EE?ANUeEEE OI?AA?NEOAO ?IAI? OA?ANA OAA*AIEA

EANUeEOI ?aaai Ieeeiiae/

OAeE 519.21

ANEIIOIOE*IA IIAAAe?IEA IANO?EEEO ?ICA’ssCE?A NOIOANOE*IEO
AeEOA?AIOe?EIEO ??AIssIUe

01.01.05 — oai??y eiia??iinoae oa iaoaiaoe/ia noaoenoeea

AAOI?AOA?AO

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa

o?ceei-iaoaiaoe/ieo iaoe

Ee?a — 1999

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia ia eaoaae?? oai??? eiia??iinoae oa iaoaiaoe/ii?
noaoenoeee Ee?anueeiai oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea.

Iaoeiaee ea??aiee: aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?

EOE?I?* A?eai??e Eiaaeiiae/,

Ee?anueeee oi?aa?neoao ?iai? Oa?ana Oaa/aiea, caa?aeoaa/ eaoaae?e
caaaeueii? iaoaiaoeee

Io?oe?ei? iiiiaioe: aeieoi? o?ceei-iaoaiaoe/ieo iaoe, aeioeaio

EIIEOEI Aiaaeai ?aaiiae/

Euea?anueeee ?inoeooo iaiaaeaeiaioo

caa?aeoaa/ eaoaae?e ?ioi?iaoe?ei? nenoaie o iaiaaeaeiaio?;

eaiaeeaeao o?ceei-iaoaiaoe/ieo iaoe

EOEEE Ieaen?e Ieoaeeiae/

?inoeooo iaoaiaoeee IAI Oe?a?ie

aeieoi?aio a?aeae?eo oai??? aeiaaeeiaeo i?ioean?a

I?ia?aeia onoaiiaa: ?inoeooo i?eeeaaeii? iaoaiaoeee ? iaoai?ee IAI
Oe?a?ie (i. Aeiiaoeuee)

Caoeno a?aeaoaeaoueny «27» aa?aniy 1999 ?ieo i 14 aiaeei? ia can?aeaii?
niaoe?ae?ciaaii? a/aii? ?aaee Ae 26.001.37 o Ee?anueeiio oi?aa?neoao?
?iai? Oa?ana Oaa/aiea ca aae?anith: 252127, i.Ee?a-127, i?iniaeo aeaae.
Aeooeiaa, 6, Ee?anueeee oi?aa?neoao ?iai? Oa?ana Oaa/aiea,
iaoai?ei-iaoaiaoe/iee oaeoeueoao, aoae. 42.

C aeena?oaoe??th iiaeia iciaeiieoenue o a?ae?ioaoe? Ee?anueeiai
oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea (Ee?a, aoe. Aieiaeeie?nueea, 58).

Aaoi?aoa?ao ?ic?neaii «25» na?iiy 1999 ?.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee I. I. Iieey/oe

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. Oai??y noioanoe/ieo aeeoa?aioe?eieo ??aiyiue aoea
caii/aoeiaaia ia ii/aoeo 40-o ?ie?a iaoiai noi??//y I.I.Aiaiethaiaei oa
I.I.E?eeiaei oa E.?.A?oiaiii ia iniia? aaaaeaiiai E.?oi iiiyooy
noioanoe/iiai ?ioaa?aeo. Oai??y noioanoe/ieo ??aiyiue, caniiaaia ia
iiiyoo? noioanoe/iiai ?ioaa?aeo ?oi, aeyaeeanue o?nii iia’ycaiith c
oai???th ia?oeiaae?a, aeaoaeueii ?ic?iaeaiith Aeae.Aeoaii, I.Iae??ii oa
aaaaoueia ?ioeie. Canoinoaaiiy oaeiai i?aeoiaeo aei ?ica’ycaiiy
noioanoe/ieo aeeoa?aioe?eieo ??aiyiue aeicaieeei io?eiaoe i?e i?e?iaei?o
iaiaaeaiiyo ia eiao?oe??ioe oai?aie ?nioaaiiy oa ?aeeiino? ?ica’yceo.

Aea/aiiy aneiioioe/ii? iiaaae?iee aeeooc?eieo i?ioean?a aoei i?aaeiaoii
aaaaoueio aeine?aeaeaiue. Na?aae ia?oeo aeine?aeaeaiue ui aoee c?iaeai?
a oeueiio iai?yieo iiaeia iacaaoe ?iaioe ?.E.No?aoiiia?/a,
?.C.Oanueieinueeiai, ?.?.A?oiaia, A.A.Nei?ioiaea.

Ia aeaiee iiiaio iiaeia aeae?eeoe ianooii? iniiai? iai?yiee
aeine?aeaeaiiy aneiioioe/ii? iiaaae?iee ?ica’yce?a noioanoe/ieo
aeeoa?aioe?eieo ??aiyiue:

1. Iaiaaeai?noue ?ica’yce?a noioanoe/ieo aeeoa?aioe?eieo ??aiyiue.

2. Iaiaiaaeai?noue ?ica’yce?a noioanoe/ieo aeeoa?aioe?eieo ??aiyiue c
?inoii /ano

3. No?ee?noue ?ica’yce?a noioanoe/ieo aeeoa?aioe?eieo ??aiyiue.

4. A?aiaee/i? oai?aie.

5. Aneiioioe/i? aeanoeaino? ?ica’yce?a noioanoe/ieo aeeoa?aioe?eieo
??aiyiue, eiao?oe??ioe yeeo caeaaeaoue a?ae ia?aiao?o.

