Study of periodic solutions of the second-order notations
Приведен численно-аналитический метод исследования существования и приближенного построения периодических решений счетных систем дифференциальных уравнений второго порядка....
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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/7798 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512761517178880 |
|---|---|
| author | Ronto , N. I. Martynyuk , O. M. Ронто, Н. И. Мартынюк, О. М. |
| author_facet | Ronto , N. I. Martynyuk , O. M. Ронто, Н. И. Мартынюк, О. М. |
| author_sort | Ronto , N. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-09-14T12:03:56Z |
| description | Приведен численно-аналитический метод исследования существования и приближенного построения периодических решений счетных систем дифференциальных уравнений второго порядка. |
| first_indexed | 2026-03-24T03:33:55Z |
| format | Article |
| fulltext |
YllK 517.927.4
H. 11. Po11TO. n-p qJH3.-MaT. nayK,
0. ~I. .\lafJT1,11110K. acn . (.Hu--r ~taTOMaTJnm AH YKpaHHJ,I, RHeB)
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nocTpOe ?IHSl nep1lO.:tH'!eCKHX p ewe m1ii C'leT!-lbl X CHCTeM i rnqxpepemm a.JJbHblX ypaBHeHHii BToporo
nop sinirn .
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pi ,,:rn'IHHX p03B'H3KiB 3Jii'leHHHX CHCTeM .!l.HcflepeHIJ,iaJibHHX piBHl{Hb npyroro IlO f)SI.!l.Ky.
O Goc HOBal-n!e npaBOMOtIHOCTH npHMeHeHHH tIHCJieHHO- aHaJIHTHtJeCKOro MeTO,l\a
nocJi e.Lio BaTeJJbHb!X nepHO.!JJitJec1mx np H6JJH)KeHlill I( 6ecKOHetJHb!M CHCTeMaM
JlHCjxpepeHUH8JlbHb!X ypaBHeHHH HMeeT p5IJJ. oco6eHHOCTeii [ l, 21.
B H8CT05Illl.eii pa6oTe 3TOT MeTO)l pacnpocTpaHHeTC5! Ha 1-lOBblll KJiacc ypaa
HeHHl'f , a HMeHHO Ha CtJeTHble CHCTeMbl )lHcp<pepem.r,HaJibHblX ypaBHeHHH BTOporo
nopH.n.Ka
if~ •
X = f (f, X, X), (1)
r.n.e X = (X1, ... , Xn, ••• ) - TOtIKa npocTpaHCTBa m orpaHHtiemmii lJHCJIOBOi
nocJJe.UoBaTeJibHOCTH c HOpMoii 11 x II sup I xnl, f (t, x),=(/1 (t, x), ... , f n (t,
/1
C H . H. POHTO . 0. M. MAPTblHIOK. 1992
ISSN 0041-605:J. ()' K.p . .MQT. ,1Cyp11.., 1992, T. 44, M 1
x), ... ) - HenpepbIBHa51, nepHo.ztHqecKaH no t c nepHo.ztoM T B o6Jiacrn
(t,x,;)E(- oo , oo) X D 1 X D2 (2)
qiyHKUHH, TipmrnMaJOUJ.aH 3HaqeHHe B npoCTpaHCTBe m; D1, D2 - orpaHHqe1-1-
J-lbl e 3aMKHYTble o6JiaCnl npocTpaHCTBa m.
P accMaTpirnaeTcH 3a.zta 1ia HccJie.uoaaHH51 cymecTBOBamrn H nptt6JIH)KeH
Horo HaxomneHHH T-TiepHO}lHlJ eCKOro perneHHH (1), T. e. peweHHH T-nepHOJl.H
q ecKoi-i KpaeBoH 3a.uatJH (!), (3)
. .
x(O)= x (T), x(O)=x(T). (3)
non T-TiepHO.llHlJeCK HM p ew emI eM CHCTeMbl (1) 6y.neM TIOHHMaTb ClJeTHy1O
ae1<rnp -cjJyHKUHIO x (t) = (x 1 (t), . .. , x" (t), . . . ), onpe.neJie1rny10 npH t E (-oo,
00 ), npHHHMaJOI.1.1,YIO 3Ha4 eHH51 B npoc-rpaHCTl3e m, .l1Ba)l{.1lbl Henpepb!BJ-1O .nmjxpe
pe11uHpye,,1y10 H Y.ll.OB.'I ernopmom.y1O ypaBHeHH!O (1), a TaK,Ke ycJIOBH!O (3).
TTpe.nnoJTO}f(l-!M, qro B o6Jiacrn (2) npaBaH lJ3CTb t (t, X, x) CHCTeMbl (])
ynoBJieTBOpH eT ycJIOBJIIO orpaH HlJ E' llllOCTH C'leTHb!M BeKTOpOM M = (M 1, ...
. . . , Mn, ... ) E m H ycJioBmo Jl11mu11ua co cL1en1bIMH MaTpttuaMH K1, K2 c HeOT
puuaTeJTbHbIMH 3Jie1e1na~m
l f(t,x,y)I ~ M,
lf(t, x',y') - f(t,x",u")I ~ K 1 !x' -x"I + K 2 jy'-y"I, (4)
tE(-00,00 ), x ',x"ED 1, y' , y"ED2 ,
rne I f (t, x, y) I = (\ f 1 (t, x , y)\, . . . , If n (t, x, y)I, ... ).
Ilpe1inoJioxmM, tJTO n apaMeTpbI M, K 1 , K2 , T, D 1 , D 2 KpaeBoH 3a,naqH
()), (3) TaKOBbl, lJTO ,II_JIH HHX BbinOJIH5110TCH ycJIOBHH:
l) MHOmec TBo Da TOlJeK x0 = (x01 , .•. , x0 n, ... )Em, co.nep,Kalll,HXCH B o6JJac-
T2
TH D1 BMeCTe co caoeii ~ = 4 M-oKpeCTHOCTbIO, HenycTO, H MHO)KeCTBO
5
Dv, o6pa3oaa,rnoe y = 6 T M-oKpeCTHOCTblO HyJJeaoro BeKTOpa npoCTpaI-!CTBa
m, Jie,KHT B 06Jiacn1 D 2
Da =I= 0, Dv c D 2 ;
2) onepaTOp Q0 , nopm1<.naeMhlH MaTpttuefl
1!BJIHeTCH BTIOJIJ-Je perym,pHb!M [31
I\Qo\l ~ qn< I.
