Стійкість стаціонарних режимів системи твердих тіл, які утворюють напівзамкнений ланцюг: Автореф. дис… канд. фіз.-мат. наук / Е.Г. Веласко, НАН Укра

IaoeIiiaeueia aeaaeaiIss iaoe oe?a?ie

Iinoeooo i?eeeaaeii? iaoaiaoeee I iaoaiIee

AAEANEI A??A?A A?aniaea

OAeE 531.36; 531.38

NOIEEINOUe NOAOeIIIA?IEO ?AAEEIIA

NENOAIE OAA?AeEO OIE,

ssEI OOAI?THTHOUe IAIIACAIEIAIEE EAIOeTHA

Niaoeiaeueiinoue: 01.02.01 — oai?aoe/ia iaoaiiea

Aaoi?aoa?ao

aeena?oaoeii ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa oiceei-iaoaiaoe/ieo iaoe

Aeiiaoeuee – 1999

Aeena?oaoei?th ? ?oeiien.

?iaioo aeeiiaii a Iinoeoooi i?eeeaaeiii iaoaiaoeee i iaoaiiee IAI
Oe?aiie, i. Aeiiaoeuee.

Iaoeiaee ea?iaiee : /e.- ei?. IAI Oe?aiie, aeieoi? oic. — iao.
iaoe,

i?ioani? Naa/aiei Ieaenie
sseiae/,

Iiiinoa?noai Oe?aiie o
ni?aaao iaoee i oaoiieiaie,

canooiiee iiiino?a.

Ioioeieii iiiiaioe:

aeieoi? oic.-iao. iaoe, i?ioani? Iethoii Ieaenaiae? Ieaenieiae/,
Oaaai?ianueeee aea?aeaaiee iaaeaaiai/iee iinoeoooo; caa. eaoaae?e
iaoaiaoe/iiai aiaeico;

eaiaeeaeao oic.-iao. iaoe, aeioeaio Eiaaeue Aieoi? iaaiiae/, Aeiiaanueea
aea?aeaaia aeaaeaiiy aoaeiaieoeoaa oa a?oioaeoo?e, aeioeaio eaoaae?e
aeuii iaoaiaoeee oa aeiiiiao?ii.

I?iaiaeia onoaiiaa:

Aeiiaoeueeee aea?aeaaiee oiiaa?neoao, eao?aae?a oai?aoe/iii oa
i?eeeaaeiii iaoaiiee

Caoeno aiaeaoaeaoueny»__24_____» ___09___________ 1999 ?. i
___17____aiaeeii iacaniaeaiii niaoeiaeiciaaiii a/aiii ?aaee Ae 11.193.01
i?e Iinoeoooi i?eeeaaeiii iaoaiaoeee i iaoaiiee IAI Oe?aiie ca
aae?anith: 340114, i. Aeiiaoeuee, aoe. ?. Ethenaiao?a, 74.

C aeena?oaoei?th iiaeia iciaeiieoenue o aiaeiioaoei Iinoeoooo
i?eeeaaeiiiiaoaiaoeee i iaoaiiee IAI Oe?aiie ca aae?anith: 340114, i.
Aeiiaoeuee, aoe. ?. Ethenaiao?a, 74.

Aaoi?aoa?ao ?icineaii «12» 08 1999 ?.

A/aiee nae?aoa?

niaoeiaeiciaaiii a/aiii ?aaee
I. A. Eiaaeaanueeee

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. I?e ?icaeyaeaiii oe?ieiai eeano caaea/ no/aniiai
iaoeiiaoaeoaaiiy aeioeieueii iiaeaethaaoe ia’?eoe, ui aea/athoueny, o
aeaeyaei nenoaie oa?ii?ii ca’ycaieo aaniethoii oaa?aeeo oie. Oae,
aaaaoi i?eeaae?a i i?eno?i?a aa?aoeieii?, ii?nueei? i eini?/ii? oaoiiee
yaeythoue niaith noeoiiinoue oaa?aeeo oie. Aei nenoai ca’ycaieo oaa?aeeo
oie iaeaaeaoue ai?inoaoe, ai?ineiii/ii nenoaie, iaiiioeyoi?e, oiea, yei
iaa?oathoueny ia no?oii aai no?oiiiio iiaeaini, eieinii aeiiaaei oa ?i.
Iiaeaeue nenoaie oaa?aeeo oie c oniioii canoiniao?oueny i aeey iienaiiy
aeeiaiiee eini?/ieo aia?aoia.