A inoaii?e /an aaeeea caoe?eaaeaiiy aeeeeea? caaea/a aeine?aeaeaiiy
aneiioioe/ii? iiaaae?iee ?ica’yce?a noioanoe/ieo aeeoa?aioe?eieo
??aiyiue c ia?aaoey?iith caeaaei?noth eiao?oe??io?a a?ae ia?aiao?o, aai
c eiao?oe??ioaie eio?? nai? ii nia?, acaaae? eaaeo/e, iiaeooue i?eoaee
ia ca?aaoenue. Iniiaiith i?iaeaiith o aeai?e oai??? ? aeine?aeaeaiiy
a?aie/ii? iiaaae?iee ?ica’yce?a noioanoe/iiai aeeoa?aioe?eiiai
??aiyiiy

(1)

aea 0t1, aT(o), T(o) ae?eni? aei??i? ooieoe?? oae?, ui

aeey aeey eiaeiiai N>0,
?ica’ycie — ?nio? aeey eiaeiiai TT00, wT(t) — n?i’y a?ia??anueeeo
i?ioean?a, ui aecia/aia ia eiia??iinoiiio i?inoi?? , a?aeiia
aeiaaeeiaa aaee/eia iacaeaaeia a?ae wT(t).

Ie aeine?aeaeo?ii ??aiyiiy oeio (1), oa?aeoa?iith iniaeea?noth yeeo ?
aeineoue ia?aaoey?ia caeaaei?noue eiao?oe??io?a aT(o), a?ae ia?aiao?o ,
a naia a?ae ieo ia aeiaaa?oueny ??aiii??ia iaiaaeai?noue ii ia?aiao?o o
e?ioeaaeo iaeanoyo ci?iieo, oa ia aeiaaa?oueny ??aiii??ia
a?aeie?aieai?noue eiao?oe??io?a a?ae ioey. Cie?aia, a aeayeeo
oi/eao ok eiao?oe??ioe aT(o), iiaeooue aooe » «- aeaeieie
iine?aeiaiinoyie, ca?aaoenue a aea-yeiio nain? aei ae?ac?a ,
(ok — oi/ee neiaoey?iino?), aai iaoe ae?iaeaeaiiy ?ioiai oa?aeoa?o.

Aia?oa ??aiyiiy c oaeith ia?aaoey?iith caeaaei?noth a?ae ia?aiao?o aoee
?icaeyioo? a ?iaioao Eoe?i?/a A.E., i?e aeine?aeaeaii? oa?aeoa?o
iiaaae?iee cii?iiaaieo iano?eeeo ?ica’yce?a iaeiiaei??ieo noioanoe/ieo
aeeoa?aioe?eieo ??aiyiue. I?e oeueiio anoaiiaeee, ui i?e aeine?aeaeaii?
iiaaae?iee ?ica’yce?a ??aiyiiy (1) aaaeeeao ?ieue a?ae?a?athoue ooieoe??

,

aea — aeayea n?i’y noaeeo.

Iaea?eueo caaa?oai? ?acoeueoaoe aoee io?eiai? a ?iaioao Eoe?i?/a A.E.
aeey eeano E1 ??aiyiue (1), a yeeo

I?e oeueiio anoaiiaeee, ui a E1 a?aie/i? i?ioeane aeey i?e
iathoue aeaeyae , aea l(o) — iiiioiiia iaia?a?aia ooieoe?y
, — neaaeee ?ica’ycie ??aiyiiy

,

— aei??ia ooieoe?y . Cie?aia, a?aie/iee eean i?ioean?a
i?noeoue ocaaaeueiai? aeeooc?ei? i?ioeane a ?icoi?ii? I.?.Ii?oaiea.

Cia/ia i?inoaaiiy a aeine?aeaeaii? aneiioioe/ii? iiaaae?iee ?ica’yce?a
??aiyiue (1) i?e aeineoue ia?aaoey?ia caeaaei?noue eiao?oe??io?a aT(o),
a?ae ia?aiao?o, i?e oiia? ??aiii??ii? a?aeie?aieaiino? a?ae ioey oa
iaiaaeaiino? eiao?oe??io?a aeeooc?? aoei aeinyaiooi a inoaii?o ?iaioao
N.ss. Iaoii.

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. ?iaioa
a?aeiia?aea? ieaio iaoeiaeo ?ia?o eaoaae?e oai??? eiia??iinoae oa
iaoaiaoe/ii? noaoenoeee Ee?anueeiai oi?aa?neoaoo ?i. O. Oaa/aiea a
?aieao iaoeiai-aeine?aeieoeueeeo oai: “Iiaoaeiaa iaoaiaoe/ieo iiaeaeae
ia?aaoey?ieo aaiethoe?eieo aeaoa?i?iiaaieo ? noioanoe/ieo nenoai oa
aeine?aeaeaiiy aeanoeainoae ?o ?ica’yce?a” ?97120, “Noaoenoe/iee aiae?c
aeiaaeeiaeo i?ioean?a ? iie?a oa eiai canoinoaaiiy” ?97048.

Iaoa ?iaioe. Aea/eoe a?aie/io iiaaae?ieo i?e ?ica’yce?a ??aiyiiy
(1) aeey eeano E2, oiaoi ??aiyiue aeey yeeo ?niothoue ,
oae?, ui i?e an?o o ??aiii??ii ii O ia? i?noea ia??ai?noue

,

Cie?aia, a E2 aeiionea?oueny i?e ca?aei?noue
aai .

Iaoiaee aeine?aeaeaiiy aacothoueny ia iaoiaeao aeine?aeaeaiiy
iaiaiaaeaieo ?ica’yce?a noioanoe/ieo aeeoa?aioe?eieo ??aiyiue c ?inoii
/ano. E??i oeueiai aoee aeei?enoai? iaoiaee aeine?aeaeaiiy aneiioioe/ii?
iiaaae?iee ?ica’yce?a noioanoe/ieo aeeoa?aioe?eieo ??aiyiue,
eiao?oe??ioe yeeo caeaaeaoue a?ae ia?aiao?o.