(5)
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K H X TI p H 6 JI H )I{ e H H H • .IlJIH paccMaTpHBaeMOH 3a,11_aqH (1), (3) TIOCTpOHM
TIOCJTe.noBaTeJibHOCTb CL!eTHb!X cpyHKUHH Xm (t, Xo) = (xml (t, Xo), .••• Xmn (t,
Xo), ... ). 3aBHC.HW:HX OT TOL!KH Xo = (Xo1, Xo2, ...• Xon, ..• ), KaK OT TiapaMeTpa,
II ynoBJieTBOp.HIOW:HX TiepHO,II_HL!eCKHM KpaeBb!M ycJIOBHHM (3) .1lJl51 acex m npH
npOH3BOJlbHOM Xo E Dr,. noKa)KeM CXO,11,HMOCTb 3TOH IlOCJJe,11,OBaTeJibHOCTH H Bbl
HCHllM CBH3b ee npe.neJJbHOH cpyHKUHH C nepHO.1lHlJeCKHM perneHHeM aa.natJH (I),
(3).
HaM no11a.no611Tc5I B ,naJJbHeflrneM JJHHeHHbIH onepaTOp L, KoTopbIH .neflcT
aye-r Ha HenpepbIBHYIO npH t E [O, T] cqeTHYJO BeKTOp-cpyHKUHIO f (t) =
= U1 (tJ , ... , f n (t), ... ) cornacno cpopMyJJe
84
I
Lf (t) = I (f (s) - Sf (t)) ds,
x
(7)
ISSN 0041-6053. YKp • .MaT. :J1CypH.. 1992, r. 44, M l
r)le S - onepaTop HHTerpaJibHOro cpem1ero
T
Sf(t)=; 5f(t)dt.
0
E cm1 onepaTopoM L no.ue11crnonm-b Ha c!Jymmmo (7) , TO
I t I
L (Lf (!)) = L2f (t) = S [Lf (t) - S (Lf (t))) dt = S [S (l (s) - Sf (t)) ds -
0 0 0 ·
I
(8)
- S(_\'U( s) -Sf(t))ds)]dt. (9)
0
Hs (7)- (9) BblTeKaIOT cJit.uy1ouu1e CBoiicrna oneparnpoB S, L, L2:
:t Lf (t) = f (t)- Sf (t), :t L2 f (t) = Lf (t) - S (Lt (t)), ( 10)
S 2f (t) = Sf (t), ( 11 J
T T T
5 Sf(t)dt = s (S, (( dt) = s, (t)dt,
0 0 0
T [ 1 T ] Lf (t)l=T = j f (t) - T f f (t ) dt dt = Lf (t}t=O = 0. <12)
L2f (t) .=T = 0 = L2f (t)f= O' S (Lt (t))i= O = S (Lt (t)) l=T = S (Ll (t)) . ( 13
O<1eBH,lll-lO, 4TO eCJIH f (t) - nepl-JO,llWJeCK35l C n epHOllOM T cpy11KUH5l, TO cpy1rn
UHH Lf (t) H L2f (t), Bbl4Hc.ryef!Hble corJia CI-10 (7), (9), TaKme 6y.uyT nepttOlltttl eC
KHMH cpyHI<UIHIMH C nepHO)l0M T.
CorJJaCHO JieMMe 3.1 [l, C. 131, K0T0pa5I cnpaBe,ILIIHBa H B CJ!y<1ae C4eT1-J0-
Mep1-1011 BeKTOp-q:iyHKUHH f (t), HMeeM
I Lf (t} J ~ a1 (t) If (t} lo, (14)
rJJ.e
If (t))8 = ( sup lf1 (t)I, ... , sup lfn (t) I, ... ),
iE [O,T] IE[O,T]
(l 5)
a 1 (t) = 2t ( 1 - ~) ~ . ; •
naJJee, B CttJiy (9), ( 14 ), ( 15) llJIH Bcex t E [O, Tl
I L2f (t) I ~ ai(t) I Lf (t) lo ~ ; a1 (t) If (t) lo ~ 7; If (t) lo· (16)
PaCCMOTpHM nocJie)l0BaTeJibHOCTb cpyHKUHH
Xm (t, X0) = X 0 + L 2f (t, Xm - 1 (t, X0 ) , ;m- 1 (t, X 0) + Pi2 + Pi,
m= l,2, ... ,x0 {t,x0) = X0 ED1:1, 07)
Xm (t, X 0) = (Xm1 (t, X0), ••• , Xmn (t, x0), ••• ),
r.ne x 0 , p 2, p 1 - napaMeTpbI, npttHa.UJiemamHe npoCTpaHCTBY m.
Bb16epeM napaMeTpbl P1 H P2 TaKHM o6pa30M, 4T06bl cpyHKUHH BH.lla (17)
Yll0BJi eTBopHJIH neptt011H4eCKHM KpaeBblM ycJIOBHHM (3) )J.JIH Bcex m nptt np0H3-
BOJ!bH0M s 11a,1e11tttt x0 E Dr,.
C Y'leT0M ( l 0) H3 ( 17) CJieJlyeT
. . .
