Development of the Bogolyubov-Mitropolsky method of integral varieties
A survey is presented of some results relating to the development of the Bogolyubov-Mitropol'skii method of integral manifolds.
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| Date: | 1992 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | Russian |
| Published: |
Institute of Mathematics, NAS of Ukraine
1992
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2318 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508244134330368 |
|---|---|
| author | Lykova , О. B. Лыкова , О. Б. |
| author_facet | Lykova , О. B. Лыкова , О. Б. |
| author_sort | Lykova , О. B. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-09-14T12:03:56Z |
| description | A survey is presented of some results relating to the development of the Bogolyubov-Mitropol'skii method of integral manifolds. |
| first_indexed | 2026-03-24T02:22:07Z |
| format | Article |
| fulltext |
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MOTpeHIIOH B [28] aBTOHOMI-1011 CHCTe Mbl l U1CjXjJepe11uaaJibl!bIX ypaB1-1e1m{1
dxldt = Ax+ f (x , y), dyldt = By+ g (x, y), ( 10)
npaBbl e lJ3CTH K OTOj)OH onpe,rre.~Cllhl II 11enpepb!BHbl I la MHmKeCTBC [Rm X V,
rne V - 1reI<oTopa51 06.n acTh npoc-rpa 11 crna 11{'', coJ1.e p)lrnu.1,an TOlfKY !J = 0;
A l1 B - fIOCTOHll ll bfe l\13TPH I!bl n opH l ll(013 Ill X 1/ l H 11 X n COOTBeTCTBeHI-10.
non np 1-16JIH:>KeI-I HblM C I-IeBH3 KO H (0, b (x)) HI-ITerpa,1bHbll\ l MIIOr oo6pa3 HCM CH·
CT e Mbf (10) 6y,ueM llOHHM3Tb mnerp a JJ b I-IOe MHoroo6pa3 11e CHCTeMbl
dxldt =Ax + f (x , y), dyldt = By+ g (x, y) + b (x ).
06o3Ha'1HM ero
(1 1)
( 12)
ECJJH BeKTOp-Q)YHK l[H H (J) (x) 1-1e n pep b!BHO nmjxj)ep e m rn p yeMa, TO OHa HBJJ HeTCH
pemeH HCM iJ.HCjxj)epeHUH3.7JbHOro y p aBHeI-IH H B tfaCTHblX npOH3B0 iJ.HblX BH,ll.a
a(J)/ox (Ax + f (x, ©)) = B© + g (x , (1) ) + b (x). ( 13)
P emeHHe ypaBHeHHH (1 3) HaXO,ll.HTCH npH CJJ eiJ.y lOLI !H X n p enn onomeHHHX:
3S ISSN 0041 -6053. Y,cp. Mar. ~ypn., 1992, T. 44 .. It& l
l. BeKTOp-cpyHKl.(lHI f (x, y), g (x, y) .norryrna10T aCHMITT0TH 1IeCKHe pa3JIO-
:>KeHH51 BH)1a
f (x , y) = f 2 (x, y) + ... + I,, (x, y) + o (II (x , y) Ilk),
g(x , y) = g ~ (x. y) + ... + g11 (x , y) + o (li (x, y) II "),
II (x, y) II --+ 0, r .ne /;, g; - cjiopMbI i-11 n e ne1m c Be1nopHb1M11 1rn::icjx j:m1me1-1TaM H:.
2. Co6cTnem11,1e :rnat1 e mrn A; (A) Ma-rp11u1,1 A JI c o 6crne1-11-1b1 e 3 1-1 a 4 e1urn
Aj (B) MaTp11Ubl B ll0J.llJIIl !e 11b1 ycJJOBIII0 p),1 (A) + ... + /JmAm (A) =I= l, j (Bl,
j = 1, ... , n , rn,e /Ji, ... , p,,, - ueJ1b1e 11eoTp111.taTeJ1bHb1e <n1c.na , CBH 3 a1I11b1e co
on10we 1111 e M Pt + .... -!- Pm = i, i 0~ 1, ... , k.
TTpe,ucTaB11M p e u1e1-rne ypaB1-1eHHH ( 13) B B11,11.e
k
<J) (x) = Y <D; (x),
.;...J
( 14
i= 2
r,ue ©; (x) = colon [([); 1 (x), .. . , CD; " (x)I. CDu (x) - <jJOpMbl i-11 CTenerrn npn
KaM;ioM j = 1, .. . , n. lloc: 1e no,UCTa11omrn ero n ypaBHe1rne (13) H 11p1-1paB1u1Ba
HHH Cj)OpM O)lHl·IaKOBOi"l CTenem1 B .lleBOtl H npaBOtl 1-IaCTII }.10 nop51J.lK3 fl, llOJ1 yqa
eM }.l.rrn 011pe}leJ1e mrn <.l\ yparrne mrn
o<D / ax Ax = B cJ\ + u; (x), i = 2, . . . , k,
r,ne cpopMa U; H e 3aBHCH T OT cj)O!)Mbl <Dj n pH j ;;?, i.
IlpHp3B[-l!IBa51 0CTaBIJIHeCH LJJ!eI-Ibl, l lclX0)lHM Bbl!)cl)l{e JIH e )1.~51 HeB>l3 1{H
b (x) = R,. icl<Dl clxf (x. CD)- g (x . <ll)I,
( I 5)
( 16)
B l<OTO[)OM Rn [ · l - OCT3T0K acHMl1TOTHLJeCKOro pa3JI0)Kel-!H>I yJ<a 33 HHOtl B.
CK061<ax cjJy1-1K 1i 1m.
,Uo1rn 3 a1-1a cJ1e; Ly 101.ua51 TeopeMa.
T e O p e M a 2. nycmb onu-I.0ClltrlC11bH.O CUC/llC.Mbl ( I 0) BbtnO,iwuomCR y cA0-
6U5l I, 2. TociJa &ma CllCll leAta UMeCl/l CChtH.Cl116CH.H.OC npu6AltJICCH.H.OC C HC6Jl3KOU
(0, b) UHl11CcpGAbH.OC MH.Oc006pa3UC euoa (12), (J KoInopo.,1t 6eKmop-cpyH.KU,U5l (])
npeocma6/.lM{l (1 6UOe ClCU.M/UnOtnll'LCCIWcO pa3AO.JICCH.U5l no KOOfJOUHama;1t ( 14).
<PyHK/.{ll5l b (x ) onpecJeAflCl/lCfl c/JOf),ltYAOU ( 16) ' a qJO/hlt!J CD; (J pa3,10JICCH.UU ( 14)
MO.JICH.O H.ai'unu MCtnOUOM HeonpeOeAeH.H.blX K09cpcpuqueH.tn06 U3 ypa8H.CH.U5l ( 15)
npu KG.JtciJo.11. i.
