SCIENTIFIC ABSTRACT SIMANOV, S.N. - SHIMANOVICH, S.V.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R001549510007-9
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: August 23, 2000
• Sequence Number: 7
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

S 14 1 tll) 1.V
SUBJECT VSSR/MATHEMATICS/Differential equations CARD 113 PG - 501
AUTHOR SIMANOV S.N.
TITLE on the determination of the characteristic exponents of a linear
system of differential equations with periodic coefficients.
PERIODICAL Doklady kkad.Nauk 109, 1102-1105 (1956
reviewed 1/1957
Let be given the system
(1) (A +IAF(t, r))X.
dt
Here x = (xl,...,x.); A - Va ji 11 (j,i-1,...,n), aji constants; F(t,/,) - hfjin
(j,i-1 .... n), fji continuous functions being 21r-periodic in t, which are
analytic in ~A for 0 < I/A I -~. Let the equation
(2) 1A - X?d- 0
have k46n roots which are eitner equal N1 or differ from A, by � N- C-11
N integral. Let the system
(3) dx (A. - 9 'AI)X
Tt- -
admit m.4,k solution groups

Doklady Akad.Nauk log, 1102-1105 (1956) CARD 2/3
X. cpi , x (i) Lpit +
2
X(i) tvi-I ... +%0 ') I-I)t + Vi)
Ii (Vi-I)I + IFZ
PG - 501
where If are periodic vectors with period 2,M and )i are integers which
satisfy the condition V 1 + V2 + "' +Vm - k' Let 'q,,j (J-1,...,m) be
linearly independent periodic solutions of the system conjugated to M-
Then
-Ifj 0 ...... Ji-11 i,j-lp2,...,M)
and
YJ) . Sij.
i
T
The characteristic exponent, which for/A- 0 changes to the root )L,, be
^I + o( ( 16,) (o4(0) - 0). Putting

Doklady Akad.Nauk 109, 1102-1105 (1956) CARD 3/3 FG - 501
ME
n
b z f (t,O)\y 'f8i dt
I j - 2-M f 894-1 06 Sj
0
CX-*4 OLV-4
B -1b ij I). momentg,the course Of i
4 - 2 06 (J clx:L ( t) at tl,.e giye, For this :reason
Re Aj = 0 determine' `~~~ t be knOvM - t of the trajectory
In order t dt t) Mus (t as elemen to
,rY&1 (t - M'
,ceding int to point "1 0. According an
on the pre not consider tt e
author does V b~ ), - U ( ) there corresponds
the jece %.(t 1. the SY13tem
but the P (RefN then to
'rasovskiY ordinary 61161tem
y
e,juivalent c,
d%t Axt + J(xt
--t~t a certain function space, (Ref -
which is co-asidered in -formation due to A. y. Lya?
the aid of a 'trans
With to the form
(1.8) is brought
Card 2/3
SJ110401601024 3 ~2120
C ill/ C /0 /05/020
Delay ability in the Critical C 333
ase Of a Zero Root for Systeos ~,ith
(4-1) Yl(yV z
dt it(
---dt Azlt( C7 ) + Z
Now Y I and Z are Ox2and i (Yo z, t
M I I ed into the power series ,z~ . .....
gly . ..... at a certain Point The author states t
the solution x = 0 of an
on the fact whether m 0.1) be asy' hat whether
and mPtOtically stable or unstable depends
Positive or neeative. lie 4e. an example,
gm are even or odd and -hether g, 91 are
N. G. Chetaye,, J. G. hialkin and 0. V I
There are 10 references: 8 Soviet, Kal3ankov are :-mentioned.
SUBLUTTED: r 1959 1 American and I Hungarian.
,ecember 21,
Card 313

81713
3/02C/60/133/01/09/069
C 111/C 333
AUTHOR: Shimanov, S. N,
TITL 70 37", cillationsvof ruasilinear Systems With Time
E: Almost Per-7 Os
Lag in the Case of Legeneration
PERIODICAL: Doklady Akademii nauk SSSR,1960,Vol.1173, Nn.,I, PP~36-39
T-.IXT: The author considers the system
dx(t)
S X(t + d-iL( c) ) + F-F(t, x(t +N"-))
dt T
,t(-- an n-dimensional matrix;
where x(t) is an n-dimensional vector, d-
ti d1j,j(.~-)11 a Stieltjes measure, fl j(,15-) functions of bounded varia-
tion. Let the characteristic equation
dit 0'
(2) + E ~' + e 0
possess m critical roots \m, while the real parts of the other
roots are smuller than -2 a,( X>O). The functionals
F(t), X(t + F )are defined in D(H) x(,3, )I S H on
the plecewise continuous functions x(& --C:S 3- S_ O)and it is:
Card 1/4