Xm (t, X 0) = Lf (t, Xm-1 (t , Xo), Xm- 1 (f, X0}) - S (Lf (t, Xm-1 (t, X0), Xm-1 (t, Xo)) +
+ 2p 2t + p 1. (18)
ISSN 0041-6053. YKp. Mar. o,cypn., 1992, r . 44, M l .85
no,llCTaBJIHH (17), (18) B nep11O,n11qecKHe KpaeBb!e ycJIOBHH (3) H npmmMaH
BO BHHMam1e paBeHCTBa (12), (1 3) , ,UJIH onpe.neneHHH P1 H P2 nonyqaeM 6ecKO
eeqHoMepHyIO CHCTeMy anre6pa11qecKHX ypaBHeHHH
Xo = Xo + P2T2 + PiT,
P1 = 2p2T + Pi,
OTKy,na HaXO,llHM P1 = P2 = 0. Cne,noBaTeJibHO, Bee qJieHbl nocne,noBaTeJibHOC
TH cpyHKU.HH
.
Xm (t, Xo) = Xo + L2f (t, Xm-1 (t, X0), Xm-1 (t, x0)),
(19)
m = 1, 2, ... , x 0 (t, x0 ) = x0 E Dti,
saBHCHill.HX OT Xo, KaK OT napaMeTpa, Y.UOBJieTBOpH!OT nep1-1O,n11qecKHM KpaeBb!M
ycJIOBHHM (3) npH npOH3BOJlbHOM Xo-
CnpaBe,llJTHBO cne,ny10mee YTBep)K,UeHHe O CXO,llHMOCTH nocJie,UoBaTeJibHb!X
np116JIH,Kei111(1 Xm (t, x 0) BH,lla (19) .
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pe1-ufua!lb1-tbtX ypae1-te1-tutl onpeoeAe1-ta, 1-tenpepbt61-ta, nepuoou'lecKaR no t c ne
puoooM T 6 o6Aacmu (2) · u npu1-tuMaem 31-ta'le1-tuJl 6 npocmpa1-tcmee m, a maK:11ce
BbtnOAl-le/-lbl ycA06U5l (4)-(6).
T ozaa 110C/lC006atne/lb/-lOCtnb cpy1-tKU,UU Xm (t, Xo) 8UOa (19), 3aBUC5lllfllX om
x0 , KaK om napaMempa, u yooeAemeopRIOUfUX nepuoou'leCKUA·t KpaeBbtM y cAoauR,11.
(3) , pae1-t0Mep1-to cxooumrn npu m-+ oo aMecme co ceoetl npoU3aooHot1 ;m (t, x0)
Otnl-lOCUtne/lbl-lO o6Aacmu
(t, X0) E [0, T] X Dti (20)
" npeoe/lbHblM cpyHKl{UJlM coomeemcmee1-t1-to x• (t, x0), x • (t, x0)
lim Xm (t, x0) = x• (t, x0), Jim ;m (t, x0) = i (t, X 0). (21)
m~~ m~ro
n pu amoM x* (t, x0} MflR.emcJl yooeAemaopJl/OUfUM npu t = 0 1-ta•taflbf-lbtM
!JCAOBUJlM
x(O) = x• (0, X0 ) = x 0 , ; (0) = ;• (0, x 0 ) = - S (Lf (t, x• (t, X 0 ), ;• (t, x,1)) ('.?.2 )
a 001-lOBpeMeHHO T-nepuoou'leCKUM ycAOBUJlM (3) peuteHUeM UHmezpaAb/-lOcO upaB
-,,.wi
x(t) = x0 + L2f (t, x (t),; (t)),
•· e. peiue1-tueM Kpaeeotl 3aoa'lu
X = f (t, X, ~) - s (f (t, x· (t , XoJ, ;• (t, Xo)).
. . .
x (0) = x(T) = x0 , x (0) = x (T) = - S (Lf (t, x• (t, x0 ), x• (t, x0)).
KpoMe mo20 , cnp aeeofluBbt cAeoy10Ufue oi1e1-tKu no2 peumucmu:
• rz q"'
IJ Xm (t, Xo)- X (t, Xo) II ,,;:;: 4 I _ q II M II,
• •• 5 qm
Xm (t, X0) - X (t, x 0) II ,,;:;: 6 T I _ q JI M 11, m = I, 2, ... ,
zae
[lx(t)I! = sup max lxn(t)!, IIQII = II T4
2
K1 + ~ r ·,t \I ,,;:;: q< I.
n IE(O , T] n
(23)
(24)
(25)
ISSN 0041-6053. J!Kp. Mar. ~ypn., 1992, T. 44, M l
.ll o Ka a a Te JI b c TB o : C1-1aqaJia 110Ka)KeM, qro ecm1 x 0 E D6 , TO Bee
~UHH Xm(t,x0)ED1, a Xm (t, x 0) ED2 np11 fE[0,T] . rloJiaraH B (19) m=O,
- OCHOBaHHH (16) 11 ycJIOBH H orpaHH'Ie IHIOCTH (4) HMeeM
X1 (t, Xo) - Xo = L2f (t, Xu, 0),
y2
l x 1 (1,x0) - x,. J ¾ 4 M.
)lmp¢€pe1m11pyH (19) C yqe TOM (I 0) , TTOJIYLJaeM TTOCJiellOBaTeJibHOCTb
. . .
(26)
X m (t . X0) = Lf (t , Xm-1 (t, X0), Xm- 1 (t, X0)) - S (Lf (i, Xm-1 (t, X 0), Xm-1 (t, x0})),
m = I, 2, .. . , x0 (t, x0) = x0 . (27)
Pia (27) 11p11 m = 0 B cm1y (4), ( 14), rrnTbIBaH, '!TO
T y2
S a 1 (t) dt = 3 ,
0
• l T 5
lx1 (t,x0)1:s:;:;;,1 (t)M+y Sa1 (t)dtM:s:;:;; 6 TM. (28)
0
EcJIH yq ecTb 11pe)IITOJIO)l\e HHe (5), TO Ha OCHOBaHHH HepaBeHCTB (26), (28) 3a K
JIJOq aeM , tITO x 1 (t, x0) E Di, a ~ 1 (t, X 0) E D 2 , ecJIH x0 E D5.