Iaoeiaa iiaecia ?acoeueoao?a ?iaioe:

— Iaea?aeai? aeinoaoi? oiiae aneiioioe/ii? no?eeino? oa iano?eeino?
?ica’yce?a noioanoe/ieo aeeoa?aioe?eieo ??aiyiue ?oi,

— Aeine?aeaeaia aneiioioe/ia iiaaae?iea ?ica’yce?a ??aiyiue, a?aie/iee
aeiaaeie i?ae eeanaie ??aiyiue c no?eeeie ? iano?eeeie ?ica’yceaie,

— O aeiaaeeo iano?eeeo ?ica’yce?a io?eiai? aeinoaoi? oiiae ca?aeiino?
ooieoe?iiae?a ?ioaa?aeueiiai oeio a?ae iano?eeeo ?ica’yce?a noioanoe/ieo
aeeoa?aioe?eieo ??aiyiue i?e ,

— Aeine?aeaeaia aneiioioe/ia iiaaae?iea ?ica’yce?a ??aiyiue ?oi i?e
aeineoue ia?aaoey?i?e caeaaeiino? eiao?oe??io?a a?ae ia?aiao?o.

Oai?aoe/ia oa i?aeoe/ia oe?ii?noue aeena?oaoe??. Iaea?aeai? ?acoeueoaoe
ocaaaeueiththoue oa aeiiiaiththoue ?acoeueoaoe ?ai?oa iaea?aeai?
Eoe?i?/ai A.E., Aiiaea?aaei A.A., Iaoii N.ss., Aeiaciaei I., Aeueoa
Ae.I., I?iaa?aith A.O.

?acoeueoaoe, io?eiai? a aeena?oaoe??, iathoue oai?aoe/ia cia/aiiy ?
iiaeooue aooe canoiniaai? aei aeine?aeaeaiiy aneiioioe/ii? iiaaae?iee
aeeiai?/ieo nenoai c ia?aaoey?iei oa?aeoa?ii cao?aiue, iio?aaa aea/aiiy
oaeeo nenoai aeieea?, cie?aia, o iaoai?oe?, aeiiii?oe?, iaoaiaoe/i?e
a?ieia??.

Iniaenoee aianie caeiaoaa/a. An? ?acoeueoaoe aeena?oaoe?eii? ?iaioe
io?eiai? aeena?oaioii naiino?eii, i?e iino?eiee oaac? oa i?aeo?eioe? c
aieo iaoeiaiai ea??aieea.

Ai?iaaoe?y ?iaioe. Iniiai? ?acoeueoaoe aeena?oaoe?eii? ?iaioe
aeiiia?aeaeeny ia can?aeaii? iaoeiaiai nai?ia?o c oai??? aeiaaeeiaeo
i?ioean?a i?e eaoaae?? oai??? eiia??iinoae oa iaoaiaoe/ii? noaoenoeee
iaoai?ei-iaoaiaoe/iiai oaeoeueoaoo Ee?anueeiai oi?aa?neoaoo ?iai? Oa?ana
Oaa/aiea. (ea??aieee /eai-ei?. AI Oe?a?ie, i?io. ssae?aiei I.E. oa
ae.o.-i.i., i?io. Eica/aiei TH.A.), ia eiioa?aioe?yo: Oinoa
I?aeia?iaeia Iaoeiaa Eiioa?aioe?y ?iai? aeaaeai?ea I.E?aa/oea (Ee?a,
15-17 o?aaiy, 1997?.), I?aeia?iaeia eiioa?aioe?y “Modelling and
investigation of systems stability” (Ee?a, 19-23 o?aaiy, 1997?.), Nueiia
I?aeia?iaeia Iaoeiaa Eiioa?aioe?y ?iai? aeaaeai?ea I.E?aa/oea (Ee?a,
14-16 o?aaiy, 1998?.).

Ioae?eaoe??. Iniiai? ?acoeueoaoe aeena?oaoe?? iioae?eiaaii a ?iaioao
[1]-[7]. O ?iaioao [1-4], iaienaieo o ni?aaaoi?noa? c Eoe?i?/ai A.E.,
iinoaiiaea caaea/ oa iaoeiaa ea??aieoeoai iaeaaeaoue aeieoi?o
o?ceei-iaoaiaoe/ieo iaoe, i?ioani?o Eoe?i?/o A.E., a eiie?aoi?
?acoeueoaoe io?eiai? aeena?oaioii naiino?eii.

Ia’?i oa no?oeoo?a ?iaioe. Aeena?oaoe?eia ?iaioa neeaaea?oueny ?c
anooio, /ioe?ueio ?icae?e?a, caaaeueieo aeniiae?a oa nieneo oeeoiaaii?
e?oa?aoo?e, ui i?noeoue 83 iaeiaioaaiiy. Caaaeueiee ianya ?iaioe 113
noi??iie iaoeiiieniiai oaenoo.

Aaoi? ue?i aeyeo? nai?io iaoeiaiio ea??aieeia?, i?ioani?o Eoe?i?/o
A?eai??th Eiaaeiiae/o ca oe?ii? ii?aaee oa iino?eio oaaao aei ?iaioe.

INIIAIEE CI?NO ?IAIOE

O anooi? iaa?oioiao?oueny aeooaeuei?noue oaie aeena?oaoe??,
oi?ioeth?oueny iaoa aeine?aeaeaiue, ei?ioei aiae?cothoueny iniiai?
i?aoe?, ui a?aeiinyoueny aei oaie aeena?oaoe?? ? iaaiaeyoueny iniiai?
iaea?aeai? ?acoeueoaoe.

A ia?oiio ?icae?e? aeena?oaoe?? aeine?aeaeo?oueny aneiioioe/ia
iiaaae?iea ?ica’yce?a eeano noioanoe/ieo aeeoa?aioe?eieo ??aiyiue ?oi,
yeee ciaoiaeeoueny ia noeeo eean?a ??aiyiue c no?eeeie ? iano?eeeie
?ica’yceaie.

Io?eiai? ?acoeueoaoe ? iiaeaeueoei ?icaeoeii oa iineeaiino? ?acoeueoao?a
oa iaoiae?a aeine?aeaeaiiy aneiioioe/ieo aeanoeainoae noioanoe/ieo
aeeoa?aioe?eieo ??aiyiue aeaieo o ?iaioao A.A.Nei?ioiaea, A.E. Eoe?i?/a,
I.I. Neiaiaeaithea oa ?io.

