Simplex analysis of dynamical systems with a perturbation. New criteria of stabilization of a homoclinic separatrix and its use
We study adiabatic invariants of nonlinear dynamical systems of autonomous and nonautonomous type on symplectic manifolds, and, in particular, give criteria for the existence of such invariants and methods for constructing them explicitly. We make a thorough analysis of the phenomena of splitting of...
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| Datum: | 1992 |
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| Hauptverfasser: | , , , , , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
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Institute of Mathematics, NAS of Ukraine
1992
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/2319 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508249492553728 |
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| author | Mitropolsky , Yu. O. Antonishin , I. O. Prikarpatsky , A. K. Samoilenko , V. G. Митропольський , Ю. О. Антонишин , І. О. Прикарпатський , А. К. Самойленко , В. Г. |
| author_facet | Mitropolsky , Yu. O. Antonishin , I. O. Prikarpatsky , A. K. Samoilenko , V. G. Митропольський , Ю. О. Антонишин , І. О. Прикарпатський , А. К. Самойленко , В. Г. |
| author_sort | Mitropolsky , Yu. O. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-09-14T12:03:56Z |
| description | We study adiabatic invariants of nonlinear dynamical systems of autonomous and nonautonomous type on symplectic manifolds, and, in particular, give criteria for the existence of such invariants and methods for constructing them explicitly. We make a thorough analysis of the phenomena of splitting of a homoclinic separatrix from a hyperbolic singular point, and propose a new exact method for constructing an analogue of Melʹnikov's function, which gives a necessary and sufficient condition for transversality, which generalizes automatically to dynamical systems in Rn, n ∈ Z+. |
| first_indexed | 2026-03-24T02:22:12Z |
| format | Article |
| fulltext |
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xyBa HHHM npe,il.CT3BJJeHH51 (2 ) Be,n.e )10 cn i BBi,IJJ!OWeHH51 Ha M:
11
<I> d~Q , '\., ( aQ )c1P
cp = ' L aj - aPj j•
i = l
(3)
aQ
5-IKlllO ssecrn Be.r1wmuH Q1 = - ai + iJP. E [/) (M), 7 = T,",i, TO cpopMa qi< 1> E
J
- n
E A 1 (M) (3) Ha6yne BHrJJRny cp< 1> = dQ- )~ QidPi, 3Bi)lKH .n,.rI5I q;opMH w<2 > =
,.;...J
j = l
= dq/ 1> EA2 (M) 31-iaxo,IUIMo KaHoniqHHH a11pa3
n
ro< 2> = L dP j I\ dQj , (4)
j = l
B cm1y nem1po.n.}Ke1-1ocTi CHMnJJeKTHqHo·i cTpyInyp11 o/2l E A 2 (M) 3HaX0.D.HM0,
IU0 BCi MlJ0)KHHH cj:JyI-IKUHi P, QC[/) (M) cjlyI-IKUiOI-IaJJbHO He3aJieRrni Ha M,
np11q0My {Qj, Q,J = 0 = {Pj, Pk}, {Pj, Qk}"' = 0;, 1< , j, fl= 1, n, T06TO iHBO
JJIOTHBHa CHCTeMa cj)y1-IKuiii QC[/) (M) K31-IOHi•rno cnpm1,eua .D.O TaK01 )K
OICTeMH cj)yHKUiH P c [/) (M), w.o i Tpe6a 6yJJo BCTaH0BHTH. 3oKpeMa, B
3M iHHHX {Q; P } eBOJIIOUiH .D.HHaMitIHOI CHCTeMH (1) JiiHiihrn, T06TO Qj =
= ci (t - t0 ) + Qio E IR, Pi= Pi E IR, j = T,n, .n.e ci E IR, j = I,n, - .ueHKHH
Ha6ip «ll!BH)IKOCTeH».
Illo6 nepeiiTH Tenep .n.o iHTerpaJJbI-IHX i1rnapiaHTiB Oyaimape-KapTaHa,
po3rJJ51 HeMo irrnapiaHTHHl1 MH0fOBH.D. M ,,c_ M y mrna;IKy JlOfO KOMTiaKTHOCTi
6iJJ bW .n.eTaJJbH0. B cnny TeopeMH JliysiJJJJH l3, 32] MH0roBH.ll M 1,, r0Me0Mopcjm11il
T" - n-BHMipHoMy rnpy. H exaii ai E H 1 (M ,,; Z:;), j = I~- 1-w6ip umrnia
Ha M ,,, w.o y-rsop!OE 6a3HC o.n.H0Mip1mi' rpyn11 roMoJJoriii MHorosu.n.y MP, i po3-
JIHHeMo TaKi iHTerpaJJH:
I t, (I) • -1-
Yi = 2n ~ rp , / = , n, (5)
Oj
JJ.e, HK i paHirne, (J)C2l = dq/ 1> E A 2 (M)- BHXi.D.Ha CHMnJJeKTHqHa CTPYKTypa Ha
M. Jlerno 6atJHTH, mo BCi cj)y1muff (5) - 33 K0HH 36epe)KeHHH Ha M JlJJH )lHHa
Mit!Ho·i CHCTeMH (1) i YTB0p!OI0Tb iI-IBOJIIOTHBHy cpyHKUiOHaJJbHO He3aJJe)lrny CH-
CTeMy c\)YHKUiH y ED (M), npH40MY, ol!eBH.D.H0, 'Vi = 'Vi [Pl, j = l,n, i ai.n.-
06pm 1<eHHH p -+ 'V - HeBHp0Jl)KeHe. 3B iJICH BHTIJIHB3€ napaMeTpH33UiH p j =
= Pi lyl, j = I~ HKa Bi.uo6pa}KaE n-BHMipHHH I<0MnaKT MP B KOMnaKT MP c➔
c _. _i\1 . Bse.n.eMo HacTym1e 6ara-ro3Haq1-1e Bi.n.o6pa:tKeHHH 3r i)].HO MeTOJJ.Y faMiJib
TOHa-51Ko6i [3]:
<D [y, Q] = I cp< I> E IR,
o(Q.Q.)
ISSN 004/-6053. Yt,;p. MaT. :HCypn., 1992, T. 44, M 1 4- 1·812
(6)
49
ne o(Q, Qo) - neHKHH rJia JI.KHH IIJJIHX 3 IIOtJaTKOM B TOtJUi QoEMv i KiH
UCM B TOtJUi QEM,,, HKHH JJe)l<HTb Ha Topi M,,. 3a norroMOro!O (6) 6yJJ..y~
€MC Ka110Hil!He rrepernopeHHH ©(y, 0)-+ (P, Q), ne 0={0iE.0(M): i=
= (ti} - HOBa iHBOJIIOTHBHa CHCTeMa cpyHKUitt Ha M, IIJ.O BH3Hall a €TbCSI
3 piBHHHb
t7)
HKi 3a6e311el!y!0Tb iHBapiaHTHiCTb CHMITJieKTHtJHOJ CTPYKTYPH ( 4), T06TO
©*ffi<2> = ffi <2> Ha M. BHKOpHCTOBY!Ol!H Terrep (6) i (7), 3HaXOJl.HMO, LUO Jl.JIH
scixj,k = l,n
r a S a t, < n avh ~ I d0j=~ d<D=r 'j' (j) =2Jt~ =2Jtuj,k,
ok 'VJ ak Y1 ak 'VJ
rn6To BeJittl!HHH 0i E [/) (M) Ha rnpi M., 3ana10-rb TaK 3BaHi KYTOBi, tJH UH
KJiillHi 3MiHHi, a Yi E [/) (M) - 3MiHHi «ni'i». B TepMiHax KaHoHil!Ho'i cttcTeMtt
3MiHHHX {0; vl Jl.HHaMitrna CHCTeMa (1) Ha M., 3anae KBa3irrepio,n:ttl!HHii pyx
3 Ha6opoM t:f3CTOT ffij = oH lyl loyj, j = G, ne H fyj C [/) (M)- BiJl.ITOBiJl.
HHH raMiJibTOHiaH CHCTeMH ( 1).
TaKHM l!IHIOM, MH rro6yn yuaJIH CHCTeMy (5) iHTe rpaJibHHX iHBapia1-1TiB
TTyaHKape - KapTaHa, eKBiBaJieHrny BHXiJl.Hitt CHCTeMi iHBapia1-1TiB P E I/J (M)
Jl.JIH Jl.HHaMitrno"i CHCTeMH ( l ). B 3araJibHOMY BHITaJl.KY, I<OJIH JJ.HHaMit:fHa CHC
TeMa (I) Ma€ iHTerpaJibHHH iHBapia1n1rnii MHOroBHJl. Mp c M, Ha HKOMY
icHy!OTb roMoTorri,rno HeTpHBiaJibHi llHKJIH o i E H 1 (Mp: ;E), j = 1, m.,, mv~ n,
11ptt1I0My CHMrrJieKTHl!Ha CTPYKTypa u/2l E A2 (M) iHBapiaHTHa, TOJl.i 3fiJl.HO 3
(5) sci cpy1rnui"i Yi E [J) (M ), j = l, m,,, ne
Y · = __!_ t, (J)O> (8)
J 2n ':l-'
Oj
- iHBapiaHTH Jl.HHaMitJHO'i CHCTeMH . .IlittCHO, CHMITJieKTHtJHa CTPYKTypa (l)( Z) E
EA 2 (M) i,rnapiaHrna BiJl.HOCHC Be1nopHoro rroJIH ( l ), KOJIH BHKOHaHa p i B
HiCTh LKffi<2 > = 0, ne LK = iKd + diK - rroxin Ha Jli 13, 321 B aJire6pi fpac
MaHa A (M) nmpepeHuiaJibHHX cpop :\-1 Ha M1-1oroB11ni M. 3 yMoBtt, LUO dLK =
= LKd Jl.JIH nosiJihHoro BeKTOpHoro rroJIH K: M-+ T (M ), i3 (8) 31-1axon1i.\.10
/ 1 r , I> 1 r
dyi dt = L KYi = 2n J LK<fJ = 2n J d£ = 0, £ E [J) (M), ne j = l, rn ,
Oj Oj
L1<cp< 1> = d'f,. TaK HK O = L1<ffi< 2> ⇒ LKd<fJ ⇒ dLKcp<ll * d2~. 3aysmKHMo TaKO)K,
LUO B (8) cpopMa <p< 1> E A 1 (M) BH3Hal!eHa 3 TOl!HiCTIO no rros1-1oro Jl.HcpepeH
uiaJia, HKHH 3aB>KJl.H MO>KHa BH6parn TaK, wo6 cpyHKUiH £ = 0. 0T>Ke, CjlyHK
nii" (8) BH3Hal!al0Tb iHTerpaJibHi iHBapiaHTH Ha M Jl.JIH (!), HKi B 3araJibHOM Y BH-
rram<y E rnanKHMH cpy1rnuisiMH 3aKoHiB 36epe)KeHHH Pi E [/J (M), j = I, mp,
mp < 2n, 511,; j He o6oB 'Sl3KOI30 llOBHHHi 6yTH B iHBOJIIOUil.
llI,06 p o :m rnpHTH KJJaC M0)1(J1HBHX Cj)OPMYJI Jl.JISl 3HaXO,[l)l(eHH51 iHTerpaJibHHX
iHBapiainiB HeJiiHiHHOi" Jl.HHaMil!HOi" CHCTeMH (1) Ha CHMITJieKTHl!HOMY MHOroBHJl.i
M, 3ayBa)l(HMO [ l ], LUO 3riJl.HO 3 cpopMyJiolO K.apTaH9 . - HhoTep LKffi<2> = 0
.ZI.OBiJibHa q>yHKUiH
n k
µ,. = s (A (1)(2))! A dPj E [J) (M), (9)
M2n-k j=I /-1
p
11.e k = 0 ,2n - l - t:fHCJIO MO>KJIHBHX iHBapiaHTiB JJ.HHaMi'IHOi" CHCTeMH ( l),
MHOroBHJl. M~n-k = { u E M 2n : Pi [u) = pi E IR,, j = 6-:,i} - rirreprrosepxHH piBH5t
iHTerpaJiiB ( 1), k = 0,2n - 1, E iHTerpaJibHHM iHBapiaHTOM Jl.HHaMit:fHOi" CHC
TeMH (I). 3oKpeMa, y BHITaJl.KY noBHoro KOMyTaTHBHOro 1-1a6opy iHaapiaHTiB
50> ISSN 0041-605;1. YK.p. Mar. :11CypH., 1992, T. 44, M I
P = {Pi E !lJ (M): j = T:n} cj:>opMyJia (9) Ha6y,lle BHrJIHJlY
n n n
µn = S J\ (u} 2>/dPi) = S J\ dQi = n f dQi
Mn j=I Mn j = I j = I CJj
f) f)
n n l[aj]
i= l
TaKHM quHOM, µn - .uo6yTOK «.UOB)l{HH» l [ail E IR.+, j = T,"n, a-UHKJJiB Ha TOpt
n
Tn = ® a i ~ M;, a6o (3ri.n.Ho 3 TeopeMOJO JiiyaiJIJJH [3]) CHM TI JleKrnqHHii
i= l
o6 'eM µn (M;) cpa3oaoro npocrnpy op6 iT M~ c: ... M. Y BHna.n.Ky n = 1 Ma€Mo
.uaa 04eBH)lHi iHBapiaHTH: ~t0 = \" u/2l = (' dcp< 1J = \ <p(IJ - o6'EM cpa30-
M1, Mh a,,,,;iJMh
aoro npocrnpy rinepnoaepxHi pia1rn cpyHKUii" f aMi.'lbTOHa H E :lJ (M 2) i µ1 =
= 1> CJ/2>/dH = l (a1]- .UOB)KHHa 3aMK1-1yrn1 op6iTH ai.n.noBi.UHOI raMiJihTOHO-
a,
BO! OlCTeMH.