3aMeTHM , 1-IT0 YC.'I0BHe 2 Bbir!0Jl[-151eTC51 B Kcl)l(Jl0M H3 CJ1e;iy1ou.1,11x Tpex
cJ1ytl3eB a) Re'J...,1(11) = 0, R e Aj(B)=!=0; 6) Re), 9 (A) ¾ 0, Re'J... j (B) > O;
B) Re 'J... ,1 (A) >, 0. R e , . 1 ( B) < 0. npB6mrnzem1bte 11 1-ITerpam,Hbie M11oroo6pa:-i HSE
B y1<a3allHblX CJ1yc1aH X Ha 3 bTBaIOTCH C00TBeTCTBeJ-11-10 u e HT!)3JlbHbl,\1 np116JI H.i1<e H
Hbl M IHITerpaJlbHbl M MHoroo6 pa 3 HeM , 11e1np - yc-ro(14 J-1 Bb!M np116JJI1 ✓1<el!Hb!M l[[[
Terpam,llb!M M IIOf006pa3 HeM , ueHTp - HeycTOHCIHBblM 11p1-16J1H.ii<eI-1Hb!M HHTerpa JJb
Hbl M MIIOl'006pa:~ 11 e '\1. 1·11-rre rpa.r1bHbie MI-IOroo6pa3H5l y1<a 3 aHHblX TH H0I3 l3 pa~rn:ax
TeopeM cy1.uecTnonaaHH fJ1>u111 wiyc1eHbI AJJ . KeJurn [5 ].
O6oc1-10Ba1rne np1rnene1-rnoro aJ1ropHTMa CBO]l HTCH I( l lOl<a3aTeJlbCTBY -reo
p e ,\1bl cy1.uecTB0Ba!IH>I Hl-1TerpaJ1bH0f0 MHOroo6pa3 H51 HCXO}.ll·IOi-'l CHCTeMbl 11 I( no
J1ylJe l·ll1 l0 Olle HJrn norpeLUH0CTH noc-rpoet·ll-I0l'0 npH0JIH A<eT-!1·!01"0 111·rre rpaJ1bH 0 fO
MHOro6pa3 H 51.
nocl<OJ!bl(Y np11Me1rneMbl l1 B Te0pHH 111nerpa.'IbHblX MI-IOf006pa3HH MeT0.ll
110Kcl33TeJ1bCTBa TeopeM cy11.1ecTB0Ba]JH H 0CH0B3H Ha TI])HH l.l.Hlle OKHMaI0UlHX
OT06pa)I<e HHH, KOTOpbl t l Tpe6yeT orpa l-lHl!e liH0CTI I H mrnu11-meB0CTll n paBbl X LJ ac
Te(1 C ll0CTaTQL[HQ M3,1b!MH K0HCTaHTaMH, TO LJT06b! He 1-1a,11aran, CT0J1b )l(eCTIUle
orpal·IHLJe !IH51 [·l a HCX0l lllYI0 CHCTeMy, 3anai-IHy10 BO BCeM npocTpaHCTBe , BMeCTO
fJ1063 JlbHOro 111-ITerpaJlbH0fO M l!Oroo6pa3HH (KaK B TeopeMaX 6oromo60Ba-
M1nponoJJbCKOro) ~'l0Ka3b!Ba!0T cyrueCTB0Bal·IHe J10 [<cl .1bH0f0 HIITerpaJJbH0rO,
MHOroo6pa3 HH. 3aMeTHM, L!T0 X0T51 fl0H>ITHe J10KaJ1bH0f0 llHTerpaJibH0f0 MH0ro
o6pa3 HH tj)aKTHl!eCKH C0)leJ))l<HTC51 B -rp yJLaX A. 1'v1. Jl>1 n y1-1ona , 0MJa K 0 o n pe,!1.eJie
HHe .nOJ<aJlbH0 f 0 Hl!TerpaJ1bJ-IOf0 MHOroo6pa3 HH, a TaJOKe TeopeMbl cymecTBOBa -
ISSN 0041-6053. YKp. M GT. :JtCypH.., 1992, T, 44, M 1
HH5l .'10!{3.JibI-lblX HIITerpa.'IbllbI X Mirnroo6pa3HH n.rrn ne.'IHHeHI-IblX )JJl(jxj1ep e 11u11-
3Jlbl-lblX ypaBT-IellHH, 6J1H3KHX Ii: HI-ITerp11py1011tHMCH (!<al( B 0KpeCTH0CTH CTaTH
qec,rnro peweI-llrn , TaK H B oKpecTH0CTH nnynapaMe-rp HLieCKoro ce1v1eikTBa Jie
p110,z1_1,1qec1rnx pewem1 I[ C00TBeTCTBY!OLU,ero l!eB03MyLU,eI-IH0ro ypaBIIelll l H) ,
Bllefll3blC 6wm npmie,uellbl B pa60 TaX [15, 16, 30-32].
non JIOKaJlbHblM ll flTerpa.'I bl-lblM MHOroo6pa3HeM CHCTeMbl (10) 6yneM ll0I-!H
M3Tb Ml-IOroo6pa3He BII.ll_a
./4l":y= cpJl(x), llx\l < r, r>0,
06.rianarou Lee Te.M Cl30HCTBOM , LITQ JJI06a 5l HI-nerpaJJbl-Ia5l KprrnaH CHCTeMbl (] 0)'
H.MeroutaH c M" xo-rH 6hr on1-1y 06uw10 T0'IKY, np1111anJ1e)KHT /4L 110 Tex nop,
noKa llx (t) II < r.
OnirnKo TI0Cl(0.'l bl(Y Jl0K3.'lbI-IOe H!ITerpaJJbI-IOe MI-10roo6pa3 1-1 e paccMaTpHBa
eTCH 11,151 llxll < r, a Hcxo;i,1rnfl c11c-reMa 3ana 1ia 1vrn x E IR.111 , TO o6bILJJIO 11cno.~b3Y-
10T H3Becmyro npmienypy np0l l0JDl(Cllll fl c\JYlll<Ill•Jr1 1331.
l lJrn c11cTeMhl (10) B 128 1 11m,a3aua -reopeMa , tj'opMy.'l11py1ou_ic1H yc.11ou11H,
np11 BblrI0JJHeHHH K0T0j)bl X 3Ta ClI C'reMa HMeeT Jl 0l<aJ]bH0e ll!-1Terpa,1b1-1oe MHOroo6-
pa311e, a Ta1O1<e npH6JIH)J<e !ll-lOe C HeBH3KO~I (0 , b) .10!«1,lbl-IOe 1-II!Terp;J,ilbfl0e MH0-
roo6pa3 ae, 3a;i,aBaeMoe BeKTop-c\Jy1-11m1161 !J = (J)J, (x). flpH 3 Tm1 o n em::a no
rpeumocTH nocTpoe111-1oro np116JJH)l(eH1-10ro HHTerpaJJhH0ro 1111-1oroo6pa3 HH 011-
pe.n.e.1HeTC51 1-1epaBeHCTB0M
II (PJ, (x)- (DJ, !x) II ~ (1 - q)- 1 Yo sup 1; b (x) /!, Yo> 0,
X
B K0T0p0M b (x) Bblpa,1<aeTCH rrocp encTB0M (jJOpMyJlbl (16), a ll0CTOH HHa5l q < 1-
nocpencTB0M I{OI-ICTaHT, xapaKTepH3YI0IUHX CBoikn,a npaBb! X 1-1acTei"1 ,!CXOllIIOil
CHC'reMbl .
