Small random perturbations in second-order oscillatory systems
The limit behavior of the solutions of a nonlinear differential equation that describes an oscillatory system with small random perturbations of the type of multidimensional “white” and “shot” noises is studied.
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| Date: | 1992 |
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| 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/2315 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508240737992704 |
|---|---|
| author | Borisenko , О. V. Борисенко , О. В. |
| author_facet | Borisenko , О. V. Борисенко , О. В. |
| author_sort | Borisenko , О. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-09-14T12:03:56Z |
| description | The limit behavior of the solutions of a nonlinear differential equation that describes an oscillatory system with small random perturbations of the type of multidimensional “white” and “shot” noises is studied. |
| first_indexed | 2026-03-24T02:22:04Z |
| format | Article |
| fulltext |
Y.UK 517.9
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MaJihie c.ny'Ia:«Hhie B03MyI.IJ,eHHJI
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JSSN 0041-6053. Y,w Mar. :JteypH., 1992, r. 44, M 1 .,
_1,fayqemuo KOJie6aTeJibHbIX CHCTeM rro.u B03.UeflcTBHeM cJiyqaHHblX B03Mylll.e
HHH TIOCBHlll.eH jpH.U pa6oT. O6w11p 1rnH 6H6JIHorpacpHH TIO .U3HHOH TeMe rrptrne
.n.etta B [ 1].
B .n.aHI-IOH p a6oTe H3yqaeTCS1 rrpe,neJibH0e TI0Be,UeHHe npH e -+ 0 KOJie6aTeJib·
HOH CHCTeMbl, OllHCbIBaeMOH ypaBHeHHeM
x" (t) + b2 x (t) = ef O (x (t), x' (t)) + f 1l 2f (x (t), x' (t)),
x (0) = x 0 , x' (0) = Yo,
(1)
r.n,e f (x (t), x' (t)) - o6o6meHH3H CJIY43HH3H cj)yHKUHH TaKaH, qTQ .UJIH Jno6oro
t> 0
t d I
Sf (x (s), x' (s)) ds = ~ S fi (x (s), x' (s)) dwi (s) +
0 i= I 0
I ~
+ ,\ .\ /d+1 (x (s), x' (s), z) v (ds, dz),
o R'
fi (x , y), i = 0, d, fd+I (x, y, z)- 1-1ecJ1yLiaHHh1e cl)YHKUHH; w, (t), i = IT.
- He3aBHCHMbie 0,11,H0MepHbie BHIIepoBCKHe npouecCbI; V (t, A) - ueHTpHpoBaI-I
HaH rryacC0H0Ba Mepa, He3aBHCHMaH OT W; (t) , i = n , V (t, A)= V (t , A) -
- tll (A), Mv (t, A)= tll (A); x0 , y 0 - HecJJy4aHHbie HaqaJJbHbie ,naHHble,
11 ( •) - Mepa Ha 6opeJieBc1rnx MH0)Kecrnax B Ri .
YpaBHeHHe BT0poro rrop 51,UKa ( 1) IT0HHMaeTCH KaK CHCTeMa CT0Xac-rnqec1rnx
ypaBI-Ie1.111ii 6e3 n ocJie.n,eiicTBHH
dx (t) = y (t) dt,
d
dy (t) = f- b2x (t) + ef O (x (t), y (t))I dt + e1l2 ~ f1 (x (t), y (t)) dw; (t) + ,_,
+ e112 .\ f d+I (x (t), y (t), z) ~ (dt, dz), x (0) = x 0 , y (0) = Yo· (2)
R•
+ e112 .\ f<1+1 (X8 (t), Ye (t) , z) ~ 8 (dt, dz), (3)
R•
x ,, (0) = Xo, Ye (0) = Yo·
B .n,aJJhHeihrreM nocT0HHHhie, He3aBHCHmHe OT e, 6y,11,eM 0603Ha4aTh C.
JI e MM a. llycmb IT (R2) < oo u Bbtn0llHflromcH. ycAoeuH.:
1) {;, i = 0, d + I , ozpaHu<teHbt u yooeAemeopRrom ll0IW/lbH0My ycAoeuro
Jlunutuu,a no nepe.AteHHbt.At (x, y);
2) cyw,ecmey,om nocmoflHHbte I( > 0 u r > 0 ma,we, 'imo Ollfl HeKomopoio
'r > 0
d
~ I ft (x ; Y)l2 + I /d+I (x, y, z)l2 ~ K (x2 + y2)'+v, x 2 + y 2 ::::;;; r 2 , v z E R2 .
i=O
12 . ISSN 0041-6053. "l/,cp, MaT, :J1CypH.., 1992, r. 44, M J
Tozoa O/l5I oocmamO'iH.O AWAbtX e > 0 u o > 0
P {x~ (t) + y~(t) ~ o} ~ Ct(o"12 + o"). (4)
1l O K a 3 a Te JI b CT B O a 1-1aJJOfHLIIIO )lOl'33aTeJJbCTBY JJeMMbl I H3 [21.