2. A )l i a 6 a TH q H i i H B a p i a HT H a BT O HOM H H X C JI a 6 -
KO 36 ypeHH X HeJJiH l!IHHX lll!HaM iqHHX CHCTeM 3
M a JI H M 11 a p a Me T p O M. Po3rJ15llleMO c, BT0H0MHe CJia6Ke 36ypeHHH .u1rna
MiqHoi" CHCTeMH (1) y BlffJlf!Jli
ut = K (u; 1o 1 = K [uj + eF Ill; ej, ( 10)
,ne 3aJJe)l{H iCTb Bi,n MaJJoro napaMeTpa IR. :l e - 0 BBa)l{a€TbCH aHaJiiTHqHolO.
5IKlll.O nptt e = 0 )lHHaMi<rna CHCTeMa r 10) )l0nycKaE iHTerpaJibHi iHBapiaHTlf
(8) ain1-10rno c11Mn:1eKTH<JHo·i CTPYKTYPH w< 2> = dq/ 0 EA2 (M), TO 11p11 necpop
Maui ·i (10) .a,eHKi UHKJIH 3 {aiE H 1 (Mp; ;z) : j= 1,m,,} Ha iHBapiaHTHHXMHO
roBH)lax Mp pyi-inyIOTbCE I1,3,Sj . 51Klll.O a(i)EH1 (Mp(e>; £')- Ti UHKJIH Ha
36ypeHoMy iHTerpaJ1bHOMY MIIOrGBH)li M p(e) c: ... M , lll,0 3aJIHlllHJIHCb, TO 3a
cpopMyJIOIO rnny (8)
I ,f •. · m< I>(e) YU>= 2:n: \.\.i "' ( 11)
CJ(j )
aHaXO.llHMO a.uia6aTHqHi i1-rnapiaHTH Yu1 E :lJ (M), )le cp< 1> (e) E A1 (M) - 36y
peHa 1-cpopMa Ha M, mm :rnnae 36ypeHy cHMnJieKTHq1-1y CTPYKTYPY ro<2> (e) E
E J\2 (M) i iHBapiaHTHa Bi)IHOCHO Bel<TOpHoro IlOJIH (10) 3 BHCOKHM nopH.UKOM
TOqHocTi lll.O.llO MaJioro napaMeTpa e - 0. 5I1mio TaKa .uecpopMauiH mono napa
MeTpa e - 0 CHMnJieKTHq~JO-i CTPYKTYPH ro< 2> E A 2 (M) icHye, To a)lia6aTttqHi'
iHBap i aHTH (11) icHyJOTb [1] 3ri.D.rIO 3 3.'.1r3o1bHHMH pe3y.TJbTaTaMH B paMKax Teope
MH KoJIMoropoBa-ApHOJib.Ua-Mo3epa . .IlJJH BH3HaqeHHH a.uia6an1qHoro i1-1aa
pi aHTa [l, 3, 271 oy :teMo KopttcTyBaTHCTb 1-1 acTynHHMH BH31-1aqe1-1HHMH.
B H 3 H a 4 e 1-1 H 51 I. cfJyHKU,iR y E :1J (M) 1lQ3UBQ€tnbCfl aoia6amutt1lUM
i1teapiaHmo1rt cAa6Ko 36ype1toi: HeAiHiuHo"i ouHaMit{,HO"i cucmeMu u, = K [u; el
1ta MH020euoi M :1 u, RKU{O OAR ooeillbHOW T] >O MO;JiCHa eKa3amu ma,ce e0> 0,
( 12)
811R 0cix e < e0 pie1to1rtipHo 3a aaHuuu K otui u0 E M. f!K,U{o eupas (12) cnpaeeiJ
llUBUU OAR 1rtai1.;)!Ce ecix u0 E M 3 mot{,Hi::rillo oo lrtflO;JiCUHU 1rta11o"i 1rtipu, mo <pyHKt.filo
,y E 2) (M) Ha3uBa10mb 1rtau;)!Ce, a6o Klia3iaoia6amuttHu1rt i1teapiaHmo1rt.
B 11 3 H a 4 e H H H 2. A a ia6amut{,Huu i11.Bapia1tm y E :lJ (M) 6y8eM.o 1tasu
eamu CUllbHUlrt, RKU{O icHye maKe e0 > 0, uio Ollfl eci x e < e0
sup I y [u; e] -y [u0 ; OJ I= O(e).
IE[O, 1/e)
(13)
H a c JI i .n. o K 1. H exatl <pyHKlfiR y E :lJ (M) ma,ca, tl{O 811R ecix t E [O,
l ie] npu e < e0
dy [u; e]/dt = O(e2). (14)
ToiJi y E :IJ(M) - cuAbHUtl a8ia6amu1mut1 iHeapia1tm 1-1.a M.
ISSN ,0041 ,6053. Ytcp. MlZT. :J1Cypn., 1992, T. 44, ,M J •
Bmu,e MH 3po6HJHI cyTTEBe npHnymeHH51: e-.!1,e<pOpMaUiH BeKTOpHoro TTOJI51
(JO) 36epirae H raMiJibTOHOBicn, y .!l,OCTaTHbO BHCOKOMY nopHJI.KY mo.!l,O MaJioro
napaMeTpa e-+ 0, mo .!l,ae MO)KJIHBiCTb cKopHcTaTHCb KAM-Teopie10 npo 36e
pe)KeHH51 iHBapi aHTHHX MHOfOBH.!l,iB [ 1, 3, 51. 51Kmo )K TaKa e-.n,ecpopMaUiH CHM
llJleKTHl!HOi0 CTpyKrypH ,!l,Jl51 (10) si,ucynrn, npo6JieMa iCHyBaHHH a,uia6aTI1'-IHHX
i1rnapia1-1TiB 33JIHW3€TbCSl Bi,UKpHTOl-0, i B 6iJibWOCTi BHna,uKiB BOHH He icHyl-OTb.
MoT11Bauie10 Uhoro rnep,il,)Ke1-nrn e To1'i cj:>aKT, mo 3 iCHysaHHH e-,ue<j}opMauit
w12> (e) EN (M) CHMnJieKTH'-IHOI CTPYKTYPH [33 ] ,uJIH (10) Ha MHorosH,ui M icHye;
i1mapiaHTHa si,UHOCHO (10) Mipa o6' eMy dµ , (u; e) = (w<2)) 11 (u; e), u E M 2"
B HpOTHBHOMY BHna,uKy T3KOI Mip11 Ha M 2" He icHye, mo npHBO,UHTb no Bi,UHO·
m e1-1mo JJ,o Mip11 dft (u) = (w< 2>) 11 (u) E i\2" (M) ,uo HBHma ,u11c1-1naui1 [34] MHO·
JK111m Tpa€Krnpiti 3 ,uaHHMH Kowi j3 06.TJ.'.lCTi oM c M np1-1 t = t0 , TaK HK se
JIHl!HHa µ ( <'3M (t)) npH t =I= to HE' 36epira€TbC51 3ri,UHO 3 TeopeMOl-0 JliysiJIJIH [ 1,
3, l 01. T11M caMHM Hei CHysa1-1Hsi c11M nJieKTH'-iHoi" .n,ecpopMauii" w<2) (e) E i\2 (M)
A1)51 ,UH l-laMil!HO"i CHCTeMH (10) MO)Ke CJIY)KHTH ecpeKTHBHHM KpHTepieM ic1-1ysairnH
AJHI (10) a,uia6aTH4HHX iHsap iaHTiB (5, 35].
1Uo6 Cq:>OPMYJIIOBaTH B 3H3JiiTH'-IHOMY Blff JIH,Ui BK333HHH BHill,e KpHTepiH,
po3rJIHHeMo eJieMe1-1T cp ET* (M) i JiiHiHHe BeKTOpHe piBHHHHH J1aKca (7, 10,
I 1 l LKcp = 0, mo eKsisaJ1e1-1n1e TaKoMy:
1· cp1 +J( •cp = 0, ( 15)
Jl,e K' - MaTpHUH 5IKo6i se1<Topa K [u; ~:J E T u (M) B TO'IUi u EM, * - one
pauiH cnpH)KeHHH si,uHocHo crn1-1,uaprnoi 6iJ1i1-lit1Hoi" rpop MH ( •, • ) Ha T* (M) X
X T (M), npH'-IOMY BBa)KaeMo , 11I,o T*(M) ~ T (M). BpaxosyJOT.JH, mo
iJ<p!at = 0 (o,uttopin.HicTb e:1eMe1na (p E T* (M)), i3 (1 5) 31-1axo,u11Mo
cpt + K'* •cp = qi' ·K - cp'* ·K + cp'* ·K + K'* ·cp=
= (cp' -cp'*)·K + grad(cp, K) = w-K + grad H = 0, (16)
Ae 3a BH3Ha'-le!IH5IM W = ,p' - cp'* - KOCOCILvleTpI--l'!Ha MaTpHL{H CHMl1JleKTI-l'!H'.l 0l
CTPYKTypu w<2>(e)EA2 (:11). mo BH3Ha'laeThCH 3a cTaH.uaprnHM n paBHJIO \1
ro12> (e) = - + (du. A w ju; e] du), A - 3B t1<1arrna onepauiH [3) 30BHilllHhOro
Jl06yrny B aJire6pi rpaCMa!-Ia A (M); (rp, K ) = HE Jj (M) - cj:>yHKUiH ra
MiJibTOHa ,UHHaMilll·IOi° CHCTeMH (10) y BHllal(l{Y p03B 'H 3LOCTi piBHHHH H .naKca
(15) . llii-ic Ho, 1-1 exa(1 { ·,·}u, =(grad(·), fJ!. ~rad( ·) ), ne :£ = w- 1: J * \ \1)-+
-+ T (M) - TaK 3Ba1-rnfl iMnJieKTH'!I-1H i'.'1 (KOCHMirneKTHl!HHH) [7] onepaTOp 1-1a
MHOroBH]li M . To.ui piBHiCTb (1 6) eKBiBaJieHTHa TaKiH raMiJibTOHOBilI c.popMi 33·
1rncy Ha M Bi,11.HOCHO nyaccOHOBOl CTpy1nypH {.' • }w ,!l,HHaMitJHOI CHCTeMH (10)
a1 = {H, u}w =0 -fl (e) g r ad H (e) = K [u; e). ( 17)
3ay1•a)KHM0, ULO HKmo piBHHHHH JlaKca (15) ,uonyc1<ae po3B'H 30K cp [u; i::I E
E T * (M) ,ueHKoro nopn,11.Ky mo,uo napaMeTpa e -+ 0, TO sinnoBill.HO iurnaMil!Ha
cttcTe~a (10) 6y,ue raMiJihTOH030fO rnro )K nopH.UKY mo,11.O n apaMeTpa e-+ 0.
Or)Ke, MO)KHa ccl)OPMYJII-OBaTH TaKe TBep.u>Ke111-1n .
T e o p e M a I . D.ocmamHbo10 y,woao,o icHyaaHHft aaia6arnu'lHOW iHaapiaH
ma asmoH0MH0 CAa6;w 36yp er1,oi" He,,iHiUHOi" duHaM i'I.HOi" cucrneMU (10) € p038' f1,3-
HiCmb y eucoKo.My nopRaKy Ma11020 napaMempa e -► 0 xapaKmepucmu'lH020 p ia-
1-LflHHR JlaKca (15).
PiBHHHHH JlaKca B CHJIY CBoE°i JiiHit11-1ocTi nonycKae p.Hn MeTo,niB "ix po3s'n3-
1CY , Hati6iJibW ecpeKTHBHHl\l cepe,u flKHX € TaK 3B3H Hl1 acH~rnTOTHl!Hl!H µ-MeTOll.
17. 111. cyTh HKoro noJIHrae B po3BHHe1rni HeBL'lo.\1oro po:rn°H3KY cp lu; el E
E T* (M) B acHMTTTOTl!l!H:-IH µ-pHn no cTeneHHX (u)-cpopM 3 BHKopacTaHHHM pe
KypeHTHoi' npoue,uypH .UJIH i"x BH3Hal!emrn . A caMe, 1-1exat1 BeK'ropHe noJie (10)
AonycKae µ-pO3KJia,u
K[u; e] = K< 0> [i/ 1>; e] + µK< 1> [u(l); e) + µ 2K<2> [u(l>; e] + ... , (18)
~e u=u"+µ,.u< 1>ET(M), µ-O-1-10B11i-i Ma.i!Hi1 napaMeTp, K<il[un>;e\E
ISSN 0041-6053. Y,cp, MIH. 'JICYPH.., 1992, T. 44, Ni l
ET(M)-Bi,ll.rroBi)].Hi (u)-cj:JopMH · j-ro rropHJ1.KY, i E:Z+, rrpHlJ0My B (18) MK
rrpHnycTHJIH ,UJH! 3pyt!HOCTi , lll,0 ,[(111-IaMilJHa CHCTeMa (10) pery.rrnp,ra B T0 'IU.i
u0 E M 2n . Po31rna,uy ( 18) ai ,unoai,uae ttacTym1e µ-npe,ucTaa.rie1rnH )].JJH BeKT0p
Horo noJJH dldi: M _. T (M), t E IR:
d d rf 1 2 d
dt = dt o + ~l ~ 1 • µ dt 2 + ... ' ( 19)
,ue HoBa i epapxiH eBomouittHHX n apaMeTpiB t j E IR, j E :Z+, BH3HaLJa E.TbC5J j3
err iBBi,UHOllleI·lb
di/ 1l /dtj = K(i+ ll [11< 1>; el, j E:Z+•
AHaJJoritrno noKJJa)leMo Tenep
(20)
cp [u; e] = (p [OJ [u <1l; e] + µcp[tl [u 0 >; e] + fI 2cp[2J [u 0 >; e ] + ... (21)
i ni)].CTaBHMO cniBBi)lHOWemrn (18)- (21) B x apa1nepHCTHlJ!·Ie piBHHIIHH J1aKca
(15) . B pe3yJJbTaT i 0,ll,ep,KHM0 p eKypeHTH y TIOCJii)lOBHiCTb JliHi fnmx Heom 10p i.ll.