MeTO)IOM MaTeMan1q ec1wi:'1 HHlIYKUHH HeTpym,o y c-raHOBHTb, tITO ll.1151 Bcex
m = 1, 2, .. . . , t E l0, Tl H mo6oro x 0 ED" 11ei:\crn1neJibHO Bee cj)yHKUirn x (t ,
Xo) B HD.a (1 9) 11J)Hila)l.rie)l<aT 00.'I aCTH D1, a TIJ)OH3B0lll-Jble Xm (t, Xo), Bb\LIBC.'15le
Mble no (27), He BblXO)IHT 113 OOJiaCTH D2:
Xm (t, x0) E D 1 , Xm(t,x0) ED2, m = l,2, .. . ,x0 ED5.
n epeXOll.51 I( Jl.OKa3aTeJibCTBY CXO)IHMOCTH TIOCJiell.OBaTeJibl-lOCTH ( 19) , ou.e
HHM pa3I-IOCTH
- .
X 2 (t, X 0 ) - X 1 (t, x0; , X 2 (t, X0)- X 1 (t, X 0).
H a OC IIOB3HI IH (1 9), (27) c ytie TOM (14), ( 16) H ycJIOBH51 Jhmumu.a (4) HMeeM
' I X 2 (t, x0)- X1 (t , X 0) I :s:;; J L 2 [f (t, X1 (t , x0 ), X1 (t, X 0))- f (t, X 0 , O)] I :s:;;
T _ · T
~ 2 a 1 (tJ If (t, x 1 (t, x0), X 1 (t, x 0)) - f (t, x0 , 0) lo ¾ 2 ai(t) [1(1 I X 1 (t, X 0)-
(29)
1; 1 I t. x0)- ; 1 (t, X 0 I :s:;; IL [f (t, X 1 (t, X 0), x·1 (t, X 0))- f (t , X 0, 0)) I+
+ S j L If (t , X 1 (t, x0 ), ~ 1 (t, x0 )) - f (t, X 0 , 0)] I¾ ( a 1 (t) + I) I/ (t, xi(t, X 0),
~1 U, Xo) - f (t, Xo, 0) lo :s:;; ( al (t) + ~) LK1 J X1 (t, Xo)- Xo lo + / (2 I ~1 (t, Xu)J.,].
(30)
n pH HHMaH BO BHHM3HHe (26), (28), 113 COOTI-IOUleHirn (29) flO.f!yqaeM
T lT2 f) J T l x2 (t . x,J ) - x 1 (t, X 0 ) I :s:;:;; 2 a 1 (t) 4 1(1 + 6 T/( 2 M = 2 a 1 (t) QM, (31)
a 1-13 (:30) -
1 ).'. ,i (/, Xo) - ; 1 (t, Xo ) I¾ ( al (t) + I) [ :2 K1 + ~ TK21 M =
ISSN 004 1-6053. Y,cp. Mar. 3/CYPH.., 1992, T. 44, M 1 81
= ( a 1 (t) + ; ) QM, (31 ')
r.ne
Ha 0CH0BaIIHH (31), (31') MeT0,L\0M MaTeM,.nw1ecKO}I HH,IJ,YKUHH M0)KH0 ,Q0K838Tb,
4T0 .U.,151 JII06oro H0Mepa m;;:;,,, l cnpaBe,IJ,JJHBbl He paBeHCTBa
I Xm + I (t, x0) - Xm (t, X0) J ~ ; a 1 (t) Q"'M, {32)
(33)
TaK KaK
\ Xm+i (t, X0 ) - Xm (t, Xo) I ~ I Xm+i (f, X0)- Xrn+i-1 (f, X0 ) I + I Xm+i-1 (t, Xo)
- Xm+i-2 (t , Xo) I+ ... + I Xm+ I (f, Xo) - Xm (f, Xo) I,
TO C yc1eT0M (32) HMeeM
T j - I .
I Xm+i (f, X0)- Xm (f, X0 ) I~ 2 a 1 {t) Qm ( ~ Q') M. (34)
i= O
AHa,10nJt1Ho B c irny (33) Haxon11M
I ;,n-J-j (l , Xo)-~m (t, X0 ) I~ ( ai(t) + ~) Qm (f Qi) M. (35)
nepeXO,ll.H B C00TH0llleHHHX (34), (35) K oue1-1KaM no H0pMe, no.riy tJaeM
y2 j-1 .
11 Xm+i (t, X0) - Xm (t, Xo II ~ T jl Q llm II ~ Q' 1111 M II,
i=O
(36)
II L+i (t' Xo) - ;m (t ' Xo) II ~ . : T II Q ,r 11 i:' Qi Ii II M 11-
i=O
(37)
T2 ~
" · e. onepaTOp, nopo)KnaeMbIH Ma Tpuue11 Q = 4 /( 1 + 6 T/(2 , HBJrneTCH Bno-
JI«e p eryJIHpHbIM. CJie,IJ,oBaTeJibHO, B CHJIY (37) H3 (36) 8b1Te1rneT, q·ro npM rn -+
-+ 00 0TH0CHTeJibl·JO o6JiaCTH (20) ,llCHCTBHTeJibHO HMeeT MeCTo CX0llHM0CTb (21).
OpHHHMaH BO BHHMa1rne, 'ITO Bee (j)yHKUHH Xm (t, Xo), ;m (t, Xo) nocne,nonaTeJib
HOCTeH (19), (27) ynoBJieTB0p moT nepH0llH4eCKHM "paeBblM ycJJ0BIIH\I (3),
JierKo 3aK : uocr HTb, lJT0 npeneJJb11b1e (l)y111<1LHH (21) TaK;Ke y.1.1.oB:1eniopH1or 11M.
TTpH cTpe.,,Memm j -+ = H3 (36) ,IJ,JIH Bcex m = l, 2, ... no.·1yciae,1 Tpe-
6yeMbie 0UCHKH (25).