Oe?iea i?aeoe/ia aeei?enoaiiy nenoai oaa?aeeo oie aecia/a? ieceo
aeooaeueieo iaoeiaeo iai?yieia, na?aae yeeo aaaeeeaa iinoea caeia?
caaea/a i?i ?oo nenoaie ca’ycaieo oaa?aeeo oie c ia?ooiiith oi/eith.
Iacaaaeath/e ia cia/i? o?oaeiiui, yei aeieeathoue i?e ?? ?ica’ycaiii,
aeeeeeaii aaeeeei ii?yaeeii nenoaie aeeoa?aioeiaeueieo ?iaiyiue, yea
iieno? ?oo iaoaii/iiai ia’?eoo, i aaeeeith eieueeinoth ia?aiao?ia oei??
nenoaie, ? aaeeei aeinyaiaiiy a oeie iaeanoi aiaeioe/ii? iaoaiiee,
iia’ycaii c iiaiaie oaeeo a/aieo, ye I.I.Aiaiyaeainueeee, E.Aiooaiao?a,
I.TH.Ioeiinueeee, A.A.Ai??, Ae.I.Eeiiia, A.I.Eioeyeia, O.Ioaeooa?,
I.ss.Naa/aiei, A.A.Noi?iaeaiei, I.?.Oai/aiei, I.A.Oa?eaiia, I.A.*aoa?a,
A.O?oeai oa ii.

Aeey ni?iuaiiy aiae?co ?iaioe i?eeaae?a oa i?eno?i?a, ui iiaeaeththoueny
ca aeiiiiiaith nenoai oaa?aeeo o?e, aeine?aeieee iaiaaathoueny
iaeneiaeueii aeei?enoiaoaaoe oa?aeoa?i? iniaeeaino? ia’?eo?a, ye?
aea/athoueny, oa ?aaeei?a ?o ?iaioe. O oeueiio ca’yceo na?aae iai?yie?a
?icaeoeo oai??? nenoai oaa?aeeo o?e ne?ae aeae?eeoe oae?, ye iiooe
?aoe?iiaeueieo oi?i ??aiyiue ?ooo ca’ycee oaa?aeeo o?e, iaea?eueo
i?enoiniaaieo aeey aiae?co oe?eaaeo oa?aeoa?enoee iiaeae?, ? iiaoaeiaa
iaei?ino?oeo eean?a ?ica’yce?a oeeo ??aiyiue, yeei a?aeiia?aeathoue
noaoe?iia?ii ?ooe nenoaie o?e, ui ?icaeyaea?oueny. E??i oeueiai,
iaei??th c iaeaaaeeea?oeo aeanoeainoae iiaeae?, ui aea/a?oueny, ?
aeanoea?noue no?eeino? ?? noaoe?iia?ieo ?oo?a, ai, ye i?aaeei, naia ?i
a?aeiia?aeathoue ?iai/? ?aaeeie ?aaeueieo i?eno?i?a. Oeei ?
aecia/a?oueny aeooaeuei?noue ciaoiaeaeaiiy oiia no?eeino? noaoe?iia?ieo
?oo?a nenoaie ca’ycaieo oaa?aeeo o?e.