HHX piBHH Hb
[OJ ' * [OJ rr 10 + Ko · cp = 0,
er LI] + K~ • . (pl I ] = - K / . (p[OJ' (22)
[2] '* [2] ' * [I] , ' * LO ] [I]' (2) C/Jt. + K o •cp = -Ki -cp - I, 2 · (P · - cp · -/( , .. .
i T. ,u., .ne J(< i+ O' = K1, j E :Z+• Dpu po3B'513a 1111i cHcTeMH p e1<ype1ITHHX pis
HHHb (21) Heo6xi.n1rn BpaxyBaTH, lll,0 B CI1JJY MclJJOCTi n ap aMeTpa µ - 0 eJJeMeHTH
u0 EM i u<O E T (M) 3a.ll.0B0JJbH5lIOTb JJiHeapH 30BaHi piBHHIIH5l (JO):
ul.,1> = K~ ,1/ 1>, duofdt0 = K< 0 > [11< 1>; el, (23)
lll,0 BHTIJJHBaIOTb 3 ( 18) i (20) npH j = 0' 5!Klll,0 BpaxyBaTH 0lJCBHJ1.HY TOT0>K
HiCTb K~-i/ 1> = K< 1> [u' I>; el, u< Il E T (M).
B 3ar aJ1bH0MY BHIT3JlKY po3KJJaJl (21) M icTHTb 111.ono CTeneHiB napaMeTpa
µ - 0 11ec1<iHlJCHl·Ie l!HCJJ0 l!JJeJ-IiB, lll,0 3 a 1-1aJJiTHlJI-I0·1 T0l!KH 3opy E. Hepo33 ' 513-
H0!O CHTyaui E.JO. AJi e Bpa xoBy I0tJ}I, lll,0 B 1rnwii1 BHXi)lHitt 3a.ll.alli (JO) MiCTHTbCSl
m e 0)lHH, OCHOBJ-JH{! Mcl JJHH n apaMeTp e _. 0, TO 0lJeBll,11.H0 PH.ll. (2 1) Heo6xi)l
H0 o6 ipaaTH irn T0MY tJJJ eHi cpl1l f uO>; el, l E J!+, JlJIH mrnro BHK0HaHa yMosa
acm,mTOTHtJHo'i MaJJocTi: cpUl [u(l >; el = 0 (e2), e _. 0. OcTaHHE. 3aysa.»<eHHS1
cnpomye 33)],aLJy aJ-Ia.TJiTHlJHOro BHK0pHCT3HH5l Ha npaKTHlli aCHMTIT0THlJ H0rO
µ-MeTo,11.y, oco6JJI1BO y T0MY BH TI 3.ll.KY, K0JJH BHXi)ll-Ia .ll.HHaMitIHa CHCTeMa (10)
np 11 e = 0 r 3M i.TJbTOHoBa 11a ' lia3oBOMY npoCTopi M.
IlpyrnM cyrreB II\•t 3ay aa)1<emrnM ,11.0 MeTo.Uy no6y,11,0BH HecKittt1e1rnoi
iepapx i'i (u)-cpopM a (2 1) E. TaKe: Tai< HK a)l ia6aTw11,;,1{1 i1rnapiawr ')' E .0 (M)
,11.JlH ( JO) BH3Ha'!a€TbCH :ia cpop:vt yJJO !O (11 }, TO 0lJeB!1,'l,H0, lll,0 Ti K0MTI0HeHTK
(pl ' J [1/ 1>; e j E T* (A1), ,1.urn Hl<HX BHI<0ua11a yMoBa rpa,lli E. HTHOCTi Bo.fJbTepH
ll I , 36! BiffJJH.ll.Y (cpl'l [u ' I>; e l)' = (rpl/J [u< I>, el)' *, i/ I > E T (M), BKJia.D.y B a.n.ia-
6arntmHH iHBapiaHT y E .0 (M) He BH0CHTb. 3BiJlCH BHTIJJHBaE, mo pH,11. (21)
e(pel<THBHO 3 T0lJl{H 3opy npaKTHlJ i!0l'0 33CTocyBa!IHH o6pHBaE.TbC5l Ha T0MY
nopHJU<yl E :Z-1- (u)-cpopMH cpl'1 [u< I \ e] E T * (M), llJJH mrnro BHK0HaHa BKa-
3al!a BHLUC YM0Ba rpa,11.iE.H TH0CTi B o,1bTepH, T3K 511{ BCi 1rncTym1i (u)-cj:JopMK
cplil l uCI>; e] E T* (M), ( ? l, B CHJlY pe1<ypNITHOCTi ·i x BH3HalJeHHH Ta 06Me
:»<e1-1ocTi ~H1opHJ1.KY ,11.JJH cpyHui'i f aM iJJbTOHa HE .0 (M) 6yJlyTb Te.»< 3allO
B0JJbHHTH yMoBy B vJJbTepH.
Bi.U3H3lJHMO Til K0}I( anp iopHy MO)l(JJHBiCTb 0TpHM3HH5l 3araJJbH0ro p03B 0 H3-
KY (J) lu; el E T * U\1) piBHHHHH J1a1<ca (1 5), mrnf1 E. Hep erym1pI IHM 111.0.no napa
MeTpa c:. _. 0, m orn rpa11HUH Jim cp [u; el fl, T* (M) 11e i cHyE. B T * (/\1) . 11,e csin.-
e-o
q11Tb, 30KpeMa , npo Te, w.o .UHHaM i tJHa CHCTeMa (10) npH i:; = 0 arrr;iopi MO:lKe
6yTH raMiJJbTOHOBOIO Ha cpa30BOMY npoCTopi M' aJJe He 33B:»<.llH )lonycKa€ CH -
n JJeKTHtJHY .uecj:JopMaui10 npH e =I= 0. 3Bi.UCH BH TIJJHBaE, lll,0 a.ui a6aTJ1q1mii i 1ma-
ISSN 0041-6053. YKp . .MQT. :HCYPH,, 1992, T. 44, M 1 5iJ
piaHT (11) ,!I.JJH mrnaMiqJ-101 CHCTeMH (10) M0)Ke 6yTH BH3HaqeHHH, TaK HK CHM·
OJ1eKTHqHa CTPYKTypa w lu; el = cp' [u; el - cp'* lu; el M0)Ke He MiCTHTH B co-
6i CHHryJJHpHHX i,-qJieHiB npH E - 0, HKHX TaK0>K He Ma€ B Ub0MY BHTTa.llKY i B
qx>pMyJJ i (11), i c11MnJ1eKTHqHa .lleq>OpMauiH w<2l (e) E A2 (M) Mae np11 e _. 0
peryJrnpHy rpaHHUIO. ,,
n p H K JI a .11 I. Po3rJIHHeMo Ha cpa3oBoMy rrpocTOpi M = IR." aBT0H0MH0
c.r: a61<0 36ypeHy HeJii1-1ifrny .llHHaMiqHy c11cTeMy THrry XeHoHa-XeiiJJeca l3, 371:
dxldt = y + i,xy (x - y) + e-ty } KI . l = X, y, e,
dyldt = - Q2x + e (x 2 - y 2)/2 - e Q 2x; 2 (x - y)
(24)
JJ.e Q E IR+ - qacT0Ta K0JJHBaHb, e _. 0 - MaJJHH rrapaMeTp. BeKTOpHe rroJie
(24), 3ri.llH0 3 (19) i (20), Ma€ µ-p03KJia,!I.
du( 1>/dt0 = K< 1> [u<1>; e] = (y(I>, -Q2x( 1>)\
du< 1>/dt 1 = K< 2> [u< 1>; e] = (ex< 1Jy(l>, e (x:< 1>2 -y(1>2)/2)\
(25)
du(ll/dt2 = K(3J (un>; e] = (ex(I>y<J> (x(I> -y(J>), - eQ2x(1J2 (x<JJ - y(IJ))'\
du(!J/dt,. = K("+l) (u(l>; e] = 0, n;;;,,: 3,
Ae 3a BH3HaqeHHHM u0 = 0 EM, u< 1> = (x< 1>, y<l)yr. ET (M). Bi.11rroai,I1.HHH it-p03-
ua.ll MaeMo TaK0)K .llJJH crrpH>Ke1rn·i MaTpHui noxi,nHo·i <l>pewe K'• [u; e]:
K~• [u(I>; e] = II O - Q211 • K;• [u(J>; e] = II ey(I> ex<:> II •
1 0 ex(I) - ey' >
en_ 2x(l) (2y'l) - 3x(l)) ll ·
eQ2x;(l)2
(26)
TTpH e-+ 0 ,ll.HHaM i q1-1a c11cTeMa (23) a11po.11myeTbCH .ll0 3BH'lai1Horo JILHIH·
aoro OCUHJIHT0pa, CHMIIJieKTH01Ha c-rpyKTypa HK0ro w< 2> = dy (\ dx - KaH0·
BitJHa. Po3rJIHHeMO p03B 1 H30K piBHHI-IHH JlaKCa (15) ,UJIH BeKTOPH0ro II0JIH
(24) TaKHH, mo rrpH e-+ 0 rropo,a.mye CHMnJieKTH'lHY CTPYKTYPY w( 2J = dy I\
I\ dx . Op11 µ-+ 0 po3r JIHHeMo arnMnTOTWIHHH po3KJia,ll. (2 I), ,ll.e
cpfoJ (e) = 0, <p[IJ (e) =-} (y' 1>, -- x(l>yr._ To,ll.i, po3s 'H3y1otJH peKypeHTH0 piB-
BHHHH (22), 3H3X0,ll.HM0
I
qi[ u; e] = 2 (y + exy2, - x + 2ex3/3//{mod (grad:/) (IR.2))} + 0 (e2). (27)
Bi,11.nosi,I1.Ha CHMrrJieKTH 0rna cTpyKTypa w <2> (e) EA 2 (M) 3a,ll.a€TbCH HBH0IO cpop-
MyJJOIO
w< 2> (e) = (1 + ex (y - x)] dy I\ dx,
lll,0 € HacJJitJ.KOM 3araJibH0! cpopMy JIH
(28)
w(2> (e) = - ·+ (du, I\ w [u; e] du), (29)
]1.e w [u; el= cp' [u; el - cp' • [u; el. TTp11qoMy y a11na,nKy (27) CHMnJJeKrnqHHH
onepaTOp w [u; e]: T (M)- r (M) 3aLta€TbCH MaTp11ue10
w[u; e] =[I +ex(y-x)]II-~ ~ll+O(e2), (30)
m.o H npHBO,ll.HTb tl.0 pe3yJJbTaTy (28).
TTepettt1.eMO Tenep ,ll.0 p03rJIHAY 3ri.llHO 3 (11) BHpa3y .llJJH a.11ia6aTHq1-1oro iH-
6f ISSN 0041-6053. ;y,cp. MaT. :JtCyptt., 19':n, r. 44, NJ 1
BapiaHTa y E f1J (M) JUIH JUll-IaMi 1rnoi" CHCTeMH (24):
'f = 2~ ~ cp< 1> (e) = 4~ ~ [- (x - 2ex3/3) dy + (y + exy 2) dx] + O (e2), (3 1)
(1 (1
.ne (p< 1> (e) = (cp [u; e], du), Kornyp a E H 1 (M1zce>; ;z) BH3Ha tJaETbC51 iHTerpaJlb
HHM MHOroBH,lJ.0M M1z(e) = {u EM : H (e) = h E lR--1- }, H (e) E f/) (M) - a i ;1.n0B i,1J.Ha
tjJy1110LiH faMiJlhTOI-Ia ,ll.Jl 51 ,lJ.HHaMi 1moi" CHCTer.rn (24). OcKiJlbKH tj)yHKLli51 faMiJlb
T0H :1 HE 'lJ (M) JlJl51 JUll-laMi'-11-IOI CHCTeMH (10) a1-131-1a1iaeThCH 3a po3n ' H:-iKoM piB
HHHI-151 ( 15) tjJOpMyJ1010 H = ( rp, K ;, Birnop11crnay10 1rn po3wupe1rnt1 rpa.n.iEHTHH
MH tJJleI-IaMH p03B' 5l30K (27), 3HaX0J]J1MO
H (e) = + [y2 + Q 2x 2 + e (xy 2 - x3/3)] + 0 (e2) = h,
u1.o e HOpMaJJbHOIO cpopMOIO 5ipKrocpa (IJYHKUii" faMiJihTOHa 1-1a IR2 i 3 pe3oHaHca
MH qacTOT TpeTboro nop,1,1iKy l3, 37].