A i?ae?icae?eao 1.1. oa 1.2. ?icaeyaea?oueny eean noioanoe/ieo
aeeoa?aioe?aeueieo ??aiyiue

(2)

aea a(o)- aei??ia, iaiaaeaia ooieoe?y, w(t) — a?ia??anueeee i?ioean,
aecia/aiee ia eiia??iinoiiio i?inoi?? . Aea/a?oueny
a?aie/ia iiaaae?iea i?e , o aeiaaeeo eiee ia? i?noea
ca?aei?noue

(3)

Iniiai? ?acoeueoaoe i?ae?icae?eo 1.1. iaaiaeyoueny a oai?ai? 1 oa
oai?ai? 2.

Iaoae

C, L- aeia?euei? aeiaeaoi? noae?, ye? ia caeaaeaoue a?ae t, x.

Oai?aia 1. Iaoae — ?ica’ycie ??aiyiiy (2) ? iaoae ia?

i?noea ca?aei?noue

.(4)

Oiae?:

1. sseui 2n0<-1, oi - a?aiaee/iee i?ioean, ? i?e 2.sseui 2n0>-1 ? , oi aeey aeia?eueii? noaei?

.

A oai?ai? 1 ia ?icaeyaea?oueny e?eoe/iee aeiaaeie 2n0, a oeueiio
aeiaaeeo a caeaaeiino? a?ae oa?aeoa?o ca?aeiino? a (4) i?e 2n0 iiaeia
aeae?eeoe ye eean ??aiyiue, aeey yeeo ia? i?noea oaa?aeaeaiiy 1 oai?aie
(no?ee?noue), oae ? eean ??aiyiue, aeey yeeo ia? i?noea oaa?aeaeaiiy 2
oai?aie (iano?ee?noue).

Ia? i?noea ianooiia oai?aia.

Oai?aia 2. Iaoae — ?ica’ycie ??aiyiiy (2) ?
aeeiio?oueny ca?aei?noue (4) i?e 2n01. Oiae?:

1. sseui ?niothoue ooieoe?y , noae? , , oae?
ui

1) i?e , , i ,

2) ,

oi aeey (t) ni?aaaaeeeai oaa?aeaeaiiy 1 oai?aie 1.

2. sseui i i?e ,
oi aeey ni?aaaaeeeai oaa?aeaeaiiy 2 oai?aie 1.

A i?ae?icae?e? 1.2 aea/a?oueny ieoaiiy aneiioioe/ii? iiaaae?iee
iano?eeeo ?ica’yce?a noioanoe/ieo aeeoa?aioe?eieo ??aiyiue. Iniiai?
?acoeueoaoe io?eiai? a i?ae?icae?e? 1.2. iaaaaeai? a oai?ai? 3 oa
oai?ai? 4.

Oai?aia 3. Iaoae — ?ica’ycie ??aiyiiy (2) ? iaoae
,

,(5)

Oiae? :

1. sseui oi aeiaaeeiaee i?ioean
i?e neaaei ca?aa?oueny aei i?ioeano ,
yeee ? ?ica’yceii ??aiyiiy ?oi

(6)

i?e ;

2. sseui , , oi
aeiaaeeiaee i?ioean neaaei ca?aa?oueny i?e aei ?ica’yceo
(6) i?e .

3. sseui , oi aeiaaeeiaee i?ioean
neaaei ca?aa?oueny i?e aei ?ica’yceo
??aiyiiy (6) i?e .

Oai?aia 4. Iaoae aeeiiothoueny i?eiouaiiy oai?aie 3.

Oiae?:

1. sseui , ? ,oi i?ioean neaaei
ca?aa?oueny i?e aei i?ioeano .

2. sseui , oa ?niothoue , ooieoe?y oae?,
ui

, ? i?e ,

oi i?ioean neaaei ca?aa?oueny aei ?ic’yceo
??aiyiiy (6) i?e .

3. sseui , oa ?niothoue
, ooieoe?y oae?, ui

, ? i?e
,

oi i?ioean neaaei ca?aa?oueny aei ?ic’yceo
??aiyiiy (0.6) i?e .

A i?ae?icae?e? 1.3 aeine?aeaeo?oueny aneiioioe/ia iiaaae?iea ?ica’yce?a
noioanoe/ieo aeeoa?aioe?eieo ??aiyiue ?oi ?c ci?iiei eiao?oe??ioii
aeeooc??. Iniiaiee ?acoeueoao aeaiiai i?ae?icae?eo iaaaaeaiee a oai?ai?
5. A oeueiio i?ae?icae?e? aoee aeei?enoai? ?acoeueoaoe 1.1 oa 1.2 aeey
?icaeyaeo a?eueo oe?ieiai eeano ??aiyiue aeaeyaeo

(7)

aea — iaia?a?aia ooieoe?y ? .

Aoa ?icaeyiooee i?ioean , aea
. Ca oi?ioeith ?oi c (7) io?eiaee

,

aea , — ooieoe?y iaa?iaia aei .
Oaeei /eiii ia?ii aiaeia oai?aie 1.

Oai?aia 5. Iaoae — ?ica’ycie ??aiyiiy (7) ? iaoae ia?

i?noea ca?aei?noue

.

Oiae?:

1. sseui , oi — a?aiaee/iee i?ioean, ? i?e

,

aea

2.sseui ? , oi aeey
aeia?eueii? noaei?

.

A ae?oaiio ?icae?e? aeine?aeaeo?oueny aneiioioe/ia iiaaae?iea i?e
ooieoe?iiaea , aea — iano?eeee ?ica’ycie eeano ??aiyiue
(2), ui ?icaeyaeaany a ?icae?e? 1, anoaiiaeththoueny oiiae neaaei?
ca?aeiino? cii?iiaaieo ?ioaa?aeueieo ooieoe?iiae?a aei “iaaieo”
ooieoe?iiae?a a?ae a?aie/ieo i?ioean?a.

A i?ae?icae?e? 2.1 aea/a?oueny aneiioioe/ia iiaaae?iea ooieoe?iiae?a
?ioaa?aeueiiai oeio a?ae aeeooc?eieo i?ioean?a ?c noaeei eiao?oe??ioii
aeeooc??. Iniiaiee ?acoeueoao aeaiiai i?ae?icae?eo iaaaaeaiee a
ianooii?e oai?ai?.