Aeei?enoaiiy iiaeo iaoa??ae?a, ye? i?eioneathoue cia/i? i?iaeie, i?e
noai?aii? ??cieo i?eeaae?a ? i?eno?i?a no/anii? oaoi?ee i?ecaaei aei
iaiao?aeiino? caaaeaoe ia ?o i?oaei? aeanoeaino?. Iaeei c i?eeiyoieo
niinia?a o?aooaaiiy i?aeaeaoeeaino? oaeeo i?eno?i?a cai?iiiiiaaii o
?iaioao G.F.Reis, W.D.Sundberg, J.E.Cochran, D.E.Christensen, aea
iaa?oioiaaii iiaeeea?noue iiaeaethaaiiy i?oaeieo noa?aeiaaeo
eiino?oeoe?e ca aeiiiiiaith nenoaie oaa?aeeo o?e, ca’ycaieo i?oaeieie
oa?i??aie. Cai?ia iaaeoi neeaaeieo aeey aiae?co iaoai?/ieo nenoai c
?iciiae?eaieie ia?aiao?aie nenoaiaie oaa?aeeo o?e aeicaiey? aeinoaoiuei
iiaii aecia/eoe aaaaoi oa?aeoa?enoee ia’?eo?a, ui aea/athoueny. Oiio ia
ii/aoeiaeo aoaiao aeine?aeaeaiiy aeioe?eueii aea/aiiy aeanoeainoae
iaei?ino?oi? iiaeae?, yea neeaaea?oueny c aeaio ca’ycaieo oaa?aeeo o?e,
ui aeicaiey? a?ae?aco ae aeyaeoe aeaye? oa?aeoa?i? iniaeeaino? ia’?eo?a,
ui iiaeaeththoueny.

I?e aeine?aeaeaii? aeeiai?/ieo aeanoeainoae i?oaeieo noa?aeiaaeo
eiino?oeoe?e (oaeeo ye ea?iaai? ia o?a?eoi??? i?oaei? ?aeaoe, i?oaei?
iaa?oia? aaee c aeaiia iii?aie ia e?ioeyo) i?aaenoaaeythoue iniaeeaee
iioa?an caaea/i, a yeeo ia’?eo ye oe?ea ?ooa?oueny ii caaeaiiio caeiio ?
i?e oeueiio cae?enith? i?oaei? eieeaaiiy. ?icaeyaeaiiy a aeai?e
aeena?oaoe?ei?e ?iaio? ne?i/aiiii??ii? iiaeaei i?oaeiiai aaeo, yea yaey?
niaith nenoaio aeaio aaaeeeo a??ineii?a Eaa?aiaea iai?acaieiaiiai oeio,
? canoinoaaiiy aei ia? aia?aoa aiae?oe/ii? iaoai?ee aeicaieeei aeae?eeoe
eeane noaoe?iia?ieo ?oo?a ai?iaaaeaeaii? iiaeae? ? aea/eoe ieoaiiy i?i
?o no?ee?noue. Oe? aeine?aeaeaiiy ? aeooaeueieie oiio, ui, c iaeiiai
aieo, naia oe? iaei?ino?o? oeie ?oo?a a?aeiia?aeathoue iniiaiei ?iai/ei
?aaeeiai ia’?eoo, yeee aeine?aeaeo?oueny, a c ae?oaiai aieo, anoaiiaeai?
aeeiai?/i? aeanoeaino? iiaeooue iineoaeeoe a yeino? a?aei?aaii? oi/ee
aeey a?eueo iiaiiai c’ynoaaiiy aeanoeainoae aeacaiiai ia’?eoo ia
ianooiieo aoaiao aeine?aeaeaiiy.

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. Aeine?aeaeaiiy
ii aeena?oaoe?ei?e ?iaio? i?iaiaeeeenue o a?aeiia?aeiino? c Ieaiaie
iaoeiaeo aeine?aeaeaiue a?aeae?e?a oaoi?/ii? iaoai?ee ? i?eeeaaeii?
iaoai?ee ?III IAI Oe?a?ie ia 1996—2000 ?iee ii aea?aeathaeaeaoi?e oai?
— 0196U002837 — “Iaoaiaoe/i? iaoiaee eiino?oeoeaiiai aeine?aeaeaiiy
no/anieo i?iaeai no?eeino?, ea?oaaiiy ? aeeiai?ee aca?iiae?th/eo o?e”,
yea aeeiioaaeanue ca?aeii iinoaiia I?aceae?? IAI Oe?a?ie.