To,ll.i ,ll.JJ51 eJJeMeHTa y : IR-+ IR 1-1a Kp1rni11 & c Mh<e> MaeMo B11pa3
!32)
B11611paIO'm rrapaM eTp h E IR+ B (32) TaK, lll,O MHoroBH)]. M1zceJ c_ M, HKHH
3a,LlaETbCH aJlre6pa"itJHOIO K])HB0!O (32), KOMIT3KTHHH i 6e3 Me)Ki, T06To Me)Ka
o Mh(e) = 0, 3HaXO,l.l.HMO, lll,O 3aMKHel-IHli KOHTYP a = Mh(e) c_ M. BHKOHaBlllH
iHTerpyaamrn no K0HTYPY a c_ M B (31), 3Haii.n.eMo
h( eh) ~ y = Q 1 - 2Q2 + 0 (e·). (33)
TaKHM lJHHOM, 3 (33) BH,1.1.HO , lll,O npH ,U:0CTaTHb0 Bemnrnx 31-latJeHirnx eHeprii" ,ll,H
ttaMillI-IOl cHcTeMH (24), -ro6To np11 h ~ Q2, e -nonpaaKa B (33) ,a:o a.n.ia6aT11 1rnoro
iirnapiaHTa y E f/) (M) E cyTTEB010, TaK 51K npH fr BiLI.CYTHOCTi He BHK01-ia1-1a yMo
Ba a,nia6aTHlJHOCTi (14). Tairnro )I{ nrny pe3yJJbTaTH OTpHMaHi B [6].
3. A ,lJ. i a 6 a T H lj ll i i H B a p i a H T H H e a B T O H O M 1-1 0 C JI a 6 -
K 0 36 ypeH H X HeJiiHiHHHX ,lJ.HHaMitJHHX CHCTeM 3
M a JI H M TI a p a M e T p 0 M. HexaA aBT0H0MHa He.1I!HllfHa JU1H3Mi 0ma
<:HCTeMa (1) Ha ct?.a30BOMY npocrnpi M HeaBTOHOMHO cna61<0 36yp10ETbCfl J1P
BHr JI51,ll.Y
lit= K[u ; .-, e] = K[u] + eF[u; .-, e], (34)
.ne .- = et E IR - rroBiJibHO 3Mimm?i eaomoui11H11ii rrapaMeTp, IR j e-+ 0 -
111a.rm?i napaMe-rp . .[lm-IaMitIHa CHCTeMa (34) B)Ke 1-1eaBTOH0MHa, T0MY npHMO 3a
<:TocyBaTH p e3yJlbTaTH nonep e!ll-lbOro po311i .1Jy MH He M0)KeM0. OKpiM Toro , Ha
HBHiCTb noBiJibI-IO'i 3aJ1e)lrnoc-ri (34) Bill 11a paMeTpa .- E IR M0)Ke cyTTEBO nopy
rnyaaTHCb B 06.'!aCTi cenap;;rrpHC [ 1, 3], q epe,l l.il0 OC06J1HBO Ba)l{J111B010 E 3aLI.alla
rro6y .LLOBH ecpeKTHBHOro l< ])HTe pi,o in1 yBal-lll H a.uiafiaTHl!l!Oro iHBapiaina npH
BKa3,HIHX BHllle YM0Bax. OcooJIHBO UiKaBUIO 3a;ratJe10 3 T0CfKH 3opy Teopi"i €
llOCJl i .•DKeHHH KJ)HTH'll·l0ro BlirlaJ.r<y J].IIIIaMi<II·IO'i CHCTeM ll (34) , lll,0 .uonyc,rnE npH
e =I= 0 aJtia6aTHl!HHH iHBapia1n v (cl E :JJ (M) , w.o 3a!l0B0Ji bHHE:: yMoBi (14) rrpH
c, -+ 0, ane np11 Ub0MY lim v (e) fJ. f/) (!H), xocia pery.rrnp1-ia rpaHHL\H (34) rrp11
e -+ 0 icHye BC!Oli.ll Ha /\,1. Ta,rn c 1nyauiH, 511( npaBHJlO, BHHHKaE TO,ll.i, K0.'IH BH
x imrn He36ype1-ia ll e.'I iHiih,a llHIIaMilll-l a CHCTeMa (I) He E raMi .rJbTOHOBOIO.
A11aJiori 0mo MeTo!ly 11 1] B oc 11oay a 11a.r1i3y 1-1eanT0II0MII0 36ype1-IO'i ,nm1a
Micino i· c11cTeMH no1<Jia.ueMo 11pouenypy po3 UJHµem-rn cjia3oBoro npocTopy M -
,.,._, 211+2 2n ') . .
-► M = /\1 ~ M X !R· 3a llOIT0MOroro .:rnox HOBHX O}U-IO])l,ll.HHX 3MIHHHX:
(1: , p)' E IR 2 (TyT .- 1-1a.u JJ.Y)1< 1<010 - onepauiH Tpa1-1crroHyBaHHH, HKY rro KOH
TeKcTy 1-1eo6xi11Ho si.upi3HHTH Bill 1-1oao'i 3Mi m-1o·i T = et E IR). THM caMHM ,ll.H-
JSSN 0041-6053. YKp. AtaT. :JICYPH.. 1992. T. 44, M 1
HaMitrna CHCTeMa (34) na M ewe aBT0J-I0M!-!0IO i 1-1a6yBaE TaKoro BHrJrn.ny:
dul dt = I( [u; -., ep I l
d-r!dt = c,fJ = f([u , -. , p: c, ],
dpldt = 0
p = 1, (35)
.no HK0ro M0}J<Ha 3aCT0CYB3TH pe3yJibTaTI1, BHKJia,ne1-1i B IT. 2. OT)Ke 1-1exai1
cp ET* (M) - po3B'H30K xapal{'rep11crn1J1-1oro pim-rnII11H J13KC3 (15)
dcp /dt + J(. " - 1.p = 0 , (36)
HKHH MO)KI-13 0TpHM3TH' 1-1aITpHirn3Jl, aCHMllTOTf!l!l-!HM µ-p03KJ!3}l0M (21 ). 51 KIJl0
po3B'5130K (36) 3Ha11.ne110 , np1rnati,111i, 3 TOLIHiCTIO O (c, 2) i np11 e --+ 0 CHMrIJleK-
~ - -
TH 1-111a cTpy1nypa w [u; e] = cp' [u; eJ - cp" [u; e] npHMYE no BHxi ; w o "i CHtMWe K·
TH•mo "i cTpy1<Typ1-1 ;<2> = w<2> + dp (\ d-r, ,ne w<2> E i\.2 (M)- c 11MnJieKTHL1Ha
CTPYKTYP3 BeKTOpHoro ITOJIH du! cit = I( [ u; 0, OJ 11a M11oroB11.ni "1, HKa 1rn3-
)1{3ETbCH 311piopi 3a.naJ-1010, y B1-1113Jll<y, H KIJJ.0 H E .0 ( M) - cl)YIIKUiH f3Mi J1 b·
TOll3 aeKTop1-1oro noJIH duldt = I( lu ; 0, OJ 113 MHoroa11.n i /\II, -ro 3CHMl1TOTH'IIIO
~ ~
np11 e --+ 0 c\1y1-1KUiH f 3MiJibT0 1-1a f-f (e) E:;) (M) JJ.,1H .n1111aMi•mo i CHCTeMH (35)
6y.ne piBHa BeJiwm1-1i H (e) ~ H + ep 2/2.
n p H K Jl a )], 2. Po3rJ151HeMO 3 HOBY HeJiiHifrny l(HH3Mi,rny CHCTeMy THITY
X eHOH3- Xei1Jieca (24), .ne ,iacToTa Q = Q (-r) E IR+ - ITOBiJibHO 3Mi1rna cpy HK-
ll. iH IT3paMeTpa -r = et E IR.: ·
dxl dt = y + exyp (x - y) + exyp I
dy/dt = - Q2 (-r) x + ep (x 2 - y 2)/2 - peQ 2 ('r) x 2 (x - y) I (37)
( = f([u; e] .
d-r! cit = p e l
dp l dt = 0
T yT BeKTOp ~ = (x, y , -. , p)1; E .M ~ IR.2 x IR.2. ITpw10My ITap3MeTp «o)J,ttop in
HOCTi» p E IR BBa>KaT11Me1v10 ..'.{OBiJibII HM, n0KJI3BUIH ~ioro 3HaLJe11HH p = I B Kinu i
po3paxy1-IKy a,ni36aTw11-1oro i1rnapi3I-JTHa
y E !lJ (M)--+ V (e) E !lJ (M).
BmrnpHCTOBYIOLJH pe3yJibTaTH IT. 2, 31-13XO,LlHM0 TaKe 3HaLJeHHH a.11, i a6aTH LJ
HOro IIIB3pia11-ra :
= h(-r) + e/2 [i _ eh(-r) + e2/2 ] O( 3)
y Q (-r) 2Q2 (-r) + e . (38)
- -
3 ayBaMHMO .:rnwe, lll.O np11 po3paxy1-rny iHTerpaJibHoro iHBapiaHTa y ED (.\[)
aa g)OpMyJIOIO
~ l +. -
y = 2n ,! cp<J> (e)
a
Heo6xi .nHO cnpoeKTyBaTH ll.HKJI crEH1 (M; zn 1!3 niJlMHOrOBH!l {u E M1t(e): Hie'=
= h E !R.+ , p = I, ,; =et= const} c M, 36epirw1-1 i1oro 3a~11rneHiCTb. Oc-1:.11; ,! :-1
YM 0 Bi:I CJ] y,KHTb KpHTep ieM Toro, mo a,uia6aTHLJHHH i1-rnapiaHT ,UJIH 33, t:1: '.'. >l
.Lll1 ll i:l:vt i LJHO"i CHCTeMH icHye, npH40MY BOH3 ,LlaE MO}l{JIHBiCTb T3KO}K .l].OC.' i i ,1,,. :l
mrnme nepexo,uy TpaEKTopif1 LJepe3 cenapaTpHcH, nos'H3a He 3 i·x pyih1 y s 2;•; 1:i\l
l5, 15, 28-31, 35, 38-451.
56 ISSN 0041-6053. JIKp. Mar. 3/Cypn., 1992, 1. 44, Ni! I
4. A)], i a 6 a T H q H i i H B a p i a H T H H e a B TO H O M H H X K B a -
sin e piO,ll,H4HO cna6KO s 6ypeHHX HeJiiHittHHX .UH-
H a M i 4 H H X C H C T e M 3 M a .'I H M TI a p a M e T p O M. Y M O B H
TpaHcB e pca JibHOro !lepeTHHY iHBapiaHT!IHX MHO
r o B H n i B i M e T o .D. B. K. M e n b H 11 K o B a. J3y.ueMo pos rm1.uan1 Ha
CKi11qe11HOBHMipHOMY MI-IOrOBH.Ui M 2' ' TaKy 11eniuiauy HeaBTOHOMHY rnaJJ.Ky ,ll,H
HaMi4HY CHCTeMy 3 Ma: rnM rrapaMeTpOM e ----+ 0:
du!dt = K (u, t; e] = K [u] + eF [u, -r, 0; el, (40)
AA ~ f JR 0 ym . ,ZJ.e, sa BH3Ha4e1rnHM, u Em , -r = e E , E - m-BHMipHoMy Topy s y Mo-
Bmo, w.o )],JIH BCiX j = T,m
( 4 1)
l3y,D.eMO BBa)KaTH, IllO BeKTOp 0 E ym e «KYTOBHM» BCKTOpOM 3MiHIIHX «)l,iH -
KYT» nemwi" acouiaoBaimi· 1-1eni1-1ia1-10i· (a6o niuii11-10i") ,D.HHaMi41-1o"i cnc-reMH ra-
MiJibTOHa Ha ,ll,HCj)eOMopcj)HOMY ym i1nerpaJlbHOMY MHOrOBII.ll.i Q;' ~ Tm, ,LJ.e BCK
TOp g E [/J"' (Q) - eneMeHT !IOBHOi" CHCTCMH "ii iHBOnIOTHBHHX 3a 1rn11iB s 6epe)KCH
HH 11a chasosoMy npoCTopi Q2"' . KpiM Toro, np1mycTH¼O, 111.0 ,D.HIIaMi4Ha cHcTeMa
(40) npH e = 0 e Ha MHOroB11,ui M 2" raMi.TibTOJ-IOBOIO s cj)yHK1.1,ie10 faMiJibTOHa HE
E[/J (M) i CHMnneKTH4Ho10 CTJJYI<'rypmo w< 2> E A2 (A1). Bpax0By1otrn BHKnanene
BHW,e, npe,D.CT3BHMO ,Zl.HHaMi4HY CHCTeMy (40) B T3KOMY 3BTOHOMIIOMY BHr JIH,ll,i:
du/dt = K [u; i::] = K [u] + eF (u], ( 42)
,_ 't "n ?m "' ?
.n.e, sa BH3I-Ial!eHHHM, u = (u, v, -r, p) EM· X Q· X IR 2 ~ M, npHtIOMY v EQ·"'
TO,Ui i T i.TJbKH TO;J.i, KOJIH e.TJeMe HT (g, 0)' E [/)2
/11 (Q) BH3Ha4aE iHTerpaJibllHtt
MHorom1,lJ. Q;' ~ T "' sr im10 s Ko11c TpyK1.de10 3M i11ru1x «niH - KYT» n. 1.