EcJIH B p aBe11crnax (19), (27) nepefrrn K np eneJ1y npH 111 -+ oo 11 y11 eCTh
C00TH0memrn (11)- (1 3), (2 1), TO J!CfK0 npoBepHTb, l{TQ npe;le.'lbllclH (ilY II KllllH
88 ISSN 004 I -6U53. YKp. ,um . :HCypt-t., 1992, r. 44, .I\!/! J
x* (t, Xo) YJlOBJieTBOp51eT ypaBI-IeHHIO (23) H HaqaJibHblM ycJIOBHHM (22), OTKY·
)la Ha OCHOBaHHH (I 0) cne.uyeT' 'lTO x* (t, Xo) - pewe1me I<paeBOH 3a.Ua'lH (24).
113 (24) BH.UHO, <ITO BCHI<Oe perne1111e x· (t, Xo) = l im Xm (t, Xo) 3TOH 3a.ua-
m➔oo
1m, .UJIH KoToporo
HBJIHeTC.H B TO :>Ke BpeMH H YJlOBJieTBopmomHM nepHO)lH'leCKHM KpaeBbIM ycno
BHHM (3) pewe1meM HCXO,UI-IOro ypaBHel-IH51 (I), npH'leM TaKHM, lJTO e ro Ha'laJib
J-IOe 3Haqe1me 3a.uaeTcH no (22).
2. l1 C C JI e .U o B a 1-1 H e C B O H C T B TI p e )l e JI b 1-1 O H rp y H J< U H 11.
B npaBylO qacTb paccMaTpHBaeMOH CHCTeMbl cqenihlX JlHqxpepeHUH3JlbHblX ypaB
HeHHH (1) Bcer)la MO)KHO a)l)lHTHBHO BBeCTH .UOTIOJII-IHTeJihHb!H cqenihlH ynpaBJIH
IOUlHH napaMeTp µ H Bbi6paTb HatiaJibl-IOe 3Ha'leime npOH3BO)lHOH Yo TaKHM,
'lTO npH mo6oM 3Ha'leHHH x0 E De, pemeHHe 3a)lal!H Kow11
0X = f (f, X, X) + µ, µ = (µi, µ2, ···, µn, ... ),
x(0) = x0 , x(0) = Yo
6y.ueT YJlOBJ!eTBOpHTb 3aO)lHO H nepHO)lH'leCKHM KpaeBh!M YCJIOBIUIM (3).
(38),
(39).
T e o p e M a 2. EcAu BbtnOAHeHbt ycAOBWl m eopeMbt I , mo OAfl A106ot1 mo•t
Ku Xo E Dr, Hat1oymc5l maKue eouHC/1lBeHHbte 3 Ha'LCHU5l C<te/1IHOZO ynpaBA5l/OU1ec0>
napaMempa
µ = - S If (t, x• (t, X 0), x• (t, x0))]
u HG'taAbHOU CKopocmu
Yo= - S [Lf (t, x• (t, x0 ), x• (t, x0))],
(40}
(41}
cOe X 0 (f, X 0), ;- (t, X0 ) Bbl.'lllCA5llOtnC5l COcAGCHO (2 J ), 1ltn0 petUeHUe X (t) =
= x· (t, Xo) = lim Xm (t, Xo) HQ'-lQAbHOU 3QOQ 1lU (38), (39) 6yoem yooe11em eo-
rn➔oo
p5lmb nepUOOU'lCCKUM KpaeBblM ycAOBUflM (3).
1l O K a 3 a T e JI b CT B 0. Ha OCHOBaHHH TeopeMbl I y6e )K.uaeMC.H, l!TO
cpyHKUHH X (t) = x* (t, Xo) 51BJIHeTCH peweHHeM KpaeBOH 3a.ua•m (24). TeM ca
Mb!M HaH)leHhl 3HalleHHH napaMeTpa µ BH)la (40) H 1-1aqaJ1bHOH CKopoCTH y (),
BH)la (41), npH KOTOpblX cpyHKUH51 X (t) = x* (t, Xo) = lim Xm (t, Xo) .HBJIHeT-
m➔oo
CH pewei-meM KpaeBOH 3a,Ua'lH (38), (3) npH KpaeBh!X ycJIOBH.HX (39).
TToKa)KeM, l!TO 3TO 31-Ia'leHHe napaMeTpa µ e,UHHCTBeI-IHO, T. e. npH JII060M
OTJIH'lHOM OT (40) 3Haqe1-11111 peweHtte 3a.ua11H Koum (38), (39) 1-1e 6y)leT y.uosn eT
BopHTh nepHO.UH'leCKHM KpaeBb!M ycJIOBHHM (3).
TTpe.unoJIO)KHM, l!TO cymecTBYIOT T3KHe )lBa 3H3lleHHH µ', µ", µ' =/= fl 11
,
'l!TO perneHHH X (t, Xo, µ'), X (t, Xo, µ ") 3a)lal!H KornH (38), (39) COOTBeTCTBeI-IHO
npH µ = µ' H µ = µ" YJlOBJleTBOpHIOT nepHO.UH<IeCKHM KpaeBhlM ycJIOBH51M
(3), T. e. HBJIHeTCH perneHHeM ypaBHeHHH (23). Tor.ua .
jx(t, x,, i>")-x(t, x0 , µ')I= I L2(f (t, x(t, x0 , µ"), x(t, X 0 , µ"))-
' -[(t, X (t, x0 , µ'), x (t, X 0, µ'))) I,
(42)
I i ii
I X (t, X 9 , µ.•)- X (t, X 0 , µ') I = IL [f (t, X (t, x,, µ"), X (t, X 0 , µ")) -
-l(t, X (t, x0 , µ,'), l(t, X 0 , ~~')))]-S [f (t, x(t, X 0 , µ"), l(t, X 0 , µ"))-
-f (t, x(t, x0 , µ'), x(t, Xv,µ')))] 1-
AHaJiormrno ycTaHOBJleHHIO oueHOK (29)-(31) H3 COOTHOllleHHH (42) TIOJiy
'laeM
Ix (t, x,, µ") - X (t, X 0 , µ')I ~ ; a 1 (t) [K1 l x (t, X 0 , µ")- x (t, X 8 , µ')I+
ISSN 0041-6053. :IK,p. AtaT. ~,cypn., 1992, x. 44, M I 89
(43t
1 ;(t, Xo, µ")- x(t, Xo, µ')I¾( al (t) + r) [K1 lx(t, Xo, µ")- x(t, Xo, µ')I+
+ K2 I~ (t, X 0 , µ")-; (t, X0 , µ') I].