Iaoa ? caaea/? aeine?aeaeaiiy. Iaoith aeaii? aeena?oaoe?eii? ?iaioe ?
ciaoiaeaeaiiy aeeiai?/ieo aeanoeainoae nenoaie aeaio oaa?aeeo o?e, ye?
ooai?ththoue iai?acaieiaiee eaioetha.

Aeey aeinyaiaiiy oe??? iaoe ?ica’ycothoueny ianooii? caaea/?:

caeiaoooy ??cieo oi?i ??aiyiue ?ooo aeine?aeaeoaaiiai ia’?eoo;

anoaiiaeaiiy oiia ?nioaaiiy o oaei? nenoaie noaoe?iia?ieo ?oo?a:
??aiii??ieo iaa?oaiue ? ?aaoey?ieo i?aoean?e;

aeine?aeaeaiiy no?eeino? oaeeo noaoe?iia?ieo ?oo?a.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a aecia/a?oueny ianooiieie
iieiaeaiiyie:

Aia?oa io?eiaii ??ci? oi?ie iae?i?eieo ??aiyiue ?ooo nenoaie aeaio
aaaeeeo ai?ineiiia Eaa?aiaea iai?acaieiaiiai oeio o aeiaaeeo, eiee o?ea
c/eaiiaai? i?ae niaith i?oaeiei oi?aa?naeueiei oa?i??ii. Aeacaii
?ioaa?aee oeeo ??aiyiue.

Anoaiiaeaii, ui nenoaia, yea aea/a?oueny, i?eionea? ??aiii??ia iaa?oaiiy
ia’?eoo ye oe?eiai iaaeiei aa?oeeae?. Iaea?aeaii iaiao?aei? ? aeinoaoi?
oiiae no?eeino? oaeiai ?ooo. Aeine?aeaeaii aieea aeaaaeaino ia
no?ee?noue ??aiii??ieo iaa?oaiue. Ciaeaeaii ?aciiainio /anoioo ia?oiai
oeio, yea a?aeiia?aea? aeiaaeeo ca?ao /anoioe e?i?eieo eieeaaiue c
eooiaith oaeaee?noth ??aiii??ieo iaa?oaiue. Aeiaaaeaii iano?ee?noue
aeacaieo ?oo?a a ieie? oe??? /anoioe i?e iiya? aeaaaeaino a nenoai?.

Anoaiiaeaii oiiae ?nioaaiiy o nenoaie, ui aea/a?oueny, ?aaoey?ieo
i?aoean?e ? aeacaii aeinoaoi? oiiae ?o no?eeino?.

Oai?aoe/ia ? i?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoao?a. ?acoeueoaoe
aeena?oaoe?? iathoue oai?aoe/iee oa?aeoa?. Aiie iiaeooue aooe
aeei?enoai? aeey iiaeaeueoeo aeineiaeaeaiue ?icieo caaea/ i?i ?oo nenoai
ca’ycaieo oaa?aeeo oie, a oaeiae ye ia?oa iaaeeaeaiiy i?e aea/aii?
?iai/eo ?aaeei?a i?oaeii? aaeee c aeaiia iii?aie ia e?ioeyo. Ciaeaeaia
?aciiainia /anoioa iiaeiia o?aoiaoaaoenue eiino?oeoi?aie o ca’yceo c
iayai?noth o ?aaeueieo aae?a iaei? ianeiao???.