Cnoco6b! l1 0CTj)0eJ-!l-rn np1-16.!JH)l(eHHhl X H!-ITerpaJJbHbL\ !1.1l!Orou6pa 3 1-1i'1
MeT0ll0M aCHMITT0 THLJ€Cl,H ,\ pa3JJO)KeJ-IH ~i no J-;00jl l lHllc!T3M llJI H HeaBT0II0M -
1-lbl X CHCTeM paccMaip 1rnaJ!HCb B [26, 27]. B q acTIIOCTH, nmi CHCTeMbl
dxldt = A(t)x + B(t;y + f(t. x. y), dy idt = C(t)y + g(t,x,y), (17)
r.n.e x - m - sewrop, y- n-BeKTop, t E IR , 1101-::a :ia 11a c ,11e:i y 1o utaH reopeMa.
T e O p e M a 3. fl ljClllb Onll-lOCU/ne/lbHO CUClllea-\lbl ( ] 7) 8bllW/lHf/10nlCfl
!JC,WBUfl:
I. BeKmop -cf1yw,,11uu l (t, x , y), R (t, x , y) HenpepbtBHbL u oonycKmo,n ac1.um-
11wmu<1ec,-;ue f)WJROJIC Cl-lllfl k-20 nopHrJ,-;o no ,-;oopUUHO/lllUl X , !J C Henpepb/81-lb/;l/ll U
02paHU'leHl·lt,L1l/l no i IW :icpcpuu,uet-una.Hu, npu &llW,H pa3.10JtCeHUfl Hll'il/HOIOll!Cfl
'{JleHOA1.U He HU.7/Ce B//W/7020 nopflOKa .
2. c/JyHOOA'lCl-ll/WllbHWl .Mampui4a X (t, s) u cpyHKU,Ufl I'puHa G (t, s) coom
eemcmBeHHO ypaeHeHur1
dxldt = A (t J x. dyldt = C (t) y
II X (t, s) II ,s;;: K eXU- sl, II G (C 5) 11 ,s;;: Ne- VJl-sJ_ V > 0, l(x < '\I.
T ozaa CUC/"IWAW ( 17) UAteem npu6AU.7/CCH HOC U!-lrlle2pallb!-loe AH-W2006pa3ue
Alnp. : y = (D (t , x) = © 2 (t, X ) + ... + CJ)h (t, x ), t E ff<.
C HCBfl3KOU (0, b (t, x)), zoe c/JopAtbl (J)i (t, x ) C HenpepbtBNblAW u 02paHU'teHHbU/U no
t K08cj)(jJU U, Ue!-lf!WA1U onpeOC/lfl/0/1lCfl paeeHC/118GAW
<1\ (t, x) = .\ G (t, s) u ;(s, X (s, t) x) ds,
IR
6 K01110pblX U; (t, X) - U36CC/111-lb/1! cj_l!JNKl{UU;
b (t , X) = Rli [o<Dlox (B (t) <D + l (t, X, CD)) - g (t, X, ©)];
eoe Rk I. ] - OC/11aJ110K acu.1-ttU/l0/1lU'leGK020 pa3/lOJICCHUfl yKa3GHHOU B CK06KaX
<pyHK/.11Ull.
C noco6 nocTpoemrn m-nerpaJJbHhI X MI-10roo6pa3m1 1-1a oc11onc \r(.,...;:, _, ,. , i.:-Vlll
T0TH'leCKH X pa3JJO)l(eHHH no K00p.ll_H HaTaM pa3Bl-!T Ta !OI<e B pa60Tl.:_i;:i4J.
38 ISSN 0041-6053. JIK.p. J.taT , 3/CYPH., 1992, T. 44, N; I
Cyn(eCTBeJ-II-Ioe p a3BHTHe KOHCTP Y KTHBHOH CTOpOH bl M eTOJJ,a mne rp a Ji b l·l b lX
MHoroo6pa3 Hi-'I CB513a Ho c p a3pa6o rK011 B r 17, 22 , 23 , 26, 35-38] cnoc o6oB no
CTpoeHHH np116 mrn<e illlblX HHre ,-pa Ji b llbl X M IIOr oo6p a3Hi'.i Ila OCI-IOBe MeTO)Ja a CHM
flTOHl'1CCKII X p a:rn o;,Ke llHll no napaMe -rp y. llp,rneneM BHcllJaJie 11 e 1<o r opb1e CBCJJ.e
l!l l5J IB -re o p 1rn acHM flTOTH'ICC KH X p aJ,il O)Ke 1rni"1 1391.
AcHM llTOT!llJCC K OH nOCJICJJ.OBa Te.ilbJ-IOCTb lO Ha3bIBaeTCH rIOC JJC)JOBaTeJlbl-lOCT b
<\)Y IIKUHH go (e) , g l ( ~: ), ... , gh (e ), . . . • onpeJJ.e.rr e irnbTX B OK p eCTI·IOCTH e = 0, ,'l-151
u ( e)
KOTo pofi BblfJOJI I-IHCTCH COOTHOll.le 1me Jim "'"+1 = 0 )]} TH Jlfoooro Ii. CyMMa
>:➔O gh (e)
" (D" (E) = )..., a;g; (e), npe1I.cTaoJIHiol.llm1 .rrH11 e i1Hy10 KOM61niau1110 n e pBbIX I? tJJie
~
i = O
HOB j (a m -10{1 rIOCJIC)l013cITeJib!-I OCTH. I-lcl3 bl BclCTCH a CHMIITOTl1lJe c1H!M pa3.70}Ke rrneM
<j)yHK LUIH (j) (e) C T0 1HI OCTblO g ,, (e) OTHOCHTCJlb!-10 31'011 rroc.1eno BaTeJi b l·IOCTH,
rp (e) - <J> (e)
e CJm Jim '' = 0 K a )l<JJ.oe c .naraei\-10e 11a1-11-10r, cyMMbI Ha3bIBaeTCH
c:➔O gh (e)
~w e 1-10M ac 11Mn-ro Tw1ecKoro p a3; 1mKe 1-rnH, •<aiK)laH nocTo H1-11-1aH a; - 1m 3cj)clmiu1-
e 1-IT0M aC HM l1TOTHLieC I<Oro p a3JIO>KC lll·IH, a pa3 I-IOCTb Rh [ cp j = cp - (D - OCTaT
K OM :noro a CHMfffOTWieC l{OfO pa ::wmKeJ-11-IH.