TTycTb .D.JJH cj)yHKUHH /;, i = 0, d + I, Bbmo.'rnHeTcH ycJJOBHe 2 JJeMMbI npH
1' > I . Tor.na, 11cnoJib3Yll (4), y6e)1<.naeMcH , 4TO K c JJy<IatlHbIM npo u eccaM
b Ye (t )
0e (t) = - et - arctg bxe (t)
MmK11O npHMeI-IHTb 0606me1rnym rj1opMyJ1y 11m [3]. ]IJJH npouecca Se (t) = (ae (t),
8e (t)) TTOJJYLIHM CllCTeMy CTOX3CTJ1l-leCJ<H X ypam-1e1m i,i 6e3 nOCJ1e.net1CTBHH
r.ne
dse (t ) = Ae (t, Sc (t)) dt + Oe (t, Se (t)) dwe (t) +
+ \ Ce (t , £e (t) , z) ~;e (dt, dz), £e (0) = (a0 , 00),
k•
We (t) = {wf (t) , i = ! , d}, Ac (t, a, 0) = (A~1> (t, a, 0), Af > (t, a, 0)),
Ge (t, a, 0) = {o1; (t, a, 0) , i = I , 2; j = 1,d}, Ce (t, a, 0, z) =
= (Ci1> (t, a, 0, z) , cf > (t, a, 0, z)), '\Jle = ..!!!._ + 0, A~l) (t, a, 0) =
e
~
- ab+ e112 sin '\Jl.JJ+ 1 (a , 'ljJe, z)} II (dz),
cos 'ljJ ~ sin 2'1jJ d ~
Af> (t, a, 0) = - ab e fo (a, '\Jle) - 2a2b/ ~ /7 (a, ,Pe)+
i = l
+ -1 ,I {arctg [tg 'ljl8 - e1/ 2
e R'
cos '4) ~ }
- 'lj\ --l- f 112 ab e f d+ I (a, 'ljl8 , z) II (dz),
sin 1jJ,, - cos'ljl8 ~ crt (t, a, 0) = - b fi a, 'Pe), crt (t, a, 0) = - ~ f i (a, ,Pe),
j = IT.
(5)
I ~
C ~1> (t, a, 8, z) = b {[(ab sin 'ljl8 - e112 f d+ i (a, 'ljl8 , z))2 + a2b2 cos2 '1Jle]1/2 - ab},
112~
c<2> t lt e r,+1 (a . 'Pe, z) l e (t, a, e, z) = arc g g '\Jle - ab cos '\Jle - '\Jle,
f;(a , '\j)) =f1 (acos'\j), -absin'\j)), i=0,d,
-. ; Y~
/J+i (a, 'ljl, z) = fd+t (a cos 'ljl, - ab sin 'ljl, z), a0 = V x~ + b2 ,
0o = - arctg bYo •
Xo
ISSN 0041-6053. Y,,;p_ MaT. xypH., 1992, T. 44, M l 13
Te o p e Ma. Ilycmb II (R2) < oo, x~ + y~ =I= 0 , t E [0, t0] , cpy1-1Kiwu f1,
i = 0, d + 1, Olpa1-1u4,e1-1bt , fo -- oou1-1 pa3, f ;, i = 1, d + 1, - oeaJ1cou 1-1e
npepbl8HO ouqxpepe1-1qupyeMbt no nepe,1teHHbtM (x , y) ,
.\ [II v /d+1 11 2 + II v 2 fd+ 11121 II (dz)~ C
R•
u cyU1,ecmeyem nocmoflHHafl r > 0 niaKafl, 4,,no oAfl HeKomopozo i' > 1
d+ l d+ l L If; 12 ~ C (x2 + y2)1+-v, L ll v {; 112 ~ C (x2 + y zp
i = O i = O
npu x2 + y2 ~ , 2 u A/060.M z E R2. 3oecb V f = ( i: , -:: ) ' v 2 f - Mam-
"
pul{a emopbtX npou3BOOl-lbtx no (x, y), II a 112 = L a; UAfl a ER", II a W =
i = I
ll
_ ~ a 2. oAfl MampUL{bt a. - .l.J t/
i ,j= l
Tozoa npOlfeCC Se (t) = (ae (t), 8e (t)) CAa6o C>:00/ttnCfl npu e -+ 0 K pe-
iue!-llt/0 [ (t) = ( a (t), 0 (t)) CtnOXQCl7U1 1teCKOcO OUr/Jrpepe1-1,t(lWAbHOZO !JPGBHCl-lUfl
cfJe
d[(t) = A (a (t)) dt + a (a (t)) dw (t), [ (0) = (a0 , 00),
-(IJ 1 2." l- ~ sin 'P ~ cos2 1Ji ]
A (a)= 2n .\ - fo (a, 1Ji)-b- + f (a , 1.Ji) 2b2 a d1j),
0
A- c2J ( ) = _1_ 2( [- ~f ( ,h) cos 1p _ ~1 ( ) sin 21JJ ]d
a 2n .l o a, 'I' ab a, 'P 2a2b2 \jl ,
0
a (a) = { 2n1b2 r f (a, 1j)) B (a, 1j)) d1Jir
2
,
rv d ~ ,_,
f (a, 1j)) = ~ f; (a, 1j))+ .\ f~+' (a, 1j), z) IT (dz),
i= l R•
(
s in2 'P
B (a , 1p) = .