Iniaenoee aianie caeiaoaa/a a ni?eueieo ioae?eaoe?yo. A ?iaioao [3, 5]
caeiaoaa/o iaeaaeaoue ?acoeueoaoe, ui iia’ycai? c aiae?cii ?ooo nenoaie
a??ineii?a i?e n = 2. A ?iaio? [4] caeiaoaa/ai iaea?aeai? oiiae
?nioaaiiy ?aaoey?ieo i?aoean?e ? caienai? ??aiyiiy ?ooo, ye?
e?iaa??ciaai? a ieie? oeueiai ?aaeeio.

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??. Iniiaii ?acoeueoaoe aeena?oaoei?
aeiiiaiaeaeenue ia:

— I?aeia?iaei?e eiioa?aioe?? “No?ee?noue, ea?oaaiiy ? aeeiai?ea nenoai
oaa?aeeo o?e” (Aeiiaoeuee, 1996);

— I?aeia?iaei?e eiioa?aioe?? “Iaoaiaoeea a ?iaeono???“ (Oaaai?ia, ?in?y,
1998);

— I?aeia?iaei?e eiioa?aioe?? “Fourth International Conference on
Difference Equations and Applications” (Poznan, Poland, 1998);

— iaoeiaeo nai?ia?ao a?aeae?e?a i?eeeaaeii? iaoai?ee oa oaoi?/ii?
iaoai?ee IIII IAI Oe?a?ie (Aeiiaoeuee, iaoeiaee ea??aiee — /eai-ei?. IAI
Oe?a?ie I.A.Oa?eaiia).

Ioae?eaoe??. ?acoeueoaoe aeena?oaoe?eii? ?iaioe iioae?eiaaii o 7
ae?oeiaaieo i?aoeyo, a oiio /ene? o noaooyo [1-4], o i?aoeyo
eiioa?aioe?e [5,6] oa o oacao aeiiia?aeae [7].

No?oeoo?a ? ianya aeena?oaoe??. Aeena?oaoe?eia ?iaioa neeaaea?oueny ?c
anooio, 5 ?icae?e?a, aeniiae?a, nieneo aeei?enoaieo aeaea?ae ?c 117
iaeiaioaaiue, ye? aeeeaaeaii ia 121 noi??ieao, aeeth/a? 7 iaethie?a.

INIIAIEE CI?NO

O anooi? iaa?oioiao?oueny aeooaeuei?noue aeena?oaoe?eii? ?iaioe,
oi?ioeththoueny iaoa ? iniiai? caaea/? aeine?aeaeaiiy, oeaco?oueny
ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie, aeeeaaea?oueny
iaoeiaa iiaecia ? i?aeoe/ia oe?ii?noue ?iaioe, iaaiaeyoueny a?aeiiino?
i?i ai?iaaoe?th ?acoeueoao?a ?iaioe.

A ia?oiio ?icae?e? iiaea?oueny iaeyae e?oa?aoo?e, yea ia? a?aeiioaiiy
aei oaie aeena?oaoe??, ie?aneththoueny iniiai? aoaie ?icaeoeo iaoeiai?
aeoiee c ianooiieo ieoaiue: aea?? ci?iieo, yeee aea? iiaeeea?noue a?eueo
aeeaieiai aea/aiiy aeanoeainoae iaoai?/iiai ia’?eoo, ui ?icaeyaea?oueny;
iiooe ?aoe?iiaeueieo oi?i caieno ??aiyiue ?ooo nenoai ca’ycaieo oaa?aeeo
o?e; ciaoiaeaeaiiy noaoe?iia?ieo ?ica’yce?a oeeo ??aiyiue, ye?
iienothoue ?iai/? ?aaeeie ia’?eo?a, ui iiaeaeththoueny; aeine?aeaeaiiy
no?eeino? io?eiaieo ?ica’yce?a. Oeae iaeyae aeaiiino?o? oe?ieee niaeo?
i?eeeaaeieo caaea/, ye? iiaeooue aooe ?ica’ycai? ca aeiiiiiaith iiaeae?
nenoaie ca’ycaieo oaa?aeeo o?e, iieaco? iaiao?aei?noue ? aeooaeuei?noue
aea/aiiy eiie?aoieo iaoai?/ieo ia’?eo?a, ui iiaeaeththoueny nenoaiaie
oaa?aeeo o?e.