"
TTy CTb c yMMa Cl)" (x, e) = )~ a; (X) g; ( e) HBJIHCTC51 aC HMJ"JTOTllLJCC l(HM p a3-
i = 0
.JIO>KeJ-IHeM cj:>yI-IKUHH cp (x, e) rrp 11 Katf<JJ.OM x ED , rne D - npo11300:11, 110e tv11-ro
; Ke CTBO llairnoe ac11MnTOT11 11ec Ko e pa3JlmKe rrne pam10Mep1-ro . ecm1 npeJJ.e.rr
). cp(x , e) - W,, (x, e) 0 " D l3
1111 ( ) = paBHOMepHbll-1 OTf-!OCHTCJluI-10 X E . l1f)OTHl3110M
e➔O gh e
CJI Y'WC a cJ-!MIITOTHlJCCl(Oe pa3Jl0tf(CJ-IHe llcl :=lb!BaeTCH He paBI-IOMepI-ll,IM
rl p11 noCTpoeHHH a cl!MllTOTlflI CC l<HX pa:v1mKe1-mi'1 pewe,-rnii 11ml)( r e p e muraJib
l-lblX y p aB!I Clll1J,! B K alJe CTBe aCHMl1TOTH'I CCJ(O h 110CJICl\OBaTeJibl·IOCTl1 UOLI'-IHO 6e
PYT c-re n e 1111 y 10 n ocJ1e 1.1.0BaTe Jib!IOCTb, T. e. 110.1ara10T g" (e) = e" .
vbJIOi KHM pa 3 B1H y 10 13 pa6oTax [26, 35- 38] c x e My rroCTpoe 1rnH rrp116JJH)l<eI-I-
1-1oro l!IITerp a .ilbHOro Mlloroo6 pa:-H!H MeTOilOM a CHMf!TOTHlJeCI<HX p a:-;JIOi KCI-IH(I no
n a paMe-rpy Ha npHMepe noCTpoe mrn u e 1-1TpaJ1b11oro np116Jim1<e1-11-wro 111-rrerpa.r1b
noro MI-IOroo6pa3HH aBTOHOMJ-IOi-i CHCTCMbl
dxldt =Ax + X (x. tJ, e). dy!dt = By + Y (x, y, e), (18)
.3a,ua1rnofi Ha np011 3Benemm IR 111 X IR" np11 1<a,KJIOM e E [O, e0 l, npu 3TOM A,
B - 110CTOHJ·Il·lble MaTpHllbl.
CrpOHTCH rrp116mnKe1-rnoe HJITerpaJibl·IOe Ml·!Oroo6pa3 1-1e CHCTeMbf ( 18) B Bl-!j(e
TIOJIH HOM a 1?-1-i CTene JIH OTIIOC IHCcl bHO na paMeTpa e 11 JJ.OKa3bIBaeTC51, LJTO flOCTpO
€J-ll·IOe npH6JlH)l(CHI-lOe HI-ITerpaJ1bIIOe Ml-10f006pa3 He Hl3JI51CTCH p a BI-IOMepHblM
'1CHMl1TOTHlJCCKHM pa3 JIOiKCHHCM u e r-npa.~bHOro MHOroo6pa3 HH C!!CTCMbl (18):
M:y=cp' (x,ei. x f IR"'. cp(0,e)=0.
TipennoJiaraeTCH, '-!TO OTHOCHTCJ!bl-10 CHCTe M bl (18) BbinOJIHHIOTC5l c.1e11y1ou111e
yCJIOBHH:
I. Be1nop-cpyHKUHH X (x, y, e), Y (x, y, e) Herrpepb!BHbI Ha M IIO,hecrne
:{Rlll X /Rn.
2. CooTBeTCTBy1omaH (I 8) 11opm1<I1a101llaH CHCTeMa (e = 0) HMee,- HHTe
rpa JibHOe MJ-!Oroo6pa3 He Mo: y = (p (X), XE IR"' , cp (0) = 0.
;3 _ Brnrnp- cj)yHI<UHH X (x, q>+ez, e), Y (x, cp +ez, e.) 1.1.onycKaroT acHMrrTo
TH'-leCKHe pa3 JIO)KCHHH rro e nopH!lKa e k , B J{OTOpblX I<03C!xllHUHCHTbl X1, Yi He
33 BIICHT OT Z .
C orJ1acuo MeToJiy B03Myl.lle1rnt1 111ue-rcH npH6Jm,Ke1-me K qi (x). llJIH 3Toro
n ocpeJJ.CTBOM 3aMeHbl y = cp (x) + z CHCTeMa (18) npHBOJJ:HTCH K BHJJ:Y
dxldt =Ax+ f (x, eZ, e), dzldt = Bz + g(x, eZ, e),
JSSN 0041-6053. :J,cp. MlZT. xypn., 1992, T. 44, J-.& l
(19)
39
,..,_, k ,..,_,,..,_, k _,
npH 3T0M f (x, e<D, e) = ~ e1f; (x) + R,, [fl, g (x, e©, e) = ~ eig; (x) + Rk [g].
i= O i = O
3a.uaqa CB0,UHTCH K TI0CTpoeI-IHI0 C00TBeTCTny1ow.ero aCHMTIT0THLieCK0r0 pa3-
JI0)KeHHH, 3ana1ow.ero npH6JIH)l<e1moe HeHTpaJibHoe tt1nerpaJib1-IOe MI·IOroo6pa3tte
CHCTeMbI ( 19) . LloKa3bIBae-rcH, LtT0 3Ta 3anaqa 3KBHI3aJ1e11-rHa 3aJlaLie 06 orpa1rn
qem10M HI-rrerpaJi bH0M MI-IOroo6pa3HH IIOJJypacnaBweikH CHCTeMbl BH)],a
dxl dt = Ax + f O (x),
( V; (x) = g; (x) - L
i+ r= i
r=,t;O
dzl dt = Bz + v; (:t), i = 0, I , ... , k
orpi f (r =I= 0, i =I= j),
OX r
V;(O)=O).
LloKa3ai-ra cJie.uyrow.aH TeopeMa.
(20)
T e O p e M a 4. flycmb OlllHOCU/llC/lbHO cucme.Mbl (20) (3btnO/lHR./OmC51. yc
Jl0(3Ufl:
1. I(Jlfl flt06btX HatJ.a/lbHbtx iJaHHbtX .-, ~ 3aiJa'1a Kowu dx/dt = Ax + fo (x).
xii=,= ~ UJ.teem eiJUHCm6eHHOe peutCHUe Ha ocu IR : X = ,y (t - ., ~).