B IlOKOMilOHeHTHOMY BHrJl5l)],i LlHHaMitJHa CHCTeMa (42) samicyETbCH TaK:
du/dt = K [u] + peF g [u, v, -r; ep l,
dvldt = G [v], d-rldt = ep, dpldt = 0, p = I , (43)'
.ue Fg c D (M) -pe.n.yKoBaHHH Ha6ip cpy11Kuii'1, eKBiBa.Tie1-1rnHtt Ha op6iTax A H·
11aMi 1rno"i CHCTeMH dvldt = G Iv] (Ha MHoroBH,D.i Q2"') i:-s6ypeTimo B (40) n p u
cj)iKCOBaHOMY BH6opi i1-ITerpa.1bHOro MHOrOBH.UY Q;' c➔ Q2"', .UHCj)eOMOpcjrnoro
Topy T "' 3 YMOB3MH (41). Tenep MO)KHa 33CTOCyBaTH JlO aBTOJ-IOMII0°i 1-1e.'1 i1-1 i r1-
l·IO·i .lUlll3Mi4H0°i CHCTe~m (43) Ha MHOrOBH.LJ.i M pesyJibT3TH Il . I 3 MeTOIO
0 Jlep)K3HHH a.uia6aTl l'!llllX i1rnapia1-1TiB . 3ayBa)KHMO Ti.TJbKH, w.o npH e = 0
c11Mn.r1eKT1-14Ha CTpy1<Typa :;<~ 1 E A2 (M) s a,11.aETbCH cyMmo w~~1 + w<;> + dp f\
f\ di: = ;;;~>, i olleBl1JJ.HO, e 1-1eBnpon,Ke1-1010 Ha T ( M). TTp11 e cf= 0 CHMIIJieK
Tl-!4Ha CTpyKTypa c:(2) -Z/2> (e), T06TO ,11.ecj)OJJMYETbCH, rrpH40My B CHJIY O(e 2)
ac11MTTTOTH '-IH00i TO'lHOCTi ,l.l.HHaMil!HO"i CHCTeM H ( 43) 1-1eo6xi.u.1-10, mo6 BHKOl·IY·
B3.fJHCb YMOBH
-
w<2> (e) IQ= wg> + 0 (e2 ) E A 2 (M) IQ'
'(44)
ro< 2> (e) I = dp /\ d. + 0 (e2 ) E A2 (M) I (,,p) (,,p)
11.JIH Bcix uE Mi (i:,p)'EIR2 npH e----+0. Pa3oM s yMoBaMH (44) necj)o p Ma UiH
Cl-!Ml1JlCKTJ1lIHO"i CTpyKTYPH ;,2) (e) 1,\1 BH3H343ETbCH YMOBOIO KapTaHa - H bOTep:
L-:<2> (e) = 0 (e2), e--+ 0. CaMa CHMrrneKTH4Ha cTpy1<Typa ;<2J (e) E A 2 (/W) BH
K
ISSN 0041-6053. Y,cp. Mar. '1Cyp1-1., 1992, r. 44, M 1
aaaqaeTbCH qiopMyJIOIO [5]
,...., I l"'V ~ l"'V -
cJJ<2l (e) = 2 ( w [u; e] dll, /\ du), (45)
,ne u> [u; e] = cp' [11; el - <p'* [u: e] i cp [u; eJ ET* (M) - po3B'5130K ni.nn0Bi,n1-10ro
piBHHHHH JlaKca (36). Ania6aTW-IHHH i1mapia1rr y E fl} (M) BHpaxoByeTbCH 3a
~ ~
q)opMy.1010 (39), npHl!0MY UHKJI a c M~. ,"le h c f/J (M)- nonna CHcTeMa iH-
11
napiarniB i1nerpaJibuoro M110ron1-1,ny M-, B11311a'-!eHa 3 Toq11icno O (e2). CaMa
h
~ ~
esomouiH (43) 33.ll,3ETbC51 cpyHKUiEIO faMiJibTOHa H (e) = (er (e), I( ) E f/J (M),
~ ~
51Ka BX0)l.HTb B CHCTeMy cjiyHKUiH h E f/J (M) . E o<IeBH,11.HHM, w,o an piop1-1e BH-
31-1alJeH1rn YM0BH KOMTiaKTHOCTi i1nerpaJibl·IOrO MH0fOBH,ll.Y, a TaK0)K CTii-iKOCTi
op6iTH B OKOJii cenapai.·p1-1CH ,ll.HHaM i lJHQI CHCTeMH (40) € npo6JieMaTHqHJ-!M y 33-
r aJibH0MY BHna;my. ToMy CK0j)HCTaEM0C5J ni,IXO,ll.OM po6iT [8, 16-24, 46], B
OCHOBi 5JKHX JJe)KHTb T3K 3BaHHJ1 MeT0,ll. B. K. MeJTbHHKOBa [4].
Hexa11 noLJaTKOBi .ua11i Kowi ;JJI5J CHCTeMH (43) TaKi, w,o npH e = 0 ,11.HHaMil!
Ha CHCTeMa Ut = K r ul, u E M' nop0.U)KYE I'OM0- a6o reTepOKc1i11itrny op6 iTy
[8, 12] 1-1a M, a .UHHaMil!11a CHCTeMa dv;dt = (i lvl, v E Q,- nepio_u1-1q1-1y op6iTy
I-Ia Q. CTaBHTbC51 3a_ua,1a ,ll.OCJi i.U1KeI-IH5J noBe.uimrn fOMOKJiil-JilJJ-101 cenapaTpHCH
Ilj)H C * 0, 30KpeMa CTpy1nyp1-1 o6JJaCT i Yi" pyi'.rnyBaJIH}l' UL0 npHB0.ll,HTb .!l,0 5-IBH
w,a nepeTHHY 6JIH3bKHMH TpaEJ<Top i HMH i1rnapimrrnoro MJI0I'OBH..'l.Y ,IlHHaM i tJHOl
CIICTeMH (43) npH e = 0. HK uac.'! iJJ.OK, o.uep,KyEM0 B pe3y.1bTaTi 11eimerpoB-
11iCTb HeJ1iHi111-1o"i .um-raMi'-lHOI CHCTe~IH (43), rn6To nopywemrn YM0B CTiHKOCTi
i 1mapiattT1-1oro MH0roBIUla 3ri ,zu -10 KAM-Teopi"i 123, 34], a TaK0H< pe,rni:-iaui10
HBH.LI.l_a CTOXi:lCTH3aui Y TpaEKTOpitt B OK0.1 i cenapaTpHCH-Ti:lh 3Bal!HH .UH I1aMi'-l
m1(1 xaoc. OCTam-1E BHMarae noBe.uemrn ic11ynamrn cneuiaJibHHX .ueqJop111auitt ro
MOKa1 iIIiL11-10·i cenapaTpHcH, w.o MaI0Tb WJ 3BY «n i .UKOBH C111e11Jia» (Smale horse
shoe) l5, 15, 21-26. 28-31, 38-401, .HKa Bi.uo6paH<ae B.'JaCTHBicTb acoui
i'.1osa11oro Bi .uo6pa)!{eH!rn TTyam<ape 112].
3 MeT0I0 nonaJiblJJ0ro BH K0j) HCT3 HH51 1-IaBe.UeMo TaJ<i BH3HaLJeHH5J [12, 13].
BH 3I1a L1e1-11-151 3. Hexau aa8Lme eeKmopHe noAe (1) Ha ,itHocaeuoi M;
<u-cpaHU<tH010 AtHOJICUHOIO cu (u) mo<1Ku uEM Ha3ueaembcf/. MHOJICUHa w (u)={w EM:
: 31 11 E IR+, lim t 11 = = i u (t,.)--+ w}; ei8noeiOHO a-cpaHU'lHOIO MHO:J/CUHOIO
11-.00
a (u) mo1tKU u EM 1-W3UBGWLbC5l MHO:J/CllHa a (u) = {w EM: 3t 11 E IR-, Jim t 11 =
n➔oo
=- = i u(t 11 )--+w} , oe u(t0) =uEM.
I-I a c JI i .u o K 2. f!Ku1o c,) (u) - w-cpaHU<tHa MHOJ/CWlG mo<tKU u E M
8AR eeKmop1wc0 nOAJl (1) u1 = K lu], mo w (u) - a-cpaHU'tHa MHOJICUHa OAR
eeKmopHoea noAR u t = -K [ul Ha AtH020euoi M.
H exatt Q c N - rJ1a.UKH11 ni.uM1-1oroBH.ll i f : M --+ N - nurpepettui110-
Ba11e Bino6pa)!{emrn .
B H 3 Haq e II H H 4. Biao6paJ1CeHHfl f: M --+ N Ha3ueaembcf/. mpaHceep
eaAbHUM ao QcN B mo•a{i uEM, RKUfO a6o f[uJqQ, a6o f.Tu(M) +
+ Tn,,:i (Q) = Tflu 1 (ii/), mo6mo f/.Ku1io o6pa3 T,, (M) npu ari eioo6paJJCeHHf/.
/ : T (M)--+ T ( N) Micmunu, e co6i nionpocmip e Tnu] (N), aonoeHJlAbHUU ao
Tr,,,1 (Q). Bionoei8Ho coeopumu-1ie,110, U(O aioo6pa:i1CeHHf/. f : M--+ N mpaHc_
eepcaAbHe 00 QC N' (f th Q), f/.K.U{O (]OHO mpaHcaepcaAbHe B KOJICHiu mo'lt{i
uE M .
H exaH ii E M - oco6mrna «1-1epyx0Ma» T0l!Ka ser<Top11oro 110.rrn (!), HKa e.
rinep6oJii4HOIO [13), rn6rn MaTpu1u1 H1<06i K': T (M)--+ T (M) 1-1e Mae BJ1aCHHX
·;rnatieHb, 111,0 J1eMaTb I-Ia yHBHitt oci KOMnJ1eKc1-1o"i nJ10111,H11H (C. B 1.1_b0My BHna.uKy
Ha seneMo TaKi BH3HalJemrn.
B H 3 H a lJ e H H 51 5. MHOJ/CllHQ lnO'lOK ws (u) C J\1 , OAR 5lKUX mO<tKa
u E M e w-2paHU'lHOIO ALHOJICUHOIO, Ha3U60€tnbCf/. cmiUKUAt ✓HHOcOBUOOJt oco6AUBOi0
mO'lKU u E M . BiiJnoeiiJHo Atl-lOJICU/-la mortOK wu (u) C M' tl{O MOIOtnb mo•tKy u E
-E .\1 B f/.KOCmi a-2paHUt/HOL A·tHOJICUl-lU, HG3ll(]G€mbC5l HecmiUKU,1! Atl-l020(]U00.M oco6-
.!1Ll80L l1W'lK.U U E M,
.58 I SSN 0041-6053. YKp. AtaT. :J1Cypn., 1992, T. 44, M l
B H 3 H a <1 e H H 51 6. fl KU{O acM -2inep6011i•t1-ta [ 13] 3G.MKHeHa op6ima
ee1onop1-to20 no11J1. (1) Ha M1-to2oeuoi M, mo eu31-ta<te1-ti cmi ii.Kuii i 1-tecmiii1wii MH020-
euou ws (a) i wu (a):
W 5 (a)= {u EM: w (u) = a},
W" (a)= {u EM: a (u) = a}.
(46)
B H3Ha<rnMo Terrep BiJJ.o6pa)KeHJ-rn TTyai-mape [8, 12, 13, 23, 24] JJ.JI51 .n.i,111aMill-
11oi· CHCTeMH (43) 33 )].OIIOMOrOIO IIOIIepe<IHoro rrepepi3y i; = {i7 E /\1 : µ [u] =
= µ lu, g [vll = 0}, JJ.e cpyHK11,i51 µ E :1J (M) 33JJ.OBOJibH5JE yMoBy <crpaHrnepcaJib
HOCTi»:
(gradµ [u], K [u; el) =f= 0 (e), e-+ 0, (47)
.n:rn scix u E }:;_ Bse.neMo TaKe rro31iat1emrn: ;; [gj = (u, Q~n x JR~i:.p)) c M.
T aK 51 K MI-IO)KHHa q)y1-mu,i11 gi : Q-+ JR, j = ~. iHBapiaHTHa BiJJ.HOCHO JJ.H
l·l aMi lIHOI CHCTeMH (43), TO Jienrn 6al! HTH, w.o MHO)KHHa }:; C M po3wapoByETbC51
Ha MHO)KHHH BHJJ.Y ; [g] C }:; , T06TO
~ = LJ ;[g], µ fu [g]] = 0. (48)
uEM
BpaxoByIOllH, w.o JJ.HHaMi<rna CHCTeMa (43) JJ.OIIYCI<aE e-,necpopMaui10 CHMIIJieK
TH'IIIO.i CTpyKTYPH w<2> (e) E J\ 2 (M), yMOBy (47) MO)KHa 3arrHcan1 B e1rniBaJie
IITHOMY BiffJI5Jni TaK: 11..1rn Bcix uE~ {H, u}- =f=O(e), e-.0 . B CHJiy eso-
, ul(e)
mo1.1,i·i JJ.HHaMiq11o"i cuCTeMH (43) rrpH JJ.e51KOMY 3Hat.Je1rni rrapaMeTpa t1 E IR. eJie-
.-.., _....., -., -., -.,
Me HT u 1 [gl = u [g] (t 1) c }:;, .ne t1 > t 0 E JR, T06TO µ [u1] = 0, i T. JJ.. rrpH 3pO
CTa HHi rrapaMeTpa t E IR,. THM caMHM BH3Hat1e1-1O sino6pmKemrn TTya1-1Kape
p f :~-~. )l.e Pe: Un [g]--+ Un+I [g], un+I [gj = u [g ] (tn-1-1> E ~. t,,+1>tn, nE:Z+
T oni Ol!eBH)I.I-1O, ll.1,O BHXimra ,lHHaMit.JI-13 CHCTeMa (40) JJ,Ol1yc1rne rrepiO,il,Ht.JHHH
po3B'5130K .rrnwe TOJJ.i, KOJIH ic1-1ye JJ.eHKe ui,1e L!liCJio n0 E:Z+ TaKe, w.o
~ ~ -., ~
p ~'.•: Uo [g]-+ u, [g] = Uo [g] C }:;, npHt.JOMy rrepiO~ll HOro JJ.OpiBHIC,E t.JHCJiy
"• T 0 =fn0 -f0 EJR+-
SlKW.O p03B'H30K ll CM JJ.HHaMit.JHOi" CHCTeMH (43) npH e-+0 E 6JIH3b·
1rnM ,u.o "ii" p03B'513KY npH e = 0, TO BiJJ.o6pa)KeHJrn TTyaHKape ~ -~ 6y,nt.
e-6JIM36KHM JJ.O BiJJ.110B iJJ,HOro BiJJ.06pa)KeHH51 TTyaHKape Po:~ - ~ rrpH e = 0.