. O aeai?e aeena?oaoe?ei?e ?iaio? aea/a?oueny oaea iaoai?/ia nenoaia o
iaei?ino?oiio aeiaaeeo, eiee n=2.

O ae?oaiio ?icae?e? aeeeaaeaii caaaeueio iaoiaeeeo aeena?oaoe?eieo
aeine?aeaeaiue, yea aaco?oueny ia o?aaeeoe?eieo niiniaao ? iaoiaeao
aiae?oe/ii? iaoai?ee oa oai??? no?eeino? ?ooo. Iaaaaeaia iaoiaeeea
iaea?aeaiiy ??aiyiue ?ooo nenoai ca’ycaieo oaa?aeeo o?e. Ia?a?aoiaai?
iniiai? oaeoe ?c oai??? no?eeino? ?ooo, ui aeicaieythoue ciaoiaeeoe
iaiao?aei? oa aeinoaoi? oiiae no?eeino? noaoe?iia?ieo ?oo?a ca
Eyioiiaei. Aeacaii oai?aie, ye? ni?yiiaai? ia io?eiaiiy aeinoaoi?o oiia
no?eeino? ioeueiaiai ?ica’yceo nenoai c ia??iaee/ieie eiao?oe??ioaie.

A o?aoueiio ?icae?e? iiaeaii iien iaoai?/ii? nenoaie, ui aea/a?oueny,
caienaii ?? aaeoi?i? ??aiyiiy ?ooo, io?eiaii ??ci? oi?ie neaey?ieo
??aiyiue ?ooo oa aeacaii ?o ?ioaa?aee.

.

Aaeoi?i? ??aiyiiy ?ooo oaei? iaoai?/ii? nenoaie ca?aeii c ?acoeueoaoaie
?.I. Aiea?aanueei? oa I.ss. Naa/aiea iathoue aeaeyae:

(1)

,

;

;

;

;

g — i?enei?aiiy neee oyae?iiy;

yeiai ia? iai?yi i?ioeeaaeiee aei iai?yio neee oyae?iiy;

;

;

;

— eiao?oe??io aei?noeino? i?oaeiiai oa?i??o;

;

e?aiea o ??aiyiiyo (1) icia/a? aaniethoio iio?aeio.

, oiaoi

.

ia?iaiaeeeoey?iee aei ieiueie oi?aa?naeueiiai oa?i??o, oiaoi

.

:

.

?c aaeoi?ieo ??aiyiue (1) aia?oa io?eiaii ??ci? oi?ie neaey?ieo ??aiyiue
?ooo iaoai?/ii? nenoaie, ui aeine?aeaeo?oueny, oa aeacaii ?o ia?o?
?ioaa?aee. O yeino? ci?iieo, ui iienothoue iieiaeaiiy a??ineii?a, ooo
ia?ai?:

;

o ?ia?oe?aeueiiio i?inoi??;

o ?ia?oe?aeueiiio i?inoi??.

O aeaio inoaii?o aeiaaeeao aeey caa?aaeaiiy neiao??? ??aiyiue a?aeiinii
o?e, ui aoiaeyoue aei nenoaie, oi?aa?naeueiee oa?i?? ?icaeyiooi ye
noa?e/iee, yeee i?aeii?yaeeiaaiee aieiiiii?e a’yc?, ui iiaea aooe
io?eiaia c oiiae i?oiaiiaeueiino? inae oi?aa?naeueiiai oa?i??o:

.