-Y
S
/Adl N)'I 333/01/09/069
C 111/ C 333
Almost Periodic Oscillations of Quasilinear Systems With Time Lag in
rho Case of Degeneration
I F(t, x( V~), 'p-kF x in D E.
k(tl
2~) F(t, x(~')) are finNeo trigonometric sum3, if one substitutes in
them a piecewise continuous function x D(H) in 45-which for its
p~-rt is a finite trigonometric sum WitA res ect to t, the frequencies
of which do not depend on 3-) F(t, X(,E~TE ) Batisfy the Lipschitz
conditions with respect to the variable;3 x in D(11) ~ DE *
To every critical root A. there corresponds a periodic solution
(t) of' (1) with F-- 0. ihen the system' conjuFate to (1) possesses
j),-Iriudic solutions 'Vj(t) which correspond to the m critical roots
/\ j,
Theorew 1-. Let XOM Mi q-1(t) + - + Mmcfm(t) be an almost periodic
solution of (1) for 0, If the parameter M M0 satisfy the
equations
Card 2/4

SIO 2016 0/13 310181YNIO 6 9
C Ill/ C 333
Almost Periodic Oscillations of Quasilinear Systems With Time Lag in the
Case of Degeneration t
ik O(t +b- O(t +C-))
(5) P (Mil.." M lim -1
M t t,x1 xn
jr
- ry (t)dt
and if tile equation
(6) d 0
0 ij
possesses no roots with vanishing real parts, then for sufficiently small
E_ (1) admits an almost periodic solution x A (t6E ) whi ,h for E - 0
transforms into the generating solution xo(t, M ).
The dij occurring in the theorem are defined by the scalar product
di j t + (t
Card 3/4

81713
S102016011331011091069
C 111/ G 333
Almost Periodic Oscillations of Quasilinear Systems With Time Lag in
The Case of Degeneration
Theorem 2 states that, if all noncritical. roots of (2) and all roots
Qf (6) possess negative real parts, then the almost periodic solution
(t,,'-- ) of theorem 1 is asymptotically stable for sufficiently Lnall
, If only one of the above roots has F- positive real part, then
X' (t.F-) is unstable.
T',-,e author mentions 11. It'. Bogolyubov, J, G. Malkin, G. J. 31ryuk and
N'. I'M Krylov,
There are 7 Soviet references.
ASSOCIATION: Urallskiy gosudarstvennyy un:Lversitet imeni A, V. Gor1kogo
(Ural State University imenitt. M. Gor1ki
PRLSENTED: March 9, 196o, by IT. 16'. Bogolyubov, Academician
LUBYITTED: March 8, 1960
Card 4/4

YEIk.'OLT~; L. *~I- !jt,.d A. D.
"~Aqbility amd of !iy-:t~-,,13 with time lag."
Paper 1,resented at the LrAl. Smposiun on %onllnear Vibrati:ms, Kiev, USSA,
9-19 Sep 61
iiesparch Technical Physic-, Low Tnstitute of the Ukrainian S.Sit,
Acad,~-y 3f Ecipnces, nar),uv

89582
S/14 611000100110061006
14-340 C1 1 1YC222
AUTHORs Shimanov, S.N.
TITLEs On the stability in the critical case of a zero root for
systems with an after-effect (singular case)
PERIODICAL% Izvestiya vysshikh uchebnykh zavedeniy. Slatematika, no. 1,
1961, 152-162
TEXTs The present paper is a continuation of an earlier paper of the
author (Ref. I s Ob ustoychivosti v k iti h skom slucbaye odnogo nulevago
kornya dlya sistem s posledeystviyem On the stabilit in the critical
case of a zero root for systems with an after-effectlyPMM v. 24, no- 3,
447-457,1960) and investigates the singular case. V/
The author considers the system
dx i(t) n x (t + 9-)d-Z j (-~-) i- X, (x,(t + .,n) (1.1)
dt L
J-1
where the integrals are understood in the sense of Stieltje S and the
fun.-tionals X i defined on the piecewise continuous functions x i01)
Card 1/9

89582
S/140J61/000/001/006/006
On the stability in the :critical case .... C111/C222
- ~":5 0 , satisfy the conditions
Ix (x") Xi(xl)l < Llix" V11 , L - LJ(1J X-1 + 0XV11 (1.2)
11 X(0)11 supox, (-9-)l , - - -, xn(D-) I ) f or -T 4 &- ~6 0
while L,, -C, are positive numbers. Let the equation x
E X + d 0 (1-3)
have a vanishing root A 0 , while for j> 2 it holds Re 2 W. (-',> 0
Let x, (t +,5) , - *V 1:!~- -9- !!~ 0 serve as an element of the trajectory xi(t) of
(1.1). In the fun-:~tional space. Bfxio-) I I - V!!!5 '&- !5- 0 to (1 .1 ) there
corresponds the ordinary system (cf. Ref. 5% W.N. Krasovskiy, Nekotorya
zadach'L teorii ustoychivosti dvizheniya. [Some problems of the theory of
stability of the motion] Fizmatgiz, 1959).
Card 219

89582
S/140J61/000/001/006/006
On the stability in the critical case ..... C111/C222
dQ (2) - Ax, (0) + R (xi (0)), (1.4)
dt
dv, (8) < 0 < 0),
d8
A? (0, 0
S xj (0) dVj (0) (0.0).
R (9 0 0