2. BeKmop-cjJyHlw,uu v; (x) Henpepbt6Hbl u oipaHUt./.eHbt: llv; (x)II ~ const <
< CXl.
3 . Cyw,ecnwyem cjJyHKU,Wl [ pt.ma Y/Ja6HeHufl dzldt = Bz, noiJ,tuHeHHafl
ou,eHKe /IG (t - s) JI ~ N e- vl t-sl, N ~ I, v > 0.
ToiiJa cucmeJ.ta (20) llJ.teem 02paHu11eHH.oe uttmezpa.n.bHOe MHOwo6pa3ue npu
KaJ/COOM i = 0, 1, ... ' fl, onpeiJeflfle1HOe 6CK1110p -cjJyHK/.{Uet1
<p1 (x)= \G(t - s)v;(\l' (s-t,x ))ds, cp ; (0) = 0.
/?
B pe3yJibTaTe onpeneJIHeTCH Be1nop -rpy1-[KUHH, 3a,!1.aIOW.3H C TOLIH0CTbI0 p/ t
npH6JIH)Ke!-JHOe C HeBH3KOH (0, eb) 11e1npaJib1I0e HHTerpaJibI-IOe MI-IOroo6pa:me
CHCTeMbl (18)
k
./4!: y = cp (x) + e L eicp; (x) = ip (x) + e(D (x) = <D (x, e),
i= O
npH 3T0M HeBH3Ka (0, eb (x , e)) onpeneJIHeTCH cpopMyJIOH
- -
eb = e (iJ(DfdxR,. lfl - R,. [g]).
I13JIO)KeHHbJH cnoco6 nocTpoe1mH nptt6Jitt)KeI-IJrnro HHTerpaJibH0ro MHoro
o6pa3HH 06oc1rnBb1Baer cJie11y1ow.aH TeopeMa.
T e o p e M a 5. flycmb KpOJ.te ycAo6ut1 meopeJ.t 3 u 4 6btnOflHmomcR. maKJ/Ce
CflecJytOU{UC !JC!l06Ufl:
I . BeK11iop-cpyHKt.{UU f (x, ez, e), g (x, ez, e) Henpepbt6Hbt Ha IR rn X 1R11
U yiJ06/lCtn60f]fllO!II !fC/106/.lflM
f (x, ez, e) = Lip {x, 1:-z; A}, g (x, ez, e) = Lip {x, ez; L},
µ>0.
Jl g(x, 0,e)JI ~µ,
2. BbtnOflHR.1omcR. H.epaeeH.cm6a
2N (feep + µ) ~ vp, 2KNfe( l + eTJ) ~ ri fv - (1 + eri ) K 1l, K 1 = KA.
T oco a cucmeJ.ta ( 18) UJ.teem t{eH.mpa.llbH.oe u1-1,mecpallbH.oe MHocoo6 pa3ue
./4!: y = ip (x) + e(p (x, e) = cp• (x , e), cp• (0 , e) = 0,
npu BtnoM cnpa6eOflU6a ou,eHKa
II qJ° (x, e) - CD (x, e) II ~ (l - q)-1 sup II b (x, e) 11.
t, e
2N{b,
q = v- A(l +eri) < l,
xapaKmepu3y10w,aR- omKJ101-1,e1-1,ue .M, om Mnp- EcAu 8eKmop-cjJyH.K1{Ufl b (x , e) pM
H.OMepHo ocpaH.utJ.eHa, mo nocmpoeHHOe ./4!np MflR-emcR. pa6HOMepHbtJ.t acu111nmo
mutJ.eCKUM pC1.3llOJ/CeHUeJ.t UHmecpallbliOW MHOW06pC1.3t.lfl M C llWt./.HOCmblO p} .
40 ISSN 0041-6053. YKp. MaT. 'J/Cypn., 1992, T. 44, M l
TTocTpoeHmo MeTO,ll.OM aCHMl1TOTHlleCKHX pa3JlO)KeJrnH 110 11apaMerpy 11pH·
6mrn<eHHblX HI-ITerpaJibHb!X MHoroo6pa3HH HeaBTOHOMHb!X CHCTeM .umpcpepeHUH
aJibHb!X ypaBHeHHll 110CBHW.eHbl pa6oTbl (26, 37] .
MeTO,U 111-ITe rpaJibHblX MHOr oo6pa3 HJ1 S1BHJ1CS1 T3K)Ke 3cjx!JeKTHBHbIM cnoco6 o M
HCCJie,ll.OB3Hl! S1 CJIHryJl51p1-IO B03M YUle HllblX CHCTeM JJ,HC\xtle p e HllH 3JlbHbl X y p a B
He HJIH. Ka1< H3 BeCTIIO, MHOI'H e CJIHI'YJI HpHO B0 3MYllle HHble CHCTeMbl MO)KHO np11 -
uecn1 1< BH,n,y
dxldt = f (t, x, y), edyldt = g (t, x , y). (2 1)
r,D.e X, f - m-BeI<TOj)bl, y, g - n-Be I( TOj)bl, e - MaJibll1 TIOJIO)KH're.TJbHblH n apa
MeTp, npH 3TOM Bewrop -cj; y1-1Ku11 11 f (t, x , u), g (t , x, y ) 011pe,neJ1e11b1 H 11e 11pep bIB
HbI ,UJIH t E IR, x E !Rm H y H3 He 1<0rnpof1 o6JiaCTH V c IR".
TToJioMHB B (21) e = 0, n oJiy<mM n o po)K.ua1ow.y10 ctt c TeMy
dxl dt = f (t, x , y), g (t , x , y) = 0. (22 )
TTycTb BTOpoe ypaB1-1e1-rne CHCTe Mbl (22) HMeeT Vi30J1HpOBamme pewe1rne
Y = ho (t, x), t E IR, XE !Rm (23)
Tor,D.a Ha ITOBep x HOCTH, 011pe.ueJ1 HeMOH p a Be HCTBOM (23), CHCTeMa (22) 3KBHBa
JieHTHa CHCTeMe
dx!dt = f (t, x. h0 (t, x)).
TTpe,U110Jl0)KHM, l!TO BblflO.'IH H!OTCH CJ1eny1ow.11e ycJIOBHH:
1. B 06.riacrn
(24}
'-PYHKI.UHI ho, f, g, a TaIOKe HX 11e pBbie 11 BTOj)hie 43CTHble 11j)OH3BO,llHb!e 110 t, x ~
lj paBHOMepHO 11enpepblBHbl H orpMmCJeHbl.