H exatt Terrep MHQ)KHHa u[g; OJ c M - rirrep6oJiit.JHa <rnepyxoMa» MHO)KHHa JJ.H
HaMilIHOi" CHCTeMI-l (43) npH e = 0. Toni rrpH e--+0 icHyE EJJ.HHa )].ecpopMaUi51
u [g ; 0 I--+ u [g; ej no rinep6oJiilfHO1 nepiO.llHLlll0°i op6iTH {u [g; el (t) : t E JR,
e-6JIH36KO"i ,no MHmKHHH u lg; OJ c M . BiJJ.IIOBiJJ.HO BiJJ.o6pmKeHH51 TTya1-1Kape
P~• : ~1•--+ ~10 Mae en11 HY rirrep6oJiit.JHy «ciJJ.JIOBy» MHO>K1-my u0 [g; el c 3;10 ,
6JIH36KY no MHO}KHHl1 u r g; O] E ~\ to E IR. TTpH UbOMy rrepio,n op6iTH u 1g; e]
e K\X:ITHHM nepio.ny ,llHHaMit.JHo·i c11cTeM11 dvldt = G [v] 1-1a MHOrOBHJJ.i Q 3riJJ.-
11 O 3 c!JOpMyJ1010 (41), a rinep6oJiillHiCTb MHO>KHHH u0 [g; e] c}:; BHTTJIHBae 3
~ ~
rnro, w.o MaTpHLL51 SI1rn6i P~ [ll0 j He MiCTl1Tb y csoeMy crreKTpi o,n:mmui, TaK
wo MaTpHU.51 Id - P ' [u] MaE o6epHeHy, 3a6e3rreqy!Ot.JH icHyBaHHH rJla)].1<01
JSSN 0041-6053. JIKp. AWT. {)(CypH.., 1992, T. 44, .M l 5.9
,..._, - - ""
JJ,ecpopMaui"i u[g; 0]-+ Uo [g; e] c ~. T06TO µ [u [g; OJ]= 0 ⇒ µ [Uo [g; e]] = 0,
e-+0.
CnpaBeJJ.JIHBa TaKa JieMa [23].
JI e Ma 1 . .lloKaAbHO cmiii.KU.U i Hecmiii.KU.U .MHOZ06UOU Wioc [u [g; el)
- -
W'i'oc (u[g ; el} e-oe<jJop.M06GHO°i nepioou 1moi u6pimu u[g; el CM e ou<pepeH-
u,iii.oamw e-6AU3bKi 00 cmiii.KOZO i HeCmiuKOcO AtHOZ061./.0i6 HeOecpop.M06aHOI
zinep60Ai<ut0'i MHOJICUHU 1.?'[g; OJ c M. EiAbUJ.e mozo, flKlUiO op6ima ;;'t [gl E
~ ~
E w· (u [g; el), a op6ima a;- [g] E wu (u [g; el) , mo 6 OKOAi nepepi3y flyaH-
Kape 1:10 c: M. t0 E IR, OAfl flKOeO 6UKOHa1-ta y.1rtoaa g [v (1)] 1= 10 = g E !Rm cnpa
eeOAU.Bi maKi pieHo.MipHi no t E IR e-po3KAaou:
(49)
~ a;[g] (t, t0 ) = 0 0 [g] (t - t0 ) + Wo, 1 [g] (t, t0 ) + 0 (e2 ), t E (- oo, t0].
DpH e = 0 3ri,ll,HO 3 np1rnyw.eHHHM Jl.HHaMi'-IHa CHCTeMa (43) Ma€ r i nep-
OOJii'-IHY oco6JIHBY MHO)KHHY u [g] c M, a TaKO)K 33MKHetty roMOKJii1-1iq1-1y
~ -
op6iTy - cenapaTp11cy Oo [g] = wu (u [g; OJ) n w· (ul g; OJ), HKa HaflOBHeHa He-
TpaHCBepcaJibHO rOMOKJiiHi'-IHIIMH MHO}KHHaMH 3 M JJ,JIH Bi,TJ.o6pa)KeHHH Dy-
rv rvt ~t
aHKape P0 : ~ •-+ ~ •, t0 E IR- DpH e =I= 0 UH CTPYKTypa 6yJJ.e, O'-leBHJJ.HO, pytt-
HyBaTHCSI 3 YTBOpeHHflM TpaHCBepcaJibHO rOMOKJiiHi'-IHHX op6iT at [g], e-+ 0,
.n.e )lJIH Bci x t0 E IR
-;;t [g] = {; C Af: Peth ;w· (~ [g; el), Pe: 2:10 n wu (~[g; el)-+ 2:10},
(50)
-;;; [g] = {; cM: Pefh-Wu (~ [g; el), Pe: ~t. n w· (~[g; el)-+ ~ 10}.
u
DpH UbOMy, O'-leBH)lHO, TIOBHHHa BHKOHyBaTHCb [14] TaKa HepiBHicTb:
dim w· (u[g; el)+ dim wu (u[g; el)> l + dim M 2n. Illo6 JJ.OCJiiJJ.HTH CTPYKTY·
py pyiiI-1yBaHHH cenapaTpHCH 0 0 [g] c M )lHHaMi'-IHoi· CHCTeM11 ( 43) np11 e =f=O,
BHKOpHCTaEMO MeTO)l MeJibHHKOBa [ 4, 17, 23].
HexaH t0 E IR - JJ.OBiJihHHH eBOJIIOUiHHHH napaMeTp. To)li 3ri21.Ho 3 JieMOIO
3HaXOJJ.HMO, w.o npH e - 0 cnpaBe)lJIHBa aCHMTITOTH'-!Ha piBHiCTb
d-;;6,1 [g]/dt = K' [~ [g] (t- t0)] ;;d:1 [g] + F [;;0 [g] (t-f0)], (51)
.n.e Bepx1-lii'1 3HaK «+» Bi ,1rnoBiJlaE 3Ha'-!eHHHM t E (t0 , oo ), a HH)KHiH «-» -
SHa'-!eHHHM t E (- 00, to], Bae,neMO cpyHKUiJO raJiy)KeHHfl MHOfOBH,niB Wu(u [g; el)
i w· (u [g; el) Ha nepepiai i 10 , to E IR, B «TO'-IUi» ;;o [g] (0) c M TaKHM BHpa
BOM:
(52 )
To.n.i cnpaBe)J.JIHBa TeopeMa, w.o y3araJibHIO€ Ha 2n-BHMipH11lf BHnanoK MHOro·
BHJJ.Y M 2n pe3yJihTaTH 14, 23].
Teo p e Ma 2. f/KlUiO eeKmop-</J!fHKU,ifl (52) ,,uae no napa.Mempy t0 E Ii\
npocmi 1-1.ylli, lUiO e He3aAeJICHi eio napa.Mempa e E IR, mo OAfl oocmamHbo
Ma.fflX SHat(.(!Hb 6-+ 0 .MHOJICUHU Ws (u [g; e] (f0)) i Wu (u fg; e] (t0)) nepem/1/W·
(iO JSSN 0041 -6053. YKp. Mar. xyp1-1., 1992, T. 44, Ni 1
IOmbCR mpal-lC8epCa/lbH0. B np0112U6H0Hlj pa3 i 6UIWHaHa YM0Ba W 5 (u [g; e) X
x Uo)) n W" (u [g; e] (lo)) =I= 0.
3IIalleHHH Teope~rn 2 3ri)JJIO 3 p e3yJibTa-raMH 6ipKrocj)a-CMeiL1a [8, 12-
14, 23-26] TTOJIHrae B TOMY, lll,0 Ha HBH iCTl, Tpa11csepca.1bHO fOMOirni1-ri 1.w11x op -
6 iT B Jl.!1 J·IaM i t.111ii'.i CHCTCMi (43) npI1BO}.JJ!Tb .ilO icuysamrn ir-mapiaHTIIO.i rinep6o
.'l i q 110·i MIIGiKHHH -r11:1y «n i.u1rns11» CMei".uia.
l~H MHO}KI•IHa MiCTI-ITb 3JiiLJer-IHe L[HCJIO II ec-ri (!J(lfX nepio,U!-Jl[H !1X op6 i-r i He-
3.'l i 11e1-1He lfHC.TJO OOMC:.>KCHHX ' rienep iOLlH'-lllllX i W. iJib!IHX op6iT' npHL!OMY ca Ma UH
i1-map ia1-rrna MHO:.>KHHa MO:.>Ke MaTH npo6ony cjipa1naJi bHY a6o xayc;iopcj)osy po3-
Mip11oc-ri.
1Uo6 ecpeKTHBIIO BHBtiaTH uyni se1<Top-cj1yHKl.li i (52), po3r:rn 1-1eMo nonep e,n: -
llbO 6 iJibW neTaJlb!-10 l3J1aCTHBOCT i p03B ' H3K iB p iBHHHHH (5 1) Ha MIIOfOB HJLi ,w.
B ciwy rnro, ULO dim M !Q X iR. 2 = 2m, niHi11 11a 1-1eon_1wpi1urn CHcTeMa (5 1)
Mae nOBI·IY CHCTe~1y po3s'H3KiB, ,UHXOTOMitrna CTpy1<Typa mrn·i B TOlflLi cr0[g ] (0) E
E 2:: 1• Mae, 3or<pel\rn, BHurn,u
{~0: J(j) [g] (t, to) ET (M): j = 1, 2n} c T (Wfoc (u[ g; el)),
.ne Jl.TIH scix j = I, 2n, t , t0 E iR.
t ~
-;;6:I(i) [g] (t, to) = ~ . i[ g) (t' to; to) ir= Uo) + ~ d1:ao, I [g] (t ' --c; to) X
±oo
~
(53)
(54)
o0 [g] (t, --c ; t0 ), t, --c, t0 E iR. , - cpy11J.J.aMe r-ITaJibI-Ia MaTpHl.lH Ji iHiihm'i Liacnum p iB-
HHH!IH (5 1), HOf):VIOBaHa Ha OJlHIHll.llO: ;=o,1 [g](t , t ; to)= l, t,toEiR.i iluo)=
= (61,j, ... , 62,,,j)\ i = i' 2n, to E u\, - c\JiKCOBaHi BCKTOpII. l.Uo6 p03B '513aTH
3a;J.aLiy -rpaHCBepca.'lbHOCTi nepeTH!·IY p03'11,CflJIC ill-!X BiTOK cenapa-rpHCH
- -
o 0 [g l (t - t 0 ) c M, B ,uem<ii'.i TQltUi t 0 = t0 E IR. 3anaMo ui BiTKH TaKHMH an-
p iopm1MH s11pa3aMH : ; ; (:rt) = ;o [g j (t - t0) + e~ff.1 (;) + 0 (e2 ) , ne
~ ~ - ~ I~ ~~
crff. 1 (:rt) = Go, 1 [g) (t, to; to) n± (to) + s d--cao, I [g) (t, --c; to) F [Go [g j (· - to); --c],
± oo
(55)
a BeKTOpH ; ± (t0), t0 E IR, BH3HatJaIOTbC5l 3 yMoBH
;t.1(;;) = ;o.t(~) (56)
B ,QeHKiH cpiKcosaHiH TOLIUi l 0 = t0 E iR. B Ben.eMo -renep TaKy xar::i1nep11CTH'l-
~ - -
11 y µ--q1ytrnui10 Tttny MeJibHHKOBa. µ-(t 0 , s ) = µ (t0 , t 0 ; s) , t0 E Il{, ne JLJ1>1 Bcix
t, to EiR.
µ- (t, t0 , ~ = det IJ ;;:1(1) +;: Cto) ; t H2riJ, ... , ;o'.1c 2n-lJ + ;;ll-1 Uo) ;ii:112nJ ,
-CJo, 1 (n) 1\ (t, t0) (57)
s- napa:v1eTPH s; (t0 ) E IR, j = I, 2n - I, nosiJibH i . 3a.uaMo citmy r i nepnosepx
HIO I:10 Bino6pa:.>KeHHH CTya1rnape Ta1<010 yMoso10: .D.JIH scix l 0 E IR µ-(t.; s)=O.
~+ . --
3 siJlCH BHllJIHBaE, w.o icHyE TaKa CHCTeMa n-cpyHKUi H ni (t0) E IR, , = 1, 2n,
ISSN 0041-6053. YKp. ,\IQT. :J/CypH., 1992, T. 44, M 1 61
ru.o JlJHl Bcix t0 E IR
2n ,..._, _ _ ,... r rrf Uo) ad, l(j) [g] Uo, to) = (Jo, I (n) (to, to)
j = I
Bpax0By101rn (55), 3 (58) MaEMO
(58)
;+(to)=;- Uo)- r d1:ao,1 (to, 1:; to) F ~o [g] (T - to); -r], (59)
)le ;+ (t0) = (;f (t0 ), ••• , ;:;j~ (t0))'. 3 (55), 30KpeMa, BHIIJIHBaE mo po3w.errJJeirn
BiTKa cerrapaTpHCH ;t, 1 (;) (t, t0) c: i1, t, t0 E IR, JlJJH HKo'i BHKOHaHa yMoBa ne
pernHy (58), icHyE TOJli i JIHille TOJli, KOJIH 3HaH)leTbC5l TaKe qHCJIO to =
= t0 E IR, mo Ha ci1rniH rirreprroBepxHi ft;;- BHKOHaHa yMoBa TpaHcsepcaJib·
HOCTi
2n ~
n (t0) = L nt (lo) = 1, (60)
i=I
T06TO CHCTeMa n-cpyHKUiH 3 rrpoeKTHBHoro rrpocTOpy IRP2n B (60) CTa6iJii30BaJ:la.
