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December 31, 1967
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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