O6osHal!HM l!epe3 z (t, Xo, µ', µ") BeKTOp
(
lx(t,x0 , µ")-x(t,x0,µ')I)
z(t, x0 , µ', µ") = . . .
IX (t, X0, µ")- X (t, x 0, µ') I
Ha ( 43) HMeeM
(44}
, ,, ( I x(t, x0 , µ")- x(t, X 0 , µ') lo)
z0 (t, x0 , µ , µ ) = . . ,
IX (t, X 0 , µ")-x(t, x 0 , µ') lo
(45)
IY (t, Xo, µ)lo= (sup I Y1 (t, Xo, µ)I,•··, sup I Yn (t, Xo, µ)I, ... ).
f f
iaK KaK MaTpHila Qo (t) BH,IJ.a (45) Ma)i<OpHpyeTCH MaTpHUeH Qo, q:mryp11py10-
lll.eH B (6), TO HTep11py51 (44), HaX0,!l.HM
(46)
B c1wy yc,1 0BHH (6) Jim Q~' = 0. no3TOMy. nepexo,1u1 B (46) K npel!e .. 1,• npH
m➔oo
m-+ 00, y6e)l(,!l.3eMC51, 'ITO fl0Jiy<JaeMoe npH 3T0M C00TH0WeHl1e B0::lMWKH0
J!Hillb npH
T. e. npH
2 0 (t, x0 , µ', µ") = 0,
x(t , x 0 , µ") = x(t, x 0 , µ') , . .
r.( t , x 0 , µ") = x(t , x0 , µ'), rt'=µ" .
TipoTHBop e 1me C npem10JJ0)l(eill! e~1 ofi 0TJJHlJHH llpyr OT llpyr a napaMeTP0B µ 1 H
µ" 3aBepwaeT ll0I<a3aTeJI bCTl30 TeopeMbl.
C Jl e ll C T B H e I . H ,-;a•teC/11./W npu 6AUJKeHHOcO 3H0'leN UJ1 ynµw,Afl /0 U1eco
.napaMempa (40) u HG'f,GAbH0U. CK0pocmu (41) l>tOJ/O-t0 npuHJ/ IIUJ C00lll8el llC1718eHli0
fl,,, =· - - S!f(l.xm (t.x0 ),xm(t. 'l'.11))],
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n 0 K a 3 a T e JI b C T B 0 . CnpaBe,IJ.JJHBOC'Tb ll0CJJellHHX HepaBe HCTB He
TI0Cpe,!l.CTBeHH0 CJJe,IJ.yeT H3 (7), (8) C y4eTOM ycJJ0BHH Jl11nu111ua (4) H 11epaBeHCTB
( 14), (34)-(36), (25).
90 ISSN 0041-6053. YKp. MUT. 3/CYPH., 1992, T. 44. M I
HaitueM Heo6xo,UHMbl e H ,U0CT3T04Hble ycJJ0B H5l , npH K0T0pblX np e,ue.llbH3 5l
¢YH l< UH5l nocJ1e,UoBaTe.r1b1J0CTH Xm (t, Xo) BH,Ua (19) C0B n anaeT C p e w e1-rneM HC·
xonHoii KpaeBoii 3a,ua crn (I), (3).
T e o p e M a 3. llyC1nb 6btnOAHeHbt ycA06Wl meope.Mbt l. T oeda peweHue
x = x* (t) saiJa,tu K oiuu
t =f (t ,x,x), x(O) = x 0 ,
( 4 7)
~ (0) = - S ( Lf (t, x• (t, x0 ), ;• (t. x0)) = Yo ,
ei)e x * (t, x0) - npedeAbHGfl cj1y1-tKlfUfl (2 1), co6naiJaem c nepuodUtteCKUAt peiue
HUe,It Kpae6ot1 sadat.f.u (1 ), (3) nweiJa u mo,1b!W moeiJa , Koeda 1-ta1taAbNOe 3NatteHue
x0 fl6A&mcfl peiueHueM onpedeAfl10t4e20 ypauHeNufl
L1 (x0 ) = 0. (48)
fl pu amoM
( 49)
u npu 0cex m = 1, 2, ... iJ11.fl omKAO/-leNUfl mo<tJ-loeo T-nepuoiJu'iecKoeo peweNUfl
x = x* (t) = x* (t, x 0) memNo!l cucme,1-tbt (1) om ee npu6AUJ1ceNr1020 T-nep uo
dwttJcKoeo peu1e1-1 ufl xm (t, x0 ) r,uda (19), a maK:J1ce dAfl coonwemcnwy,ow,ux npous-
80J/-lbtX cnpaeeo,iuebt ou,eHKU
• . r2 qm
ll x (t) - xmCt,x0 >II ~ TT-q IIM 11,
(50)
n 0 K a 3 a T e JI b C T B 0 . UJJH YCT3H0BJleHH51 ll0CT3T0LJI-I0CTH YCJJ0BH5l
(48) Y LJTeM , LJTO x* (t, x 0) 5lBJJ5leTc5l T-rre puo,un4ern11M pewe 1-rne111 (23), rrpoxo
,um.1.1.11M rrpH t = 0 •1e p e3 T0LJKY x 0 • Ho (23) 3KBHBaJJeHTH0
. . .
x = Lf (t, x, x) - S (Lf (t, x, x)),
. .
x (0) = - S (Lf (t, x, x)), (51)
, , . .