3 MeTOIO e<peKTHBHoro 3anHcy YMOBH (60) npOJlH<pepeHuiIOEMO µ-cpyHKUiIO (57)
no napaMeTpy t E IR:
dµ- (t, t0 ; ;)/ dt = (tr K.' [ 0 0 ]) µ- (t, t0 ; ;) + det II ;t, 1(1) + ;1 (t0 ) ;: l(2n) -
-(I+ ;1 (to)) ;o,i(n), ... , crf1(2n-l) + ;n-1 (to);d.t(2n)-(l + ;2n-1(tu)) X
-
X cro,1 (n)), F [00 ] II (t, t0), (61)
T06TO
-., ,,...,, ,..._,, ,...._, _ ,..., 2n- 1,...._, _ ,...,, _
dµ- (t, t0 ; s)ldt = (tr K' [o0 ]) µ- (t, t0 ; s) + L £ [cr0 ; n - l (t , t0) Sj (t0), (62)
i = O
ne soUo)= 1, toEIR, 'IHCJIOBi cpyHKUi'i slcro;n-l(t,to), j= 0,2n- l ,
t, t 0 E IR, BH3HatJalOTbCH 3 (61) OJlH03HatJHo.
BpaxoBy10qH Tenep, w.o BeKTOp-rjiyHKUi'i ;ti, j) + ;1 (to) ~t.1(2n) - (1 +~ X
,_ -., ,_ -
X (t0))ao,1(n)=ai+si(f0)a2 nET(M), J= 1,2n- l,-po3B'H3KHpiBHHHIIH
a 1 = K' · a, t E IR, (63)
MO)KHa 3anHcarn eKBiBaJieHTHy YMOBi (57) piBJ-IiCTb:
',;- (t, to;;)= det Ii ~t + ~tR, ;1 + "'s1K, ... , ;;;t_1 + ~/2-tK, a;-_1 +
- -+ s-;;_1K, a 0 + s0K, cro, 1 (n) II - (64 )
T yT {;;:T (t, t0), j = 1, n - 1; K, a} c: T (M)- 6a3HC piBHHHHH (63), y3rO)l)Ke
HHH 3 arrpiopHHMH aCHMIITOTHKaMH IIPH t-+ + oo:
lim ~t (t, t0 ) = 0,
1-+:l:oo
~
-
lim J a 0 (t, ! 0 ) I= oo,
1-+±oo
lim I a, (t , t0) I = oo, t0 E IR,
1-+±oo
;,, ~r E IR, j = 1, n - 1, - JlOBiJlhHi rrapaMeTpH .
51.KW.O MH no6yJlyEMO Terrep aHaJioritJHO (64) ; +-cpy1rn11i10
(65;
..... + ,_, ....., ,,...,, ...., ,..._, -., ,_ ,..._, ......, ......., ,..._, - ,_
µ (t, t0 : s) = <let II a,t + stK, a,+ s,K, ... , a;;-_1 + st-1K, a;;-_1 + s-;;_if(.
62 ISSN 004]-6053. ~IKp. Mar. xyp1t., 1992, T. 44, M 1
(66}
TO, Ot:JeBHJlHO, YMOBa TpaHCBepca.'lbHOCTi po3w.enJremrn rOMOKJJi Hi•rno"i cena paT-
~ ~
pHCH (56) 6y.ue e1<BiBaJ1eHTI-Ia Ta1<ii1: .UJJH Bcix s0 , sf E IR, j = 1, n_-;:- I ,_
. µ-cpyHKIJ,iH BHJlY
(67)
npH neHKOMY npocTm,1y 3Hat:Jem,i t 0 = t0 E IR rrepeTBopIOETbCH B TOTO)KHHH
HYJJb. Bpax0By10,rn aHaJJori 'IHO (62), mo
~ ~ ~ + ~(±)
.ne s; = s0 E IR, s0 = 1, a ~i (t , t 0 ), j = 0, n- 1, - cj)ymrnii' BHrJJHJlY
iJ+J (t, t0 ) = det II;:~, ~1, ... , K, ;f, ... , a 0 , F II (t, t0 ),
'it> (t, to)= det II ;ii,;-;-, ... , ;tK, ... , a 0 , F II (t , t0),
16+1 (t, to)= det II ;t, ~I, ... ' ;t, __ J, ;;_J, ;o, F II (t, fo),
1~-) (t, to)= det II ;t, ;!, .. . , ;;;:--_J, ;;_J, K., 'i II (t, to),
-'iOMY
~ t ,_, ,_, ~
µ'f= (t0 ) = .\ dt exp ( .\ d-dr K') ~} ± > (t, t0 ) + fiµ t (t0),
IR lo
aµ+ (t, to; 0)
ast
I
(69),
(70)·
JlJJH Bcix l = 0 , n - I. On1<e, HKW.O t = t0 E IR, TO JlJJH Bcix j = 0, n - I
llOBHHHi BHKOl·IYBclT!ICb p iBHOCTi µf ((0) = 0.
Hexai:i Te11ep BllX i).l,Ha ).1,HHaMi•rna CHCTeMa ( 42) npH e = 0 raM im., TOIIOB;J ,
T06TO tr K' [a0] = 0, t E JR. Toni B c11JJy i1rnapiai-1rnocTi BCJlH'IHHH q (t0) =
= det I a (t, to) I .UJIH Bcix t E IR BHTIJlHBa€, 3OKper.rn , mo tiµ! Uo) = 0 .UJJ51 ecix
t0 E IR, j = 0 , n - 1, T06To cniBBi;:uic 1uemrn (70) 31-1a•m o cnpow.yeTbC5I. ,Umi
BeJJH'mH !':,.µt (t0), j = 0, n - 1, B (70) "ix po3paxyttoK Telli: cyTT€BO cnpow.y
€TbCH, HKW.O BpaxyBan1, w.o
(71)
~
.ue 3a BH3Ha•1eHHHM TOt:JKa u 0 + di (n) + 0 (e2) EM - oco6JJHBa rirrep6oJJi,ma
TQl!l{a JlHI-IaMi'IH0 °i CHCTeMH (42) npH e -- 0. TIIM caMHM, MH Ccj)OpMyJJIOB3 J1 H ne
oox i.n.Hi i .UOCTaTI-Ii YMOBH (70) iC1-1yeaHH51 B 3a.UaHiH .UllHaMit:JHii-i CllCTeMi (42)
:rn11ma TpattcBepcaJJbHoro po3merrJJemrn Ti roMOKJJi1-1i•mo·i cerrapaTpttcll. A1-ia.10-
riL1111-111 ninxin MO)!<e 6yT1-1 po3BHHYTHi-i i y e1rn a.u1<y HBttma po3m ennemrn r eTepo
K.11 i II jtmo'i cenapaTpllCH .UHHaMit:JI-IO"i CHCTeMH (42) rrpH e -+ 0, Ha t:JOMY TYT He 3Y.·
TIM II H€MOCb.
ISSN 0041--6053. JIKp. MaT. :NCypH.., 1992, T. 44, M l IS3
n p H K JI a Jl 3. Po3rJI5-IHeMO 6 iJlblll JleTaJibHO BIIITcl,UOK dim M "----' 2. - -
<Pymrn.i51 rnny Me.rJbHHKoBa µ(t0) = µ (t0 , t0), t0 E il\, Bll3Hcll-raEThC51 TaKHM BH-
pa3oM:
(72)
~
.ne s0 E IR - JJ,0Bim,HH11 napaMeTp. I3 (72) i piBH51HH51 rnny (68) 3HaXOJJ.HMO,
~ ~
LUO 1<0.Trn tr I<' [ 0 0 ] = 0, t E IR, TO
µ (t0 ; ;) = .\ dt <let II ; 0 , F II (t, to) +"; .\' dt det II K., F II (t, t 0 ) + ~;;; (to)- (73)
R R
Y BIITI3,D,Ky HB111U3 TpaI-ICBepcaJJbl-!Oro p03LUen.1emrn cenapaTpHCH 7i [g;
Ol (t - t0) i3 (73) o)lep)KyEMO 1110,umj.JiKoBaire cniBBi.ZI.IIoweHH51 MeJibHHKoBa [4]
µ Uo, ~ = 0, t0 = to E R, a6o
5 dt <let ll~o, F 11 (t, t0 ) = - ~µ;;- {)0),
IV
(74)
5 dt <let II K., F II (t, f0 ) = 0.
IR
3Bi,UCH BHTIJJHBaE, IUO y BHTI3.ZI.KY TIJiaHapHo'i HeJiiHiHHOl JlHHaMitmo'i CHCTeMH (42)
B. K.. M e.'lbHHKOB [4] BCTaHOBHB ectieKTHBHY Heo6xi,uHy yMoay TpaHcBepcaJih·
IIOCTi po3Ll.l_eJlJlelllrn fi rQMOKJi iHi'-IHOl cen apaTpHCH, IUO € 31-l3 '-IHHM BKJia,ZI.OM I
po3po6Ky 33,llal!i A. Dyarmape.
3 a y B a )K e II 11 51 I. Po3BHHYTY BHIUe MeTOJlHKY BHB'IeHH5I 51BHma po3•
UlCTIJieJ-111,1 rOMOK.'TiHilfl·!O"i cenapaTpHCH ,!I.HHaMilIHO"i CHCTeMH (42) MO)KH3 33CTO
cyaarn 14, 23] i JlO BHB'-lemrn 51BHLUa po3111.enJieHHH anpiopi nepio)],1-I'-IHHX a6o
1rnacii nepiOJlH'-IHIIX op6iT. A ca111e, BBeJleMO y3araJibHeHy cy6rapMOHi'-IHY µ-cpy1m-
ui10 T1my M eJibHHKOBa, BH3HatJeHY a 01<0.1i nepio,utttrno'i Tparnrnpi 'i ~o lg; 01 X - .- ,,.., ,_
X (t - t0) c M .ZI.HHaMitrno·i cHcTeMH (42) (npH e = 0 µ (t0 ; s) = µ (f 0 , t0 ; s) ,
to, tE IR):
µrn;n (t, f 0 ; 0 = <let II ;t + -;t K, ;, + ~1 K., ... , ;;,o + ~K, ;o, 1 (t + mT, t0 ) -
(75)
ne {-;;T, J= I ,n- 1; -;;0 ,K}ET(M), ";;rEIR,j =0,n- I, s; =S0 E IR,
st = I , i BHKOHaHa yMoBa mT = nT h, Jle m, n E ~. T,. c IR+ - 1:epioll op6i
TH u [g; OJ (t - t0) c M i3 3HatJeHI-I 51M ct)y1rnuiI faMi.7JbTOHa H (u) = h E IR i
T E IR+ - nepio.n 30BHilllHbOro s6ypeum1 F ju; ti, t E IR, npH e-+ 0. BHKOpHC
TOBy1o•m piBI-IHHHH THTIY (68) )lJIH cpyHI<Uii' (75), 31-l3XO)].HMO
rv rv to+mT _, "" ,.._,,,...,,, ,,_ ,.._,, rv rvrv
µm;n (t0 ; s) = 5 dt <let II at+ st I( , a-;- + S1 K, .. . , a0 + s0K, F /I (t, t0 ) (76)
1.