2. Xapa1nep11 c rnLJec1<11 e KOIJHH 'A; (t, x), j = 1, ... , n, MaTpHI.-lbI B (t, x) =
= gy (t, x, h0 (t, x)) y.uonJiernop mor ycJIOBHIO Re"-; (t, x) < -2y, y > 0,
j = 1, ... , n.
l lJrn CH CTe MbI (21) c rrpaBenmrna H3Bec THaH Teop e Ma A . H. T11 x o1-10Ba (40]
o 6.n 113 o crn ee p e we1rni'.i 1< p 0: 11 e 1111n:v1 n opm1<,na 1ow.eif cncreMbI (24). U e HTpaJ1b
Hoe npennoJIO,Ke HHe 3TOl1 Teop e~11,J l'.OCTOIIT B TOM, '-ITO H3 0 JlH!)OB3HHbiti 1<0p eHb
y = ho (t, x) ypas1-1emrn g (t, X , Ol ~= 0 HBJIS1eTCS1 aCHMTITOTHLJeCI<H YCTOll'-lH BblM.
TIOJlO,KeHHeM paBHOBeCHH np11coen11He l·IHO(! CHCTeMbl
dy!d. = g (t, X, y),
B KOTopo/:r t 11 x paccMaTpHBa10-rcS1 KaK 11apaMeTpb1. flpe,n110J1omeH11e 2 o6ecne
l!HBaeT aCHMllTOTHl!e CKYIO ycTOHlJHBOCTb no n e p BOMY np116JimKeHH10.
B [41 J noKa3a HO, LJTO TaKaH CHCTeMa HMeeT npHTHrilBaIOW.ee HHTerpaJJ h I-IOe
MHoroo6pa311e
.M,: y = h(t, x, e), h(t, x, 0) = hn (t, x),
110BeneHHe perneHHH Ha KOTOJ)OM OllHCbIBaeTCH ypaBHeHHeM
dxldt = f (t, x, h (t, x, e)).
(25)
(26)
B [8] TIOK333HO, '-ITO ,UJIH 3TOro MHOroo6pa3HS1 c11paBe,llJ1HB npHHUHl1 CBe,D.eHHH.
B CBH 3 H C B03 MOJKHOCTblO 3 aMe Hbl CHlffYJIHpHO B03 MYW.eHHOH CHCTe Mbl (21)'
peryMtpHOH CHCTeMOH MellbUiei:i pa3 MCj)HOCTH (26) 3 a,lla'-la rIOCTpoe HHH cpyHKUHH
h (t, x, e) np1106 pe-rae r o c o6oe 3 Ha'-leHHe . B (22] H3 JIO,h e H cnoco6 noCTpoe 1111H
c!JYHKUHH Ji (t, x , e) MeTOll0 \-1 MaJJoro napaMeTpa . B 11penrroJ10,Kel-IHH, '-ITO BeKTOp
c!JYHKUHH f, g H h0 nocTaTO'IHOe '-lHCJJO pa3 nmjxj)epe1-n~11pyeMhI, c.pyHKUHH h (t,
x, e) npeuc-raBJIHeTCH B B Hlle
h (t , x, e) = h0 (t, x) + eh1 (t, x) + ... + e"h" (t, x) + ek+'h1<+1 (t, x, e). (27)
ISSN 0041-6053. J/,cp. Mar. »:ypH,, 1992, T. 44, M l 4)
1<o9qx\:muHeHTbl 9T0ro pa3JI0)KeHH51 0.UH031-Ia'-IH0 onpe.ueJ151IOTC51 H3 qK>pMaJlbHoro
"T0)K,UeCTBa
e {~>' ~~; + ~ e' ~;; f (t, x, ~>'h;)} = g (t, x, L di;) (28)
i .
nyTeM npnpaBHHBamrn K03CprpHUHeHTOB npH 0,UHHaK0Bb!X CTeneI-! 51X I:'.,.
(.; OycTb Ha11.ue1-1b1 nepBbie k '-!JleHoJJ h;, i = 0, 1, ... , k, pa3JI0)Ke!IH51 cpyHKUHH
h (t, x, e) H nycTb H (t, x, e) = h0 (t, x) + e h1 (t, x) + ... + ekh1, (t, x). B p e -
3 YJibTaTe 3aMeHbI y = z + H (t, x, e) noJ1yqaeM
dx/dt = f (t, x, z, e), edzldt = g(t, x, z, e), (29)
r.ue
~ ( ~ iJH iJH ~ ) ,f = f t, X, z + H (t, X, e), e), g = - eat- - e c5x f + g (t, X, z + H (t, x, e) .
Tipu z = 0 Q.JYHKUH51 g 011.eHHBaeTcH no HopMe Be.1H'-IHH011 C1 ek+I, r.ne C1 -
IT0JI0)K!UeJlbHa51 noCT0HHHaH. DoKa3aHO, '-!TO CHCTeMa (29) ·HMeeT mnerpaJlbH0e
'MHoroo6pa3He M : z = iok+ t hk+t (t, x, e}, r.ue hk+ I (t, x, e) - orpaHH'-!e1rna51 no
H0pMe Q.JYHKU.H51. OTCIO,Ua CJle.uyeT, '-!TO HCX0.IUia51 CHCTeMa HMeeT HHTerpa JibHOe
MHoroo6pa3 He /4(, : y = H (t, x, e) + ek+1 hk+ t (t, x, e).
3. 0 p H JI 0 )K e H H 51 n p H 6 JI H )K e H I-I bl X H I-I T e r p a JI b H bl X
- M H 0 r 0 0 6 p a 3 H 11. U Jrn npaKTHKH 60JibWOH HHTepe c npe.llCTaBJIHeT nOirn
)Ke~me nop51,UKa paccMaTpHBae MOH CHCTeMbl .UHQ.JCjJepeHUHaJibHb!X ypaBHeHHH.
B TeopHH YCTOH'-IHBOCTH 3Ta 3a,lla '-l a B npoc TettlllHX KpHTI-fl!eCIUIX cJiy'-laHX perne
Ha A. M. J1HnyHOBb1:v1 [3]. KaK 0TMe'-laJioCb BO BBeneHHH, ee 11ccJie.UoBaJI TaK)Ke
H . H. 6oroJI1060B .llJlH ypaBHemrn 13 CTaH LWPTIIOl1 Q.JOpMe, K paCCM0Tpe HHIO K0T0-
poro, KaK noKa3aHo B I I 01, np1rnoL1HT MHor11e 3 ana'-IH I-1eJI11Hefrno11 Mexa1rnKH.
