlean be written down in the form.of a matrix equation (5).
The differential equation system (13) is obtained from this matrix
equation for determining the expected values of the random processes; the
solutions (15) are obtained from system .(13) in first approximation. A
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On the Motions of a Gyroscopic Compass Under S/020/60/133/005/003/019
the Action of Random Forces B019/BO54
thorough discussion of (15) shows that in the rolling of the ship a
gyroscopic compass has an azimuthal deviation. This deviation can be
determined by formula (25), The determination of this deviation is ex-
plained by an example. There are 1 figure and 3 references: 2 Soviet and
1 British.
ASSOCIATION: Moskovskiy gosudarstvennyy universitet im. M. V. Lomonosova
(Moscow State University imeni M. V. Lomonosov)
PRESENTED: April 12, 19609 by A. N. Kolmogorov, Academician
SUBMITTBDi April 6, 196o
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26730
S/040/61/025/003/007/026
173 D208/D304
AUTHOR: Roytenberg, Ya.N. (Moscow)
TITLE: On certain indirect methods of obtaining information
about1the state of a controlled system in a phase
space
PERIODICAL: Akademiya nauk- SSSR. Otdeleniye tekhnicheskikh nauk.
Prikladnaya matematika, i mekhanika, v. 25, no. 3,
1961P 440 444
TEXT: For an op-timum system of automatic control, the control algo-
rithm (obtained,by the method of dynamic programming, or by the
maximum principle of L.S. Pontryagin (Ref 2: Optimallnyye protses-
sy regulirovaniy'a, Usp. matem. nauk, .1959: t. XIV, vyp. I, str. 3,)).,
is expressed in basic information concerning the state of the c6-vi-.,
-trol system in a phase space. In many cases it is diffi,cullu to ob-,
tain this information, as it is not always possible',to measure all
the phase coordinates. The article,deals with one of the possible
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262-130
S/040/61/025/(4 /,",07/026
On certain indirect methods D203/D304
methods of obtaining information indirectly in the.case of statio-
nary and non-stationary systems. Stationary systems: The motion of
a stationary controlled system may be written
(D) y,
k=1
where are the generalized coordinates of the system, x.(t) are
Yk 3 1 1
the external,forces acting on the system, f jk(D) is a polynomial in,
D, and D d/dt. (101) is transformed into
Yj F] (yII-0.... I Y1, Y,, ('11n Yn) _F
Bn (1,4)
j X~ (1)Y
2~12 X, (t) ++ -A'.
where the upper indices (m ... n) dnote the order of
k) (k
the derivative of Yk with respect to time, F is a linear ftmction
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26730
S/040/61/025/003/007/026
On certain iniirect methods D208/D304
(1.12) is soivt-d by operator methods to give
r
(1.28)
'Vjk (1) -_k (0) 11 A (i T).1-1 (T) d1r (j=
where (1.27)
Nj. (t)
//I,,jk(-t)// (1-19) ayA the elerne-itt3 of N(t) are knovn from
B.V. Bulgalcov (Ref. 3: Kolebanlya- Gostekhizdat, 1949P t. I,,str.
164)- A riew arbitrery orig)n i5 cbosep and the deviations S(t 1)0
S(t J S ( tr43 )of r.11c, phase coordinate z. referred to the new
orjgj-,i some- iiistant of- time 'r:i' " tr+1 are measured. S* is
h de v i avi ori of, the -new or igin referred to the original origin.

26730
S/040/61/025/003/007/026
On certain indirect methods ... -D208/D304
I (/,)I Zk (0) L
32)
n 11~+j tIL
T) xt (-r) A
where S(ti) S* + Z.(ti) (i 19 so-9 r + 1) (1.29), S(t
S(tu L11 (Ii = 19 ..., r) (1-30). Hence, from (1.28) and
(1.32), given the initial values of the phase coordinates z
k(0)'
the values z (t) at any subsequent instant of time t may be calcu-
lated. Non-stationary systems: In this case the equation of motion
is
(D) (2.1)
k=1
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M
26730
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On certain indirect methods D208/D304
where the coefficie 9nts of the polynomial f (D) are not constantsp
jk
but are themselves functions of time. Transforming as before gives
(2.2)
Z) + aik Zk Xj (t)
and the solution is
r
'Y N IVjj T) X1 (T) dT (2-3)
zj W I-j jk (t, 0) Zk (0) + YI Y
k=j
where (2.4)
Vif V, r) A* (-c)
J=
and the matrix N(t, -r) = 0(t) 9-1 (T), where 9(t) is the fundamental...
~matrix of the homogeneous matrix equation which may be obtained
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On certain indirect methods D208/D304
from (2.2) by putting X (t) 0. Similarly to (1-32), one obtains
n 1'+1
V
A k t0) - NA, (tv, 0) Zk (0 Lj I "I T) -q-1 (r) A(2-5)
I I (t,, -1) _Tj (T) dt OL r) (2.5)
+
Evaluation of the functions of (2.~) by previously established con-
ditions leads to a solution as before. The author observes that
electronic computers may be used to solve these equations. There
are 4 references: 3 Soviet-bloc and 1 non-Soviet-bloc. The referen-
ce to the English-language publication reads as follows: R. Bell-
man, Dynamic Programming, Princeton University Press, 1957.
SUBMITTED: February 27, 1961
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M5004;
6f fS
uroptl _~tha!;zr
oii; tbW
cess,--., h"-..A"~bil
wit
the zti - adyr! stat .C-'~- VAIU'~
-compass --li
n -0
the- -foll'6w
~-""Of'- the':'_Shij~.' a j~bsitidj
~'bf -:_tbdAiatiiral~.6dd-A-i'
_:'and.", conSequej
at