Oai?aia 6. Iaoae — ?ica’ycie ??aiyiiy (2),
,

, aea ,

,

aea — eieaeueii iaiaaeaia oa caaeiaieueiy? iaei?e c oiia

1). — ?aaoey?ii-ci?iia ia iane?i/aiiino? ooieoe?y ii?yaeeo
, oiaoi aeey an?o

.

2). x- iia?eueii-ci?iia ia iane?i/aiiino? iiiioiiia ooieoe?y, oiaoi

.

3). x- iia?eueii-ci?iia ia iane?i/aiiino? iaiaaeaia ooieoe?y , oa
.

Oiae? aaee/eia neaaei ca?aa?oueny i?e
aei , aea — ?ica’ycie ??aiyiiy (6).

I?e aeiaaaeaii? oai?aie aoa aeei?enoaiee ianooiiee aeiiii?aeiee
?acoeueoao, io?eiaiee aaoi?ii, yeee ia? naiino?eia cia/aiiy c oi/ee ci?o
oai??? ?aaoey?ii-ci?iieo ooieoe??.

Eaia 7. Iaoae — eieaeueii iaiaaeaia oa caaeiaieueiy? iaei?e
c oiia 1)-3) oai?aie 6. Oiae? aeey

.

A i?ae?icae?e? 2.2. aeine?aeaeo?oueny aneiioioe/ia iiaaae?iea
ooieoe?iiae?a ?ioaa?aeueiiai oeio a?ae iano?eeeo ?ica’yce?a noioanoe/ieo
aeeoa?aioe?eieo ?c ci?iiei eiao?oe??ioii aeeooc??. Iniiaiee ?acoeueoao
aeaiiai i?ae?icae?eo iaaaaeaii a oai?ai? 8.

Aoaeaii aaaaeaoe, ui , — ooieoe?y iaa?iaia aei .

Oai?aia 8. Iaoae — ?ica’ycie ??aiyiiy (7), —
aaniethoii iaia?a?aia ooieoe?y, , ? iaoae ia? i?noea

, aea ,

oa

,

Oiae? i?ioean neaaei ca?aa?oueny i?e aei i?ioeano
, aea — ?ica’ycie ??aiyiiy (6), —
a?ia?ianueeee i?ioean.

A o?aoueiio ?icae?e? ?icaeyaea?oueny eean noioanoe/ieo aeeoa?aioe?eieo
??aiyiue

(8)

aea — iaia?a?ai? ae?eni? ooieoe??, ui
caaeia?eueiythoue oiia? ?nioaaiiy oa ?aeeiino? ?ica’yceo,
— iaia?a?aiee c eiia??i?noth 1 eaaae?aoe/ii ?ioaa?iaaiee ia?oeiaae,
aecia/aiee ia eiia??iinoiiio i?inoi?? ? aeey yeiai eaaae?aoe/ia
oa?aeoa?enoeea ? iiiioiiii ianiaaeiei iaia?a?aiei c eiia??i?noth 1
aeiaaeeiaei i?ioeanii, — caaeaia ii/aoeiaa oiiaa, ui ia caeaaeeoue
a?ae , .

Iaaeae? aoaeaii aaaaeaoe, ui , eaaae?aoe/ia oa?aeoa?enoeea
i?ioeano ia? aeaeyae — aeiaaeeiaa
ooieoe?y.

A i?ae?icae?e? 3.1 aeine?aeaeo?oueny aneiioioe/ia iiaaae?iea ?ica’yce?a
noioanoe/ieo aeeoa?aioe?eieo ??aiyiue c aeeoa?aioe?aeii ii iaia?a?aieo c
eiia??i?noth iaeeieoey ia?oeiaaeao. Iniiai? ?acoeueoaoe i?ae?icae?eo 3.1
iaaaaeaii a oai?ai? 9 oa oai?ai? 10.

?icaeyiaii i?ioean , aea i?e , i?e
, . Ca oi?ioeith ?oi c ??aiyiiy (8) io?eia?ii

(9)

aea , — ooieoe?y iaa?iaia aei .

Iaaeae? aoaeaii ?icaeyaeaoe ??aiyiiy (9). E??i oiai, aoaeaii
i?eo?eioaaoenue ianooiieo iicia/aiue:

, ,

C, L- aeia?euei? aeiaeaoi? noae?, ye? ia caeaaeaoue a?ae t, x.

Oai?aia 9. Iaoae — ?ica’ycie ??aiyiiy (9), ? iaoae
aa??aoe?y aeiaaeeiai? ooieoe?? oaea,
ui i?e , , ? iaoae ia?
i?noea ca?aei?noue

i?e .

Oiae? aeey aeia?eueii? noaei?

.

Oai?aia 10. Iaoae — ?ica’ycie ??aiyiiy (9),
, oa iathoue i?noea

i). aeey c eiia??i?noth iaeeieoey ?

i?e ca eiia??i?noth;

ii). ,

aea .

Oiae? aeiaaeeiaee i?ioean i?e neaaei ca?aa?oueny aei
i?ioeano , yeee ? ?ica’yceii ??aiyiiy ?oi

.(10)

A i?ae?icae?e? 3.2 aeine?aeaeo?oueny aneiioioeea ooieoe?iiae?a
?ioaa?aeueiiai oeio a?ae iano?eeeo ?ica’yce?a iaeiiaei??ieo noioanoe/ieo
aeeoa?aioe?eieo ??aiyiue c aeeoa?aioe?aeii ii ia?oeiaaeao. Iniiai?
?acoeueoaoe aeaiiai i?ae?icae?eo iaaaaeai? a oai?ai? 11.