BnocJie,UCTBJm 3TOH 3a,llal/ett 3a HHMaMICb MI-!Ortte aBT0pbl (4- 9, 22, 42), HCIT0Jib-
3YH npH 3T0M mne rpaJib I-lbie MHOroo6pa3HH. Ba>KHbIH pe3yJibTaT 3,lleCb np1rna.u
Jie:1iHIT B. A. n .rrnccy, l<OTOp bIH, HCIT0Jib3YH Jl0KaJibHble HHTerpaJibl-lbie MH0r0·
o6pa3HH, BnepBbi e ccpopMym,1poBaJI H 110Ka33JI npHI-!UHIT CBe,UeHHH B o6meM
(TpancuenneHTHOM) cJiyl/ae. Om-ia1<0 y1rn3a1rnaH npo6JieMa He noTepHJ1a a1<Ty
aJ1hHocrn 11 no 1-iacTOHIUe ro BpeMeHH . B pa6oTax [42- 471, a TaK)Ke (22] mia 1-1c
CJ1e.uyeTCH ,UJI51 CHCTeM o6mero BHL!a , npH 3TOM, KpoMe HHTerpaJibHb!X MHOroo6-
pa3HH, np11MeHHf0TCH TaK>Ke npH0 !III tKe H!Ible HHTerpaJibHbie MHoroo6pa3HH.
TioKa)KeM 3T0 Ha np11Mepe CHCTeMbl BH.lla
dxldt = X (t, x. y). dttldt = Y (t, K, y), (30)
r.ue t > 0, x - m-BeKT0p, y - n-BeKTOp.
OpennoJIO>KHM, '-!TO CHCTeMa 130) l!MeeT Hy.nesoe peweHHe H Tpe6yeTCH HC
CJie,lloBaTb ycTOH'-IHBOCTb 3T0ro p e ,ueH 1151 . 4 T06bI y MeHbWHTb Tpy .llH0CTH' CB51·
3aHHb!e C pe1u e HHeM 3TOH 33.ll3l[H, u e .rJ ecoo6pa :mo IT0HH3HTb nopH.U0K paccMaT
p1rnaeMOH CHCTeMbl. UJIH 3T0ro, KaK y 1<a1b!BaJI0Cb BbIWe , M0>KI-10 npHMeHHTb
IIHTerpaJibI-IOe MHOroo6pa3He A1 : y = <p (t, x). cp (t, 0) = 0 H npH 11eK0T0pblX
YCJI0BHHX .ll0Ka :laTb, l[TO 33.lla '-! a on ycTOl141IBOCTH LIJI 51 CHCTeMbl (30) 3KBHBaJieHT·
Ha 3a).l.al/e 06 ycTOHlJHBOCTH 11Jl5l ypaBHeHH5l.
dxldt = X <t. x , qi rt. K)), (31)
·noJiy'-!a10merocH H3 (30) cBenetmeM 1-1a HH-rerpaJib H0M MHoroo6pa3HH.
OnHaKo .ll.151 peaJIH3a UHH cse11e HHH rpeoyeTCfl HaHTH HHTerpaJibH0e MH0ro
-o6pa3He. HaHTH ero T0'-IH0 ynaeTCfl T0JlbK0 B pelllUIX CJIY'-1351X. B CB5!3H C 3THM
Ba»rno yMeTb ocymecTBJlf!Tb CBeLieHHe C IT0M0IUbl0 np116JIH>KeHHblX HHTer
paJibHblX MHOroo6pa3HH.
Ope11ITOJIO KHM, '-!TO yaaJI0Cb noCTp0HTb LlJl5l CHCTeMbl (30) HeKoTopoe npH6·
.fil:>KeHH0e C HeB513KOH (0, b) HHTerpaJibH0e MHOroo6pa3tte
M.np. : y = © (t, X), (1) (t, 0) = 0. (32)
ISSN 0041-6053. YK.p. MaT. 3/Cypn., 1992, T. 44, M 1
TTpe.ucTaBJIHeT HHTepec Bbl5lCHHTb , 1<0r.ua 3 a.ua•rn 06 ycTOfltIHBOCTH .UJIH CHC
~eMbl (30) 3KBHBaJieHnJa 3a)la4e 06 ycTOH4HBOCTH .11.JIH ypaBHeHH5l
dxl dt = X (t, x, CD (t, x)). (33)
L(JIH perneHHH 3TOH 3a)laLJH 3am1rneM ypaBHeHHe (33) B BH.Ue
dxl dt = X (t, x, <D (t, x)) + R (t, x ),
r )le
R (t, x) = X (t, x, q;, (t, x ))- X (t, x, <D (t, x)).
Bo MHOrHX cJiy4a5lX
II IR. (t, x) II ~ c II cp - <DI ~ C sup 11 b (t, x) 11-
t,x
TipH 3TOM, eCJIH .UJI51 TIOCTpoe1-11rn npH6JIHiKeIII1blX mnerpa.nbHb!X MHoroo6pa3HH
np1-1Me1-1HeTC51 ~1eTO}l acmvll1TOTH4eCKH X pa3J!On.<el-lHH no Koop.11.111-iaTaM, TO .11.JIH
tteBH3 KH Bbmo,111HeTCH acHMTITOT11ciec1rne paBeHCTBO 1/bll = o (l!xl/P), x --+ 0,
a eCJIH TipHMe1rneTC.H MeT0./1 aCHM TITOTltL!eCKI-IX pa 3J1 0iKeH!-IH TIO napaMe-rpy ,
TO acHMTITOTHLJec1<0e paBeIICTBO l\b\\ = 0 (e1'), e--+ 0. npe)lnOJJaraeTCH, 4TO 3TH
paBel-ICTBa paBHOMepHbl TIO t.
LloKa3aHa TeopeMa (TeopeMa 6), YTBepiK.11a1oll1aH, 4TO 3 a)la<Ja 06 ycTOH4H
BOCTH .11.JIH CHCTeMbl (30) 3KBHBa.~eHTHa 3a_ua4e 06 YCTOH4IIBOCTH )lJ151 ypaBHeHHH
{33), eCJIH
I . .Llm1 CHCTeMbl (30) BbITIOJIH51!0TC.H / ycJIOBH.51 npHHUHTia CBe)leHHH.
2. l13 BeCTHO TipH6JIHiKem1oe Hl·l'rerpaJJbHOe MHOI'C06pa3 He CHCTeMbl (30).
3. 3a.ua<Ja 06 ycTOH4HBOCTlt 11,1H ypaBHel-IH51 (33) pernaeTCH He3aBHCHMO OT
'-!JleHOB TIOJ).51,/lKa Bblllle p (OHIOC(tTeJibHO X - B CJiyciae aCHMnTOTH4eCKoro pa3-
JIOiKNIH51 no KOOJ).UHHaTaM 1-1 OTHOCHTeJibllO c - B CJiyLJae aCHMTITOTH4eCKOro
pa3JIO)Kemrn no TiapaMeTpy).
4To6bl TIJ)OHJJJIIOCTJ)HpOBaTb 11 3 .~0iKeHJ-!Oe Bb!We, BbI)leJIHM B CHCTeMe (30)
JIHIIe111-1y10 4aCTb. B pe3yJibTaTe nOJIY'IHM CHCTeMy
dxldt = A (t) x + B (t) y + F (t, x, y),
(34)
dyldt = C (t)y + D(t)x + H (t, x, y).