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~.tbe hatu
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I
1A
t
i/3

16 1-31
S/040/61/025/004/010/021
KOOO D274/D306
AUT HOR Roytenberg, Ya. 11. (Moscow)
TIME: On the theory of alternating systens
PEERIGDM~L: Pril-Ladnaya matematika i mckhanika, v. 25, no. 4,
1961, 691-704
TEXT: A system of difference equations is derived which describe
the alternating system. The. system is solved Lind the weighting
,function determined. A method is proposed for determining the
position of an alternating system in phase space, It is corvenient
to pass from alternating, systems- of differential equations to a sys-
tem of difference equations, which is derived as follows. During
the time interval n,-. + E ~ t < 11-r + the motion is described
by a system of differential equations with variable coefficients:
~j bjk(t)zk x,(t) Q
M
zj Q = -1,, r) arc, the phase coordinates, xj(t) are given exter-
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13 1
S/040/61/025/004/010/021
On the theory... D274/D306
nal forces, After some trans f antiations, one obtains, for E (0
whose interval corresponds to oneperiod of alternation), the sought-
Eor difference equations.
r
zv (t + T + YE, c"vk W Zk(t) XV (t) (v
k=1 (1.30)
,whe re r r-,
Lv-((n + UT + E,(n + 1)r)mj;,((n + 1)
avk* (11-r + 6
J=l kL=l
nr + -r1) x L)4k(n-r, +T 1, n"r + (1.15)
L and M being matrix weighting fiinctions. Further, the time depen-
dence Of z9(t) in the interval 0 < t < T is derived. The solution of
a system f difference equations with variable coefficients is un-
dertaken, The scalar system (1.30) is equivalent to the matrix
equation Z(t +'0 + a (t) z (t) = X W
(a 11 avk (t) 11 , X M Xv (t) (2.1)
The fundamental inatrix of the honogeneous matrix equation Z(t + Y), +
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S/040J61/025/004/010/021
On the theory... D274/D306
+ a(t)z(t) 0 is denoted,by 6(t). The matrix weighting function
N (t, JT) - E) (t) 9-1 (t + T) (2.11)
is introduced. The solution of (2.1) is
r
ZS (t) Nsk (t, 0) z'k* (t
+ N (t, jT^) X1, (t + jr - 'r (s (2.1-3)
t 7- s k
kmL JW I
it coincides in the interval 0< t,,~-V with the given matrix z*(t).
On deteimiining, weighting functiOn Nsk, it is noted that this weicrh-
tincf -function is constructed after solving the conjugated Gystem
of difference equations
r
Zk(t) + alk(t) ZI (t + -C 0 .(k r) (3.3)
For a fixed s one obtains
N sk (tl, j zl,~(tl -,511y +j,,(-) (k- r) (3.12)
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denotes the integral part of t/~r The relationships
n 1,2,
ZV Un + 1)--r) + avl,(nT) zk(rLr) (ra) V Ira.
(4.1)
avk(nT) MVju I')T t nT + T3) L (n-r + x 1, nT) (4.2)
r
X,(n It',,((n + + LPk(nT + Tl,~)x d~ +
nr k=I
+
(n + ]Jr,~) s~,(t-) d~ (4.3)
M +Tl 41
are set up; theseare obtained by setting E 0 in foregoing differ-
ence equations; (thus, Eq. (4.2) is ontaincd front Eq. (1.15). Eq.
(4.3), which are valid f or integral n only, are difference equations
.with discrete argument. Hence, the solutions to Eq. (4.1) determine
a sequence of phase coordiuu.-Ltes zV at discrete points which are the
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limit points of alternation periods, i.e. at moments t We (n
2,...). These solutions can be obtained by the method exposed
above, by replacing Eq. (4.0 by a system of difference equations
(v 0(4~4)
ZV (t +,r) + aYko W zk (t) Xvo (t)
where avko(t) and XVo(t) are step functions:. (see Eq. (1.30)). The
solution to Eq. (4.4) for values of t which arc multiples of'r, is.
z r +
E Nvk(I~Z_,O) zk(O) Nv k (1$ Tj'r) Xk ( i ' V - 'V
k=1 k=l 3.1
(v I,-, r) (4.5)
With regard to the alternating systems of linear differential equa-
tions with constant coefficients, it is found that the coefficients
avl-(t) of Eq. (1.30) are periodic functions of time with period
equal to T . Eq. (4.1) will become equations witti constant coeffi-
cients, Expressions are also given Eor the form of the functions
X,(nT) (which enter the right-hand side of Eq. (4.1)), and for the
.time dependence of the phase coordinates in the interval -L~T