X = f (t, X, x)- s (f (t, X, x)), x (O) = Xo.
E cJJ H Bbin0JIHHeTCH ycJJ0BHe (48), TO H3 (51) noJJyqaeM, LJT0 x* (t , Xo) 51BJl51 eT
C5l Ta lOKe pewe HHeM 38)1.8'-IH Koum (47) . }fa e)I.HHCTBeHH0CTH ee pewe HH H CJJe
JJ.yeT c rrpaBe)I.JII-IB0CTb p aBeHCTBa (49).
Heo6XO)I.HMOCTb YCJJ0B H5l (48) CJJe,uyeT H3 T3KHX paccy)f(lleHHJ1. EcJII J
X = x * (t) - p eweHHe rrepHO)I.HLJeCK0H KpaeBOH 33,1!3'-I H (1) , (3) , yJJ.0BJJeTB0j) H·
IOu te ' MaLJaJJbHblM ycJJ0BHHM (47), TO pewemie x = x (t, µ) 3 a..ua crn Kow11 (38) ,
(39) 6y,ueT y,uoBJJeTB0p51Tb nepH0,11.H'-leCKHM KpaeBb!M ycJJ0BHHM (3) rnrna , Korna
3l!cl'IeI-JHe rrapaMeTpa µ, Bbl'-IHCJJe 1rnoe ITO (40), p aBH0 HYJJI0 .
H a ocHoBaHHH TeopeMbI 2 peweHHe 3a..ua 1rn Kowu (38), (39) HBJJHeTcn Ta 1<
llie H peweHHeM nepHO,UHLJeCKOH KpaeBOH 33,1!3'-IH (1 ), (3) T0JJbK0 rrp11 e,UHl!CT·
Bel !I-I0M 3H3'-leHHH n apaMeTpa ~t BH)l.a (40) H HaLJaJJbHOH c1rnpoCTH (4 1). T a 1rne
s11aLJem-1e rrapaMeTpa M b! y)l<e yKa3 aJJH rrpu µ = i'1 (x0) = 0 H 11p 1-1
3T0.Vl Bb!IT0JIH5leTCH paBeHCTB0 (49), H3 K0T0poro, HCIT0J!b3 YH (25) , rroJiy qaeM
oue HKH (50).
3. Jl O C T a T O q H bl e y C JI 0 B H H C y W. e C T B 0 B a H H H IT e ·
p H 0 )I. H LJ e C K H X p e ill e H H ii. Hapn,uy C T0LJHblM orrpe,ueJIHIOW.HM yp aBHe
HHeM (48) BBe)l.eM B paccM0TpeHHe 11 rrpH6JIH)l{eHHOe BHlla
(52)
.n e M M a 1. n pu 6cex x 0 E Dr3 OAR onpedeAR1oi4eeo yparJJ-leHUfl (48) u npu6-
1tuJ1ce1-t1w20 8Uda (52) 8 1-wpMe npocmpaNcnw a m cnpMeOAU6a OlfeNKa
II L1 (Xo) - L1m (Xo) 11 ~ II : 2 K1 + 2_; K2 II l:: q II M II- (5,'3)
ISSN 0041-6053. }lt,p. MaT. :JtCypn., 1992, T. 44. M 1 91
fl o K a 3 e Te JI b c TB o. n o orrpe.ueJiemno
. ,
!'1 (x,i) - !'1m (X 0 ) = S If (t. x• (t, X0 ), x• (t, x0)) - f (t, Xm (t, x 0 ). Xm (t. ¥ 0))1
C y'-leTOM ycJIOBHH JIHIIWHUa (4) H HepaBeIICTB (34), (35) 113 nocJi e.UII e ru 11epa
BeHCTBa no.1y'-!aeM
/ !'1 (x0 ) - !'1m ( X 0 ) I~ S II< 1 Ix• (t, x0 )- Xm (f, Xo) I+ I<2 Ix• (t, Xo)-
_ ;m (t, x0)1J ~ l ~ K 1S(ai(t)) + l<2S(a1 (t)+ [)]Qm(E-Q)-1M,
HJIH
OTKy.ua cJie.uyeT oueHKa (53).
T e o p e M a 4. 17 ycmb cucme.Ma (I), sa8m-t1-taR e o6Aacmu (2), y8oeAemeo
pRem YC/lOBUflM (4)-(6) U Clle8y1ow,uM npe8nO/lOJ/CeHUflAt:
I) 8AR 1-teKomopoca 4eAoc0 m omo6paJ1Ce1-tue
!'1m: !'1m = !'1m (Xo) = f (t, Xm (f, X0 ) , Xm (f, Xo)) (54)
o6Aacmu D r, B o6Aac/Tlb !'1m (Dr,) uMeem cAa6y10 mOf.£KY
(55)
2) cyw,ecmeyem saMKHymaR, 02pa1-tuf.£eH1-taR 06/lacmb D 3 , npu1-tauAe:>KaU1,a.fl
D r,, co8epJ1Cau1aR 11204,Ky x 0 u maKaR, 4,,no onepamop !'1m monoA02u4,ecKu omo6pa
J1Caem D 3 I-la !'1mD3;
3) Ha 2pa1-tu4e r o . o6Aacmu D 3 BbtnO/lHRemcR 1-tepaee1-tcmeo
(56 )
Toiaa cuc111e.Aw (I) uMeent nepuouu'-tec,we c nepuo8oAt T pe111eHue
x = x (t), 8AR Komopoc:o x (0) E D 3 .