TTycTb H3BeCTHO aqxpmmoe HHTerpaJibHOe MHoroo6pa3He COOTBeTCTBYIOW.eii
JIHHe111-1oii c11cTeMbJ: /4l0 : y = Q (t) x. BBe.UH 3aMe1-1y
/1 = Q(t) ~ + z,
noJiy'IHM CHCTeMy CTieu11aJibHOI'O Bl-1!.lcl
dxldt = Ao(t)x + B(t)z + f(t, X, Z ) = Ao(t)x + r (t,x, z).
(35)
dzldt = C 0 (t) z + h (t, t, z).
O TI p e Jl e JI e H 1-1 e 3. EydeM caeopumb, 'tmo oJIR cucmeMbl (35) u.ueem
.Mecmo 1'pumu'lecKut1 c11y'ial1, ec11u [/Jy1-tda.ue1ww11bHbte 1rtampUU,bt X (t, s), Z (t, s)
ypae1-te1-tut1
dxldt = A0 (t) x, dzldt = C0 (t) z
nOO'-tUJ-teHbt OU,el-tKaAt
I/ X (t, s) II ~ KeXlt-st, JI Z (t, s) II~ Ne-v<t-s), t > s,
2iJe nocmoR1-t1-tbte x < v, v > 0.
T e o p e M a 7. llycmb om1-tocumeAbH0 cucmeMbl (35) Bbtn0AHR/OmcR c11e
ay10w.ue ljCA08UR:
I . H Meem .uecmo KpumuttecKut1 c11yttal1 e CMbtc11e ·mpeae 11eNuf1 3
2. BeKmop-{/JyHKl(UU r (t, X, Z). h (t, X, z) yooe11emeop,i1om ycA08UI0 Jlun-
wuu,a
11 r (t, x, z)-f* (t, x, z) 11 ~ 'A II x-x I!+ A II z- z 11.
ljh (t, X, z)- h (t, X, z) 11 ~ z II x- x11 + ~ U z-z .l
u, Kpo.ue moeo, II h (t, x, 0) II ~ µ < =.
ISSN 0041-6053. YKp. MaT. OtCgpH.., 1992, T. 44, M I
3. flocmoR.HHble 1., !l. A, l yooeAemeopmom ycAoewo ("- , !I,, A, l)-+ 0 npu
11 x II-+ 0, II y II-+ 0 pa(mOMepHO no t.
T oioa cucme.Ma (35) uAteem (p, ri)-A1Ho2006pa311e /4{. : z = cp (t, x), a. 3aiJa,1u
06 ycnwt1 1weocmu OAR curmcA1bt (35) u ypae1-1.e1-1.uft
dxldt = Ao (t) x + t (t, X, cp (t, x)) (36),
3K.6U6a,ieHlnHbl.
KaK CJi e,UCTBHe TeopeM 6 11 7 c npaBe,'lJlHBa C.'le)ly!OLL~aH Teop eMa [ 4 7] .
Te O p e Ma 8 . riycmb om1-1.oc1.1111ellbHO cucmeAtbl (35) 6bllWRHCHbl !JC1706Ufl
meope,Hbl 7, HC6fl3Ka npu6,1llJICfHH020 UHtnecpG,1bHOi'O J.tHOc006pa3llfl ll,1teem no
pR.aoK 6bll llC p, a aaaa<ta 06 !JCll10l/)11160Ctnll allfl lj/7G6Hef1Ufl
dx/dt = Ao (t) x + /* (t. x , <1) (t, x )) (37)
peuwemcfl He3cwucu,110 om 1weHoe nopRaKa Gbu.ue p .
Tozaa saaa,iU 00 !JC!lIOU'lU60C/11/l a.1fl Cl/Cllle,\tbl (35) l/. !JfJG6HeHtlfl (37) 3K(lll
(laJleHtnHbl.
M eTO)l HHTerpaJlbHbl X MllOl'OOOpa:rni1 13 CO'l eTa llHH C ll,Ueef1 CBe,UeHHH TTOJiy-
4HJ1 U!HpOKOe npHMe1-Ie 1rne 11 npll llCCJi e,UOBaI-ll1H CHHry.'IHJJIIO B03 MyULeHIIblX CHC
TeM .umtx1Jepe11uHaJlbl-lblX ypaBHC l lliti.
KaK H3BeCTI-IO, Ta K He KJiaCCbl CH CTeM )lHCjXj)epemu1aJibHbl X ypaBHCI-IHii 1303-
HHKaIOT ll[)l·l MO,Ue.111p0Bal!Hl-l 061,eETOB pa3.'1HLllIOh npnpO,'lbl, cone pUJaIOULHX
0.l]JIOBpeMe! Il-10 OblCT[)ble H Me)Mellllble )llllHKelliiH. Ta101e MO,lle.n1 ecr eCTIJe!Illbl
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| id | umjimathkievua-article-2318 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T02:22:07Z |
| publishDate | 1992 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b4/74f00111b921a8c06bc3af72508be1b4.pdf |
| spelling | umjimathkievua-article-23182023-09-14T12:03:56Z Development of the Bogolyubov-Mitropolsky method of integral varieties Развитие метода интегральных многообразий Боголюбова — Митропольского Lykova , О. B. Лыкова , О. Б. Integral Manifold Skii Method Інтегральні многовиди A survey is presented of some results relating to the development of the Bogolyubov-Mitropol'skii method of integral manifolds. Приведен обзор некоторых результатов, касающихся развития метода интегральных многообразий Боголюбова—Митропольского. Наведено огляд деяких результатів, які стосуються розвитку методу інтегральних многовидів Боголюбова—Митропольського. Institute of Mathematics, NAS of Ukraine 1992-02-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2318 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 1 (1992); 33-46 Український математичний журнал; Том 44 № 1 (1992); 33-46 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/2318/1539 Copyright (c) 1992 O. B. Lykova |
| spellingShingle | Lykova , О. B. Лыкова , О. Б. Development of the Bogolyubov-Mitropolsky method of integral varieties |
| title | Development of the Bogolyubov-Mitropolsky method of integral varieties |
| title_alt | Развитие метода интегральных многообразий Боголюбова — Митропольского |
| title_full | Development of the Bogolyubov-Mitropolsky method of integral varieties |
| title_fullStr | Development of the Bogolyubov-Mitropolsky method of integral varieties |
| title_full_unstemmed | Development of the Bogolyubov-Mitropolsky method of integral varieties |
| title_short | Development of the Bogolyubov-Mitropolsky method of integral varieties |
| title_sort | development of the bogolyubov-mitropolsky method of integral varieties |
| topic_facet | Integral Manifold Skii Method Інтегральні многовиди |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2318 |
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