sm 2,p
2a
sin 2,JJ )
2a
cos2 1P
a2
w (t) = (w1 (t), w2 (t)),
(6)
W; (t) , i = 1, 2, - H.e3a8UCU,Ubte OOHO,ltepl-lote 8UHep08CKUe npou,eccbl.
)l OK a 3 a Te JI b CT B 0. B ycJIOBHHX TeopeMbl HMeeM
1 II Ae (t, a, 0)11 + II CTe (t, a, 8)11 + ---W II Ce (t, a, 0, z)\\ ~ C (7)
c
1-1 Bb!TIOJIH HIOTCH [4) YCJIOBHH TeopeMbl cyw.ecTBOBaHHH H e,ll.HHCTBeHHOCTH pewe
H11H c11cTeMbI (5). TTpouecc se (t) HMeeT BHJJ.
I
~e (t) = So + S Ae (s, Se (s)) ds + ~e (t), (8)
0
rJJ.e
. t t
· ~e (t) = S CTe (s, Se (s)) dwe (s) + S J Cs (s, Se (s), z) ; e (ds, dz),
O O R•
So = (ao, 00)-
14 . ISSN 0041-6053. YKp. Mar. [)/Cyp1t., 1992, T. 44, M I
vfa HenpepbIBHOCTH npoueccoB l (t) .H ( (t) , JieMMbI 2 [2] II cooTHorneHIIA
(7), (8) , ( 11) cJie.nyeT
t
[ (t) = so + .l A (a (s)) ds + ~ (t) ,
0
r.ne f (t) - Herrpepb!BI-lh!H KBa,npaTHl!I-10 HHTerpttpyeMbIH BeKTOpHh!H MapTHHfaJI
c MaTpHLJHOH x apaKTepucn11<01"1
t
(~, [) (t) = s T3 (a (s)) ds.
0
3 .nech B(a) = {Bu (a), i, J= 1,2}.
}fa [61 c.11e.nyeT, l!TO cyw.ecrnyeT Btt11epoBCKHH npouecc w (t) = (w1 (t),
w2 (t) ) TaKOH, l!TO
t
"( (t) = .) a (a (s)) dw (s),
0
- - 1;2
r.ne <J (a) = B (a). _
3 ttal!HT, npouecc s (t) y ,noB.r1ernopHeT y p aBHeHHI{) (6). Ho y paBHe1-rne (6)
HMeeT e.a.m-1crne1-1Hoe pewerm e B ycJJOB HHX TeopeMhl. CJJe.noBaTeJJ hHO, npouecc
f (t ) He 33BHCHT OT BbI6opa nocJJe,UoBaTe.'lbHOCTH en - 0 . no3TOMY KOHel!HOMep
Hhie pacnpe.neJJemrn npouecca se (t) CXO/lHTCH K KOHel!H0MepHhlM pacnpe.neJJe
HH HM npouecca 1 (t) npu e -+ 0. OpouecchI Se (t) H 1 (t) - MapKOBCKHe npo
u eccb1. D,JJH cJJa6oi\: cxo,nHMOCTH Se (t) -+1 (t) npH e - 0 .nocTaTOl!HO [7] BhI
noJrneHHH YCJJOBHH
sup P {II £e (t, x, s) - x II> <5} = 0
xER.'
O~ s- t~h
(12)
.l{JJH Ka)l<.noro <5 > 0, r.ne Se (t, x, s)- perneHHe CToxacTHl!ecKoro ypam-1e1mSI
s s
£e (t, X, s) = X + S A e (T, Se (t, X, T)) d.- + S <Je (T, Se (t, x, T)) dwe (T) +
t I
s -
+ S S Ce (T, Se (t, x, T), z) V e (dT, dz).
t R.•
AHaJIOfHl!HO oueHKaM (9) noJiyl!HM
M II Se (t, x, s)-x II~ C(! s-tl + I s-t 1112), (13)
r.ne C - IlOCT0SIHHaSI, He3aBHCHM35l OT t, s, X H e. 113 HepaBeHCTBa 4e6billJeBa H
(13) cJJe.nyeT (12). TeopeMa .n0Kasa1-1a.