Oai?aia 11. Iaoae — ?ica’ycie ??aiyiiy (9).
, oa iathoue i?noea

i). , ui aeey c eiia??i?noth iaeeieoey ?

i?e ca eiia??i?noth,

ii). ,aea ,

ooieoe?y -?ioaa?iaaia, aeey aeia?eueieo noaeeo ,

, i?e ,

aeioiaiae iaoae aeeiiothoueny ca?aeiino?

i?e

,

aea — eieaeueii iaiaaeaia ooieoe?y oa caaeiaieueiy? iaei?e c oiia

1). — ?aaoey?ii-ci?iia ia iane?i/aiiino? ooieoe?y ii?yaeeo ,
oiaoi aeey an?o

.

2). — iia?eueii-ci?iia ia iane?i/aiiino? iiiioiiia ooieoe?y, oiaoi

.

3). — iia?eueii-ci?iia ia iane?i/aiiino? iaiaaeaia ooieoe?y ,
oa .

Oiae? aaee/eia neaaei ca?aa?oueny i?e aei
, aea — ?ica’ycie ??aiyiiy (0.10),
?ia?ianueeee i?ioean.

A /aoaa?oiio ?icae?e? aeine?aeaeo?oueny aneiioioe/ia iiaaae?iea
noioanoe/ieo aeeoa?aioe?eieo ??aiyiue ?oi i?e aeineoue ia?aaoey?i?e
caeaaeiino? eiao?oe??io?a a?ae ia?aiao?o. Cie?aia, a a?aieoe?
eiao?oe??ioe iiaeooue iaoe oi/ee a yeeo aiie ia ? eieaeueii
?ioaa?iaaieie aai iaoe ae?iaeaeaiiy ?ioiai oa?aeoa?o.

Iaoae aeey eiaeiiai ?nio? ?ica’ycie noioanoe/iiai
aeeoa?aioe?eiiai ??aiyiiy

(11)

aea ae?eni? aei??i? ooieoe?? , ui

aeey i?e eiaeiiio — n?i’y a?ia?ianueeeo i?ioean?a, yea
aecia/aia ia eiia??iinoiiio i?inoi?? a?aeiia aeiaaeeiaa aaee/eia
iacaeaaeia a?ae .

O aeaiiio ?icae?e? aeine?aeaeo?oueny a?aie/ia iiaaae?iea i?e
?ica’yce?a ??aiyiiy (11) aeey eeano E2 , oiaoi ??aiyiue aeey yeeo
?niothoue iaai? noae? oae?, ui i?e an?o o ??aiii??ii ii O ia?
i?noea ia??ai?noue

,

aea

,

— aeayea n?i’y noaeeo.

A i?ae?icae?e? 4.1 aeine?aeaeo?oueny neaaea eiiiaeoi?noue n?i’?
?ica’yce?a ??aiyiiy eeano E2. Iniiaiei ?acoeueoaoii oeueiai i?ae?icae?eo
? oai?aia 12.

Iaaeae? i?ae aoaeaii ?icoi?oe aeia?euei? eiinoaioe iacaeaaei?
a?ae .

Oai?aia 12. Iaoae — ?ica’ycie ??aiyiiy (11), ? iaoae —
n?i’y iaia?a?aieo ooieoe?e, aeey yeeo i?e eiaeiiio ?niothoue
eieaeueii ?ioaa?iaai? ocaaaeueiai? iio?aei? aei ae?oaiai ii?yaeeo
aeeth/ii,

sseui ?nio? noaea oaea, ui aeey an?o T, x aeeiiothoueny oiiae

1). ;

2). ,

oi n?i’y i?ioean?a — neaai eiiiaeoia.

A i?ae?icae?e? 4.2 anoaiiaeai? aeinoaoi? oiiae neaaei? ca?aeiino? n?i’?
?ica’yce?a ??aiyiiy eeano E2. Iniiaiei ?acoeueoaoii oeueiai i?ae?icae?eo
? oai?aia 13.

Oai?aia 13. Iaoae — ?ica’ycie ??aiyiiy (11) ?c eeano E2, ? iaoae
?nio? n?i’y ooieoe?e , aeey yei? aeeiiothoueny oiiae oai?aie 12 ? oaea,
ui i?e aea .

E??i oiai iaoae ?niothoue iaia?a?ai? ooieoe?? oae?, ui

1. ;

aeey aeia?eueiiai , aea

,

?

,

2. ??aiyiiy

(12)

ia? ?aeeiee neaaeee ?ica’ycie. Oiae? i?ioean neaaei ca?aa?oueny
i?e aei ?ica’yceo (t) ??aiyiiy (12).

A i?ae?icae?e? 4.3 iaaaaeai? caoaaaeaiiy oa i?eeeaaee aei oai???
aneiioioe/ii? iiaaae?iee ?ica’yce?a iaeiiaei??ieo noioanoe/ieo
aeeoa?aioe?eieo ??aiyiue ?oi i?e aeineoue ia?aaoey?i?e caeaaeiino?
eiao?oe??io?a a?ae ia?aiao?o.

AENIIAEE

O aeena?oaoe?ei?e ?iaio?:

— io?eiaii aeinoaoi? oiiae a?aiaee/iino?, iano?eeino? ?ica’yce?a
noioanoe/ieo aeeoa?aioe?eieo ??aiyiue ?oi,

— aeine?aeaeaia aneiioioe/ia iiaaae?iea ?ica’yce?a ??aiyiue, a?aie/iee
aeiaaeie i?ae eeanaie ??aiyiue c no?eeeie ? iano?eeeie ?ica’yceaie,

— o aeiaaeeo iano?eeeo ?ica’yce?a io?eiaii aeinoaoi? oiiae ca?aeiino?
ooieoe?iiae?a ?ioaa?aeueiiai oeio a?ae iano?eeeo ?ica’yce?a noioanoe/ieo
aeeoa?aioe?eieo ??aiyiue,

— aeine?aeaeaia aneiioioe/ia iiaaae?iea ?ica’yce?a ??aiyiue ?oi i?e
aeineoue ia?aaoey?i?e caeaaeiino? eiao?oe??io?a a?ae ia?aiao?o.

NIENIE IIOAE?EIAAIEO I?AOeUe CA OAIITH AeENA?OAOe??