IloKa3aTeJibCTBO. Y,1111brna51 oueHKY (53) tt JieMMY [l, c. 82], 3a•
(. q"' II T2 27 II) ) Me'-laeM, llTO MHO)Kecrno !'1mD3 "'-s. II . 1 - q ( T 1(1 + 3 1<2 II M II conep-
)KHTCH B !'1D3. EcJIH 3TO MHO)KeCTBO conep)KHT Ha'-laJIO CHCTeMbl Koop1111 HaT
f'1 = (!'11, ... , 6.", .. . ), TO 11 MHO)KeCTBO !'1Da co.uep)KHT ero. 3Toro Jl.OCTaT04HO,
llT06hI CHCTeMa (I) HMeJia nepHOllH'-leCI<Oe no t C nepHOllOM T pewe,rne.
IlJIH .UOKa3aTeJibCTBa TeopeMbl Jl.OCTaTO'-IHO llOKa3aTb COOTHGJJJeJrne
( qm (II T2 2T 'I\ ) OE !'1mDa ""-II 1 _ q 6 Ki + 3 I< 2 I) II M II .
TaK KaK OT06pa>1<e1-me !'1m (x) - TonoJIOrH'-lecKoe. TO !'1mD3 - o6JiaCTh H
rpaHHUa f A D 06.naCTH 6.mDa eCTb o6pa3 rpaHHUbl I' D 06JiaCTH D3 ms - 3
(57)
B CHJIY yCJIOBHH } !l'eopeMhl MHO)l{eCTBO !'1mD3 conep)l{HT Hy.'leByJO TOllKY CH
CTeMbl 6. = (!'11, .. . , 6.n, ... ). 2:ha T04Ka np11HanJ1e)1<I1T MHO)l<eCTBY 6.D3"'- Il(r),
eCJIH BCe TO'-IKH 2 rpaHHUhl I'6mD3 MHO}J<eCrBa 6.n,D~ 11aXOll5lTC5l OT Hee Ha
paccTOHHHH, He MeHbllieM f. I-fa 3T0f0 CJienyeT, '-!TO H'/Jl b 11p1rnanJie)KHT MHO
>KecTBy
npH
(58)
92 JSSN 0041-6053. YKp. MGT. 31eyp11.., 1992, T. 44, M 1
B CHJIY paBeHCTBa (57) cnpaBe,D_JIHBO COOTHOUJer-me
z =- ~m (x) lx EI'D, '
c ytieTCM KOTOporo HepaBeHCTBO (58) npmrnMaeT BH,U
(59)
(60)
f,,fa CK333HHoro C.TJenyeT , LJTO np H BbIITOJII-IeI-mH 1-1epaBeHCTBa (60) CHCTeMa
( I) I IMeeT n e pH01IHLJ ec1<0e C n ep HO,llOM T p ewetrne , ' ITO H Tpe6oBaJIOCb ,llOK333Tb.
I. Ca.uotiAeHKO A. 1\1 ., PoHmO H. H. 4HCJieI-l!-JO·aHaJIHTH'!ec1rne MeT0J.(bl HCCJieJ.(0BJHHH n e pH0J.(H·
4ec 1rnx p e w e 1u1f1. - Kne u : Bmua urn . , 1976.- 180 c.
2. M umponoAbCKuti 10. A . , Ca,1wtiAertKO A. M ., M apmblHJOK 11. H. CHcTeMbr :;uomou1101-rnbIX
ypa1me1rn ii C n e pH0J.(H 4 eCKHMll ll )'CJI0BH0-fle p110J.(1-14eCKI-IMH K03qXpI-11!He 1-rraMH . - K HeB :
I-I ay,c J.l)'MKa, 1984.- 214 c.
3 . Ka1-tmopoeu't JI. B., A1wAoB r. fl. <l>ynKUHOHaJib HbJH auaJIH3.- 1\1. : H ayK a , 1977.- 742 c .
O 0J1y,-1e110 04.07.9 1
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| id | umjimathkievua-article-7798 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:55Z |
| publishDate | 1992 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fc/375febcdf73a89c405f3403735438dfc.pdf |
| spelling | umjimathkievua-article-77982023-09-14T12:03:56Z Study of periodic solutions of the second-order notations Исследование периодических решений счетных систем второго порядка Ronto , N. I. Martynyuk , O. M. Ронто, Н. И. Мартынюк, О. М. Приведен численно-аналитический метод исследования существования и приближенного построения периодических решений счетных систем дифференциальных уравнений второго порядка. Наведено чисельно-аналітичний метод дослідження існування та наближеної побудови періодичних розв’язків зліченних систем диференціальних рівнянь другого порядку. Institute of Mathematics, NAS of Ukraine 1992-02-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7798 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 1 (1992); 83-93 Український математичний журнал; Том 44 № 1 (1992); 83-93 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7798/9466 Copyright (c) 1992 N. I. Ronto, O. M. Martynyuk |
| spellingShingle | Ronto , N. I. Martynyuk , O. M. Ронто, Н. И. Мартынюк, О. М. Study of periodic solutions of the second-order notations |
| title | Study of periodic solutions of the second-order notations |
| title_alt | Исследование периодических решений счетных систем второго порядка |
| title_full | Study of periodic solutions of the second-order notations |
| title_fullStr | Study of periodic solutions of the second-order notations |
| title_full_unstemmed | Study of periodic solutions of the second-order notations |
| title_short | Study of periodic solutions of the second-order notations |
| title_sort | study of periodic solutions of the second-order notations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7798 |
| work_keys_str_mv | AT rontoni studyofperiodicsolutionsofthesecondordernotations AT martynyukom studyofperiodicsolutionsofthesecondordernotations AT rontoni studyofperiodicsolutionsofthesecondordernotations AT martynûkom studyofperiodicsolutionsofthesecondordernotations AT rontoni issledovanieperiodičeskihrešenijsčetnyhsistemvtorogoporâdka AT martynyukom issledovanieperiodičeskihrešenijsčetnyhsistemvtorogoporâdka AT rontoni issledovanieperiodičeskihrešenijsčetnyhsistemvtorogoporâdka AT martynûkom issledovanieperiodičeskihrešenijsčetnyhsistemvtorogoporâdka |