1. MumponoAbCKutl !O. A., KoAoAmeii B. r . 0 ao3neiicTBHH cJiyqaliHEIX CHJI Ha HeJIHHeiiHbie KO
Jie6aTeJibHb1e CHCTeMbil/ MaT. q:>H311Ka H HeJI HHeHH . MexaHHKa.- 1986.- Bblll. 5. - C. 23-
::l4.
2. E opuceH.KO 0 . B. H eJIHHeHHble KOJie6aHHH C M3Jlb!MH cJiyqai\HblMH B03MyW.eHHHMH / / ActtMil
TOTH'leCKHe MeTOl<bl o 32,!1.a<iax MaT. cj>H 3HKH: C6. Hay<rn. Tp.-Kttea : Hu-T MaTeMaTHKH AH
Y CCP, 1989.- C. 19-27.
3. Tux.•ia11. H. H., C,wpoxoiJ A. B. CToxacTH'lecKHe nHcjxpepeHTIHaJihHble ypaaueHHH.- KHen:
H avK_ ll.YMKa, 1968.-- 354 c.
4. ru u ia11. H . 11., CKopoxoi) A. B. YnpaBJIHeMbie cJ1y'l a fiMb1e npouecchJ.- KHen : 1-! ayK. AYMKa,
1977.- 252 C.
· 5. C,rnpoxoiJ A. B. HccJienoaarrnH no TeopHH CJiy'laHHhIX npoueccoa .- KHeB: KHeB. yH-T ,
196 1. - 216 c.
6. r ux ..iw11. If. H_, CKopoxoiJ A. B. T eopHH CJiy'laHHblX npoueccon: B 3 -x T.- M .: HayKa,
1975.- T. 3.- 496 C.
7. rux.,w11. H . 11., CKopoxoiJ A. B. TeopHH CJiy'laiiHhlX npoueccoa: B 3-x T.- M .: HayKa,
1971,- T. i.- 664 c,
TTOJJy'leHo 02.11.90
0011-1
0012
0013
0013-1
0014
|
| id | umjimathkievua-article-2315 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T02:22:04Z |
| publishDate | 1992 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/78/6dcdbb48b3c9eb19e180bb8ea7c02378.pdf |
| spelling | umjimathkievua-article-23152023-09-14T12:03:56Z Small random perturbations in second-order oscillatory systems Малые случайные возмущения в колебательных системах второго порядка Borisenko , О. V. Борисенко , О. В. Differential Equation Nonlinear Differential Equation Oscillatory System Random Perturbation Limit Behavior Диференціальне рівняння Нелінійне диференціальне рівняння Коливальна система Випадкове збурення Гранична поведінка The limit behavior of the solutions of a nonlinear differential equation that describes an oscillatory system with small random perturbations of the type of multidimensional “white” and “shot” noises is studied. Изучено предельное поведение решения нелинейного дифференциального уравнения, описывающего колебательную систему с малыми случайными возмущениями типа многомерного «белого» и «дробового» шумов. Вивчена гранична поведінка розв’язку нелінійного диференціального рівняння, що описує коливальну систему з малими випадковими збуреннями по типу багатовимірного «білого» і«дробового» шумів. Institute of Mathematics, NAS of Ukraine 1992-02-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2315 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 1 (1992); 11-16 Український математичний журнал; Том 44 № 1 (1992); 11-16 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/2315/9450 Copyright (c) 1992 O. V. Borisenko |
| spellingShingle | Borisenko , О. V. Борисенко , О. В. Small random perturbations in second-order oscillatory systems |
| title | Small random perturbations in second-order oscillatory systems |
| title_alt | Малые случайные возмущения в колебательных системах второго порядка |
| title_full | Small random perturbations in second-order oscillatory systems |
| title_fullStr | Small random perturbations in second-order oscillatory systems |
| title_full_unstemmed | Small random perturbations in second-order oscillatory systems |
| title_short | Small random perturbations in second-order oscillatory systems |
| title_sort | small random perturbations in second-order oscillatory systems |
| topic_facet | Differential Equation Nonlinear Differential Equation Oscillatory System Random Perturbation Limit Behavior Диференціальне рівняння Нелінійне диференціальне рівняння Коливальна система Випадкове збурення Гранична поведінка |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2315 |
| work_keys_str_mv | AT borisenkoov smallrandomperturbationsinsecondorderoscillatorysystems AT borisenkoov smallrandomperturbationsinsecondorderoscillatorysystems AT borisenkoov malyeslučajnyevozmuŝeniâvkolebatelʹnyhsistemahvtorogoporâdka AT borisenkoov malyeslučajnyevozmuŝeniâvkolebatelʹnyhsistemahvtorogoporâdka |