1. Eoe?i?/ A.E. Eanueeoi ?.I. I?i aneiioioe/io iiaaae?ieo ?ica’yce?a
iaeiiaei??ieo noioanoe/ieo aeeoa?aioe?eieo ??aiyiue ?oi// Oai??y
?iia??iinoae oa iaoaiaoe/ia noaoenoeea.1997, Aei. 56, n.96-104.

2. Eanueeoi ?.I., Eoe?i?/ A.E. I?i a?aie/io iiaaae?ieo ?ica’yce?a
noioanoe/ieo aeeoa?aioe?eieo ??aiyiue// Iaoa??aee eiioa?aioe?? “Oinoa
I?aeia?iaeia Iaoeiaa Eiioa?aioe?y ?iai? aeaaeai?ea I. E?aa/oea” (15-17
o?aaiy 1997 ?.),- Ee?a, n.238.

3. Eanueeoi ?.I., Eoe?i?/ A.E. Oace aeiiia?aeae i?aeia?iaeii?
eiioa?aioe?? “Modelling and investigation of systems stability”. -Ee?a.-
1997.- n. 62.

4. Kulinich G.L., Kaskun E.P. On the asymptotic behavior of solutions of
one-dimensional Ito’s stochastic differential equations with
singularity points// Theory of Stochastic Processes, -1998, vol. 4(20),
?1-2, p. 189-197.

5. Eanueeoi ?.I. I?i aneioioeeo ?ica’yce?a eeano iaeiii??ieo
noioanoe/ieo aeeoa?aioe?eieo ??aiyiue // Iaoa??aee eiioa?aioe?? “Nueiia
I?aeia?iaeia Iaoeiaa Eiioa?aioe?y ?iai? aeaaeai?ea I. E?aa/oea” (14-16
o?aaiy 1998 ?.),- Ee?a, n.198.

6. Eanueeoi ?.I. A?aie/i? oai?aie aeey iaaiiai eeano iaeiiaei??ieo
noioanoe/ieo aeeoa?aioe?eieo ??aiyiue// A?niee Ee?anueeiai oi?aa?neoaoo,
Na??y: o?ceei-iaoaiaoe/i? iaoee. — 1997.- 3, c.54-61.

7. Eanueeoi ?.I. I?i aneiioioeeo ooieoe?iiae?a ?ioaa?aeueiiai oeio a?ae
aeeooc?eieo i?ioean?a// A?niee Ee?anueeiai oi?aa?neoaoo, Na??y:
Iaoaiaoeea, iaoai?ea. — 1998.- 1, c.16-24.

AIIOAOe??

Eanueeoi ?aaai Ieeeiiae/. Aneiioioe/ia iiaaae?iea iano?eeeo ?ica’yce?a
noioanoe/ieo aeeoa?aioe?eieo ??aiyiue -?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa
o?ceei-iaoaiaoe/ieo iaoe ii niaoe?aeueiino? 01.01.05 — oai??y
eiia??iinoae oa iaoaiaoe/ia noaoenoeea. — Ee?anueeee oi?aa?neoao ?iai?
Oa?ana Oaa/aiea, Ee?a, 1999.

?icaeyaea?oueny eean noioanoe/ieo aeeoa?aioe?eieo ??aiyiue

o aeiaaeeo eiee ?niothoue iaai? noae? oae?, ui i?e an?o o
??aiii??ii ii ia? i?noea ia??ai?noue

,

aea

,

— aeayea n?i’y noaeeo.
ae?eni? aei??i? ooieoe??.

Iaoith ?iaioe ? aeine?aeaeaiiy aneiioioe/ieo aeanoeainoae ?ica’yce?a
oeacaiiai eeano noioanoe/ieo aeeoa?aioe?eieo ??aiyiue ?oi oa ??aiyiue c
aeeoa?aioe?aeii ii ia?oeiaaeao. Iaea?aeai? aeinoaoi? oiiae
a?aiaee/iino?, iano?eeino? ?ica’yce?a noioanoe/ieo aeeoa?aioe?eieo
??aiyiue ?oi, O aeiaaeeo iano?eeeo ?ica’yce?a io?eiai? aeinoaoi? oiiae
ca?aeiino? ooieoe?iiae?a ?ioaa?aeueiiai oeio a?ae iano?eeeo ?ica’yce?a
noioanoe/ieo aeeoa?aioe?eieo ??aiyiue. Aeine?aeaeaia aneiioioe/ia
iiaaae?iea ?ica’yce?a ??aiyiue ?oi i?e aeineoue ia?aaoey?i?e caeaaeiino?
eiao?oe??io?a a?ae ia?aiao?o.

Eeth/ia? neiaa : aneiioioe/ia iiaaae?iea ?ica’yce?a, iaeiii??i?
noioanoe/i? aeeoa?aioe?ei? ??aiyiiy, iano?eee ?ica’ycee, noioanoe/i?
aeeoa?aioe?ei? ??aiyiiy ?oi.

Eanueeoi Aaaaiee Oeeeiiiae/. Anneioioe/aneia iiaaaeaiea iaonoie/eauo
?aoaiee noioanoe/aneeo aeeooa?aioeeaeueiuo o?aaiaiee -?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.01.05 — oai?ey
aa?iyoiinoae e iaoaiaoe/aneay noaoenoeea. — Eeaaneee oieaa?neoao eiaie
Oa?ana Oaa/aiea, Eeaa, 1999.

?anniao?eaaaony eeann iaeiiia?iuo noioanoe/aneeo aeeooa?aioeeaeueiuo
o?aaiaiee

a neo/aa, eiaaea nouanoaotho iinoiyiiua oaeea, /oi aeey ethaiai o
?aaiiia?ii ii eiaao ianoi ia?aaainoai

,

aaea

,

— iaeioi?ay naiuey iinoiyiiuo, — aeaenoaeoaeueiua ecia?eiua
ooieoeee ii?aaeaeaiiua i?e anao — aeia?ianeee i?ioeann.