JLJIH BCix ";;:j= E IR, J = 0, n - 1, npH'-IOMY BeJll1l!HHH ~µmin (to;:), IUO I!aPBHi B
(70) , TOTO)KJ-1O piBI-li HYJIIO. 51K HaCJii)lOK (76) GT() l-1MYEMO, mo KpttTep i eM TpaHC
nepca.'lbHOro po3menJieHHH nep iOJlllllH0°i opfi in1 )].HH3M il!HO"i CHCTeMH (42) npH
e-+ 0 e npocTOTa HYJihoBoro posa'51s1<y (76) npn t0 = fuE lR TaKHX cniBBiJ.lHO
llleHb:
~ 10+m T t ~ ~
µf (/0) = 5 dt exp ( S d-r: tr K') ~f (t, t0 ) = 0, j = 0, n - I. (77)
to to
ISSN 0041-6053. YKp. AIQT. :J1Cypll., 1992. T. 44, M I
3 a YB a}!< e II 1-1 H 2_ B po60Ta X [8, 19, 20] 3po6,1e11 a cnpo 6a n ep eHeCTH
M eTO.llH .llOCJii,U}!{eI-11-lH po6iT 14, 23, 25] Ha a 6cTpa,niry cx eMy, 3aCTOCOI311 Y no He
Jiin iihrnx LlHllaMilllll-l X CHCTC'M n111 y (42) 13 6 a1 1a xo13Hx HeCKiH'Ie IIIIOBl1Mip1-rnx
npocTopax . 3 01, peMa, 13 (25) 6y.ny nbCH y c rreniaJibI-IOMY BHI"Ia.llKY awrnor cj)yHK·
1.1.ii" MeJibHHKOBa (73) µM (l0 ) -= µ,11 (t0 , t0 ) , 11e 11p11 £ - 0 )lJrn 13Cix t, f0 E IR
~ 1 ~ ~~ ~ ~
µM (t, to) = e (1/ 2> (0) (I( [CTo r g] (t - tn)L at I g) (t, to) - a;::- [g] (t, to) + 0 (£), (78)
;<2> (0) E A 2 (111 ) - c11MnJie1<T11 1IH<l CTpy1<Typa a11xi ;rno"i 11e,1iHi i'.moi· .llHHaM iq1mi
CHCTeMH (42) npH e = o_ 6epy•rn noxin1-1y Jl i B3Jl013}!< BeI<TOp Horo TTOJIH K:
: M _,._ TM) Bi.II. B11pa3y (78), aHaJ1or iL11-10 <j :opMyJiaM (68) 3HaXO.llHMO nprr e -
-o:
;M Cto) = .\ dt;<2> (0) (K [;;0 [g] (t - to)J, F [;;0 [g] (t, f0)]) (79)
-oo
)lJIH scix t0 E IR. B (25] CTBep;:n1<yeTbCH, Ill,O cpopMyJia (79) Jl3e KpHTepiH
TpaHCBepcaJibHOCTi roMOKJ1i11i 1rnux op6iT B rnqui a 0 [gJ (0) EM c[)a3013oro npo
c rnpy npII 3Hat1eHHi napaMeTpa t0 -= fa E IR, npH HKOMY µM (fo) = 0 - npocTHH
HYJib c[)y1-mui"i MeJibHHKOl3a. AEaJiori,1110 , HK i B 3ayaaMe1-mi 1, MmKIJa n0Ka
sanI, w,o 1313e,ue1-1a B [25] cpy1-mui11 1vle.'Ib11111<0Ba (78) He sa6esnet1y e_ noCTaTI-1bo'i·
y:vI0 BH TpaHCBepca.1bHOCTi roMOI<.TiiHilJHHX op6iT ;;± [g; O] (t - to) , Tell( HK ciq-
Ha nJIOU.J.HI-Ia 'i:'1•c:➔ 1\1, t0 E IR, HI<a y a i ;106pm1<e1-rn i
BI!6Hp3E'TbCH lIOBiJibHHM, Tpa11caepca,'IbHHM 11p11 I',- 0
r o ll0Jl51 ( 43) LJHI-JOM, y BH!l3LU<y ouimrn neJJHl !HJ-IH
.MeJ1bHHKOBa (73) rIOBHI!Ha 13H611pan1Cb 13 Y3 f'O,ll)[{el-IHi
TTyai-mape Pe: ~10 - ~ 10
,uo 3a;_i:aHoro se1nop1-10-
y3arcw1,1-1e1-1oi· cj)yHJ<ui"i
3 (53), T06TO Ta i(, mo6
;,g=_ 1 [g ] (t, t0) ET (~10 ) B TOtJlli 1; [g; OJ iO] E 1\1 .11.JJH acix 311a11e1-11, napaMeT-
pa t0 E IR.
3 a y B a )I( e H 1-1 51 3. Po3BHIIY'rH{i B UbOMY p03}.J. i Jli MeTOLl )lOCJJi)lMeI-1 11 51
:H BHU-Ia po3merwemrn cenapaTpHc .D.JIH rinep60.r1i,11111x oco6JJmrnx TOtJOK £-ne
<i iopM,u1ii:\ raMiJibTOII013HX .llHI-IaMiLHfHX CHCTeM Ha MH0f0131!,Ui ,-\1 2", 11 E ;;z + •
I l013I-!iCTIO 3 3CTOCOBHHH .!I.JIH )lOCJJ iLI)KeHI-IH uie_i" )K 3 a1tal!i i ,ll. .1H BHIT3)1KY
)lOBiJihJ-!Oi" HeJiiHii-ii-101 .lIHHaMiqI!Ol CHCTCMH Ha MHOfOBH,Ui Mm, mEZ+, )lO
BiJibHOI BHMipHOCTi .
I. 5 mwil A_ C., Cmena1-to1JcKutl !O. n_ A1m a6aTHttec1rne HHBapnaHTbl. - K1-1eIJ: HayK. nyMRa,
198 1.- 284 C.
2 . K p11cKaA M_ A1J.11a6an1ttecirne 1111BapHaIn1,i: nep. c anrn. - M.: l fan-Bo HIIocrp. JIHT.,
1962.- 92 C.
3 . Ap1-t0AbO B. H. Ma-reMaT1P1ec1\11e' ~1 e-ro111,I 1rn acnI11ec1wii Mexa11HKH _- M_: H ay Ka, 1984.-
432 c _
4. ;'vleAbflUKO/J B. K. 06 yc-roii'IHBOCT11 ucIrrpa np11 Iw p1101J.ntteCKHX no npeMemI B03MYlllCHHHxl/
Tp. MocK. MaT. o -na. - 1963 _- 12, .N!> ! _- C. 3- - 52.
5. KosAOI/ B. B. HnrerpnpyeMOCTb H 11emITeqrnpyeM0CTb B raMHJlbTOHOB0 11 Mex aHHKe // Ycne
XH M3T_ lia YK- - 1983 .- 38, .N9 1.- C. 3-67.
6. MumportOAbCKUll [0_ A. K nonpocy 06 amrnoaTH'I CC IW M 1rnnapuanTe // M aL cpH 3HKa H HC
.1 l!Herrn. Mex aH11Ka.- 1990.- 81,m_ 14.- C. 1-30 .
7. llpuKa pnamCKuli A. K-, MuKum,oK H. B _ A:1re6pa1-l'ICCKHe acneKTbl HHTerp11pye~10cnr HeJIH
Hej:1111,t x clHJl3MH'-leCKHX CHCTeM Ila MH0rooti pa :rn H x .- K1-1en : HayK. J!YMKa , I 99 l.- 287 c.
8. Marsden J. E. Four applications of nonlinear an alysis to p hysics and Eng ineering / / New
directions in AppL Math.- 1982. - IO. - P. 85- 107_
9 . Holmes Ph. J., Marsden J.E. Horsephoes and Arno ld diifusion for 1-lamillonian syst em s
on Lie groups I I Indiana Univ _ M a t h _ J .- 1983 .- 32, N 2_- P_ 273- ,10'.l
10. Hnmeepupye,11.b1e ll.H1-IaI1rn,IecIrne CHCTe~n,I / 10. A _ /111-1-rponoJibCKHii, H. H. oorc,mo6oIJ (Mn.).
A _ K. OpHKapnaTCKHii, B. r. CaMofl .1eHKO. - K1-1 eu: Hay1c ll.YMKa, 1987.- 2% c.
11. MumponoAuCKUtl !O . A., A,unoHuuwH H. 0., llpuKapnamc1-:uli A. K- Am1 aGan1 •; ecK1-1e HH•
BapH a JJTbl 1-1eJ11-11-1 e11HblX l(JlllaMH'leCJ<H X CHCTeM C Ma,1blM napaMeTp0M / / H emmei"111b1<: 31J0JIIO
IU!0lll1Lle ypanHeI-I1151 IJ npHKJJaJlllblX c1aJla•iax_- K1-1 en: 1111 -T MaTeMaTHKH AH YCCP, 1991.
C. 90-93 .
12. /-/um c1(KU 3. BBell.Clrne Il ,1\HcjxjlepeHUH3JlbHYIO JUHJaMHKy. - M. : /1'1Hp, 1975.- 304 C.
13. llaAUC )f(._, JJu,1teAy B. f eoMeTpHtteCKa51 Te0pHH JlHHaMntteCKHX CHCTeM. BBelleHHe.- M. :
MHp, 1986.- 304 C.
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14. AHocoa Jl. B. f JJa,'\KHe .n 1HaMw1ec1rne c1 1cTeMbI.- M. : Bl-1Hl-1Tl1, 1985. - C. 152-203.
(Krnrn nayKa H TeXHHKH. CoBp. npo6.q . MaTeMaTH1rn; T . !).
15. \\7iggins S., Ho lmes Ph. 1-Iomoclini c orbits in s lowly va r ying oscillators I I SIAM J . Math .
Anal. - 1987 .- 18, N 3.- P. 6 12-629.
16. Holmes Ph . .! . , Marsden .l. E. Horseshoes in Perturba tions of H amiltoni an systems w ith
Degrees of Freedom/! Comm uns. Math. Phys.- 1982. - 82 , N 3.- P . 523-544.
17. Holmes Ph., ivlarsden .l . E. Melnicov's met hod and Arnold diffus ion fo r pert urbat ions of
inte_grab le Ham iltoni an systems// J . Math . Ph ys .- 1982 .- 23, N 4.- P . 669-675.
18. Holmes Ph . .l . Averaging a nd Chaotic moti ons in forced oscilla tion II SIAM J. App . Math.-
1980. - 38, N 1.- P. 65-80; - 198 1. - 40. N 1. - P. 167- 168.
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d ic orb its: Chaoti c Osci lla tions of forced beam// Arch . Ration. Mech. an d Ana l. - 198 1. -
76, N 2.- P. 135- 165 .
20 . Holme, Ph . .l ., M arsden .l. E . Bifurca tion to di vergence and flutter in flow-induced Osci
lla tions : An infi nite dimensiona l analysis /IA utomatica .- 1978 .- 14 , N 3.- P . 367-
384.
21. Chow S.-N. An examp le of bifurcation to homoclinic orbits // J. Different. Equnt.-
1980 .- 37, N 3.- P. 351-373.
22. Chow S. -N., Hale J. K. Method of bifurca tion theory.- New York etc. : Springer , 1982.-
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| id | umjimathkievua-article-2319 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T02:22:12Z |
| publishDate | 1992 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c8/f421d0d9338fef5549f1eb80b9c7e5c8.pdf |
| spelling | umjimathkievua-article-23192023-09-14T12:03:56Z Simplex analysis of dynamical systems with a perturbation. New criteria of stabilization of a homoclinic separatrix and its use Симплектичний аналіз динамічних систем з малим параметром. Новий критерій стабілізації гомоклінічних сепаратрис та його застосування Mitropolsky , Yu. O. Antonishin , I. O. Prikarpatsky , A. K. Samoilenko , V. G. Митропольський , Ю. О. Антонишин , І. О. Прикарпатський , А. К. Самойленко , В. Г. Dynamical System Small Parameter Type System Exact Method Symplectic Manifold Динамическая система Малый параметр Точный метод We study adiabatic invariants of nonlinear dynamical systems of autonomous and nonautonomous type on symplectic manifolds, and, in particular, give criteria for the existence of such invariants and methods for constructing them explicitly. We make a thorough analysis of the phenomena of splitting of a homoclinic separatrix from a hyperbolic singular point, and propose a new exact method for constructing an analogue of Melʹnikov's function, which gives a necessary and sufficient condition for transversality, which generalizes automatically to dynamical systems in Rn, n ∈ Z+. Исследуются адиабатические инварианты нелинейных динамических систем автономного и неавтономного типа на симплектических многообразиях, в частности, критерии их существования и методы их явного построения. Проведен тщательный анализ явления расщепления гомоклинической сепаратрисы с особой точкой гиперболического типа, а также предложен новый, точный метод построения аналога μ-функции Мельникова, что дает необходимое и достаточное условие трансверсальности, который автоматически обобщен на случай динамической системы Rn, n ∈ Z+. Досліджуються адіабатичні інваріанти нелінійних динамічних систем автономного і неавтономного типу на симплектичних многовидах, зокрема, критерії їх існування та методи їх явної побудови. Проведено грунтовний аналіз явища розщеплення гомоклінічноі сепаратриса з особливою точкою гіперболічного типу, а також запропоновано новий, точний метод побудови аналога μ-функпії Мельникова, що дає необхідні та достатні умови трансверсальности який автоматично узагальнений на випадок динамічної системи в Rn, n ∈ Z+. Institute of Mathematics, NAS of Ukraine 1992-02-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2319 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 1 (1992); 46-66 Український математичний журнал; Том 44 № 1 (1992); 46-66 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/2319/1540 Copyright (c) 1992 Yu. O. Mitropolsky, I. O. Antonishin, A. K. Prikarpatsky, V. G. Samoilenko |
| spellingShingle | Mitropolsky , Yu. O. Antonishin , I. O. Prikarpatsky , A. K. Samoilenko , V. G. Митропольський , Ю. О. Антонишин , І. О. Прикарпатський , А. К. Самойленко , В. Г. Simplex analysis of dynamical systems with a perturbation. New criteria of stabilization of a homoclinic separatrix and its use |
| title | Simplex analysis of dynamical systems with a perturbation. New criteria of stabilization of a homoclinic separatrix and its use |
| title_alt | Симплектичний аналіз динамічних систем з малим параметром. Новий критерій стабілізації гомоклінічних сепаратрис та його застосування |
| title_full | Simplex analysis of dynamical systems with a perturbation. New criteria of stabilization of a homoclinic separatrix and its use |
| title_fullStr | Simplex analysis of dynamical systems with a perturbation. New criteria of stabilization of a homoclinic separatrix and its use |
| title_full_unstemmed | Simplex analysis of dynamical systems with a perturbation. New criteria of stabilization of a homoclinic separatrix and its use |
| title_short | Simplex analysis of dynamical systems with a perturbation. New criteria of stabilization of a homoclinic separatrix and its use |
| title_sort | simplex analysis of dynamical systems with a perturbation. new criteria of stabilization of a homoclinic separatrix and its use |
| topic_facet | Dynamical System Small Parameter Type System Exact Method Symplectic Manifold Динамическая система Малый параметр Точный метод |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2319 |
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