A aeaiiie ?aaioa enneaaeothony aneiioioe/aneea naienoaa i?e ?aoaiee
oeacaiiiai eeanna noioanoe/aneeo aeeooa?aioeeaeueiuo o?aaiaiee.
Oa?aeoa?iie iniaaiiinoueth yoeo o?aaiaiee yaeyaony aeinoaoi/ii
ia?aaoey?iay caaeneiinoue eiyooeoeeaioia io ia?aiao?a , a eiaiii
io ieo ia o?aaoaony ?aaiiia?iay ia?aie/aiiinoue ii ia?aiao?o a eiia/iuo
iaeanoyo ia?aiaiiuo, e ia o?aaoaony ?aaiiia?iay ioaeaeaiiinoue
eiyooeoeeaioia io ioey. E oiio aea, a iaeioi?uo oi/eao
eiyooeoeeaiou iiaoo auoue » «- ia?aciuie
iineaaeiaaoaeueiinoyie, noiaeeoueny a iaeioi?ii niunea e au?aaeaieyi
, ( — oi/ee neiaoey?iinoe), eee eiaoue au?iaeaeaiey eiiai
oa?aeoa?a.

Aeenna?oaoeeiiiay ?aaioa aeeth/aao aaaaeaiea, /aou?a ?acaeaea, auaiaeu
e nienie eeoa?aoo?u ec 83 iaeiaiiaaiee.

Ai aaaaeaiee iainiiauaaaony aeooaeueiinoue oaiu aeenna?oaoeee,
oi?ioee?oaony oeaeue enneaaeiaaiee, ei?ioei aiaeece?othony iniiaiua
?aaiou, /oi ioiinyony e oaia aeenna?oaoeee, i?eaiaeyony iniiaiua
iieo/aiiua ?acoeueoaou.

Iniiaiui iauaeoii ?anniio?aiey a ia?aii ?acaeaea yaeyaony
aneiioioe/aneia iiaaaeaiea ?aoaiee eeanna noioanoe/aneeo
aeeooa?aioeeaeueiuo o?aaiaiee Eoi, eioi?ue iaoiaeeony ia nouea eeannia
o?aaiaiee n onoie/eauie e iaonoie/eauie ?aoaieyie. Enneaaeiaaii
iiaaaeaiea i?e ?aoaiey noioanoe/aneiai aeeooa?aioeeaeueiiai o?aaiaiey n
iinoiyiiui eiyooeoeeaioii aeeooocee e eiyooeoeeaioii niina, /oi
oaeiaeaoai?yao oneiaeth

.

Ai aoi?ii ?acaeaea enneaaeoaony aneiioioe/aneia iiaaaeaiea
ooieoeeiiaeia eioaa?aeueiiai oeia io iaonoie/eauo ?aoaiee noioanoe/aneeo
aeeooa?aioeeaeueiuo o?aaiaiee. Iieo/aiu aeinoaoi/iua oneiaey neaaie
noiaeeiinoe ii?ie?iaaiiuo eioaa?aeueiuo ooieoeeiiaeia e “ii?aaeaeaiiui”
ooieoeeiiaeai io i?aaeaeueiuo i?ioeannia. Iieo/aiu oneiaey ioeaiee
yeno?aioiia ?aaoey?ii iaiythueony ia aaneiia/iinoe ooieoeee.

O?aoee ?acaeae iinayuai enneaaeiaaieyi aneiioioeee ?aoaiee
noioanoe/aneeo aeeooa?aioeeaeueiuo o?aaiaiee n aeeooa?aioeeaeii ii
iai?a?uaiuo n aa?iyoiinoueth aaeeieoea ia?oeiaaeao. Eco/aii
aneiioioe/aneia iiaaaeaiea ooieoeeiiaeia eioaa?aeueiiai oeia io
iaonoie/eauo ?aoaiee iaeiiia?iuo noioanoe/aneeo aeeooa?aioeeaeueiuo
o?aaiaiee n aeeooa?aioeeaeii ii ia?oeiaaeao.

A /aoaa?oii ?acaeaea iniiaiui iauaeoii ?anniio?aiey yaeyaony o?aaiaiey
Eoi i?e ia?aaoey?iie caaeneiinoe eiyooeoeeaioia io ia?aiao?a. Aeieacaia
neaaay eiiiaeoiinoue ?aoaiee o?aaiaiee. Iaeaeaiu oneiaey neaaie
noiaeeiinoe ?aoaiee. Iaaaaeaiu caia/aiey e i?eia?u e oai?ee
aneiioioe/aneiai iiaaaeaiey noioanoe/aneeo aeeooa?aioeeaeueiuo o?aaiaiee
Eoi i?e ia?aaoey?iie caaeneiinoe eiyooeoeeaioia io ia?aiao?a.

Eeth/aaua neiaa: anneiioioeea ?aoaiee, aneiioioe/aneia iiaaaeaiea
iaonoie/eauo ?aoaiee, neaaay noiaeeiinoue ?aoaiee, noioanoe/aneea
aeeooa?aioeeaeueiua o?aaiaiey Eoi.

Kaskun Yevgen Philipovich. Asymptotical behavior of unstable solutions
of the stochastic differential equations. — Manuscript.

Thesis for a candidate’s degree (physical and mathematical sciences) by
speciality 01.01.05 — theory of probabilities and mathematical
statistics. — Kiev Taras Shevchenko University, Kiev, 1999.

The object under investigation is the class of the stochastic
differential equations

with coefficients a , satisfying the inequality

,

uniformly by , here

,

, , are some family of constants.

In this work we investigate the asymptotic properties of solutions of
Ito’s stochastic differential equations, of this type, driven by Wiener
process and, more generally, continious martingales. The sufficient
conditions of an ergodicity, instability of solutions of Ito’s
stochastic differential equations are obtained. In case of unstable
solutions the sufficient conditions of convergence of functionals of an
integral type from unstable solutions of the stochastic differential
equations are obtained. The asymptotical behavior of solutions of Ito’s
equations in the case of the irregular dependence of coefficients on a
parameter is investigated.

Key word: asymptotical behavior of solutions, one-dimensional stochastic
differential equations, unstable solutions, Ito’s stochastic
differential equations.

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