Aea?? ??cieo ci?iieo aeey iieno iieiaeaiiy a??ineii?a iaoiiaeai
iaiao?aei?noueth iiooeo ??ciiiai?oieo noaoe?iia?ieo ?aaeei?a ?ooo
nenoaie o?e, ui aeine?aeaeo?oueny. Oey caaea/a ae??oo?oueny o ianooiieo
?icae?eao aeena?oaoe??.

A /aoaa?oiio ?icae?e? anoaiiaeaii oiiae ?nioaaiiy ??aiii??iiai iaa?oaiiy
nenoaie, ui aea/a?oueny, ye iaeiiai oe?eiai iaaeiei aa?oeeae?;
iaea?aeaii iaiao?aei? oa aeinoaoi? oiiae no?eeino? oeueiai ?ooo.
?icaeyiooi nenoaio aeaio o?e iai?acaieiaiiai oeio, o yei? iaeia c o?e
iaei a?ae??ciy?oueny a?ae a??ineiia Eaa?aiaea. Aecia/aii e?eoe/io
oaeaee?noue, a ieie? yei? iayai?noue iaei? ianeiao??? i?ecaiaeeoue aei
aeieeiaiiy ?ioa?aaeo iano?eeino?.

A i?ae?icae?e? 4.1 anoaiiaeaii, ui ??aiyiiy ?ooo nenoaie aeaio aaaeeeo
a??ineii?a Eaa?aiaea iai?acaieiaiiai oeio i?eioneathoue ?ica’ycie

(2)

. ??aiyiiy cao?aiiai ?ooo caienaii o oi?i? ??aiyiue Eaa?aiaea ae?oaiai
?iaeo.

) ??aiyiiy

, (3)

,

;

aei eiai oeaio?a ian.

?c aiae?co ei?ai?a ??aiyiiy (3) aeieeaa?, ui:

o aeiaaeeo:

;

a) aei?noe?noue i?oaeiiai oa?i??o caaeiaieueiy? ni?aa?aeiioaiith:

;

iaiao?aei? oiiae no?eeino? ??aiii??ieo iaa?oaiue iathoue aeaeyae:

. (4)

?c (4) aeieeaa?, ui

. (5)

Oea cia/aiiy aoei ii??aiyii ?c e?eoe/iith oaeaee?noth e?eoa??th
Iae?anueeiai no?eeino? ??aiii??ieo iaa?oaiue iaeiiai oaa?aeiai o?ea c
ia?ooiiith oi/eith, yea io?eiaii ?c aeaio, ui ?icaeyaeathoueny, /a?ac
cai?io oa?i??a ia aei?noea cauaieaiiy:

. (6)

, oiaoi iaiao?aei? oiiae no?eeino? ??aiii??iiai iaa?oaiiy iaaeiei
aa?oeeae? nenoaie aeaio aaaeeeo a??ineii?a Eaa?aiaea, ye? ooai?ththoue
iai?acaieiaiee eaioetha, aeeiiothoueny i?e iaioeo oaeaeeinoyo iaa?oaiiy,
i?ae oiiae no?eeino? oeueiai ?ooo aeey iaeiiai a??ineiia Eaa?aiaea c
ia?ooiiith oi/eith.

A i?ae?icae?e? 4.3 ca aeiiiiiaith iaoiaeo *aoa?aa iiaoaeiae ooieoe??
Eyioiiaa o aeaeyae? a’ycee ?ioaa?ae?a ??aiyiue cao?aiiai ?ooo io?eiaii
aeinoaoi? oiiae no?eeino? ?ica’yceo (2):

. (7)

? ia? o oeueiio aeiaaeeo oaeee aeaeyae:

.

, oi

.

, aeinoaoi? oiiae aeeiiothoueny i?e aeia?euei?e oaeaeeino? iaa?oaiiy ?
ca?aathoueny c iaiao?aeieie.

— a??ineii Eaa?aiaea. Ca?aeii c aeai?eoiii, ui cai?iiiiiaaiee ?.I.
Aiea?aanueeith ? I.ss. Naa/aieii, ciaeaeaii e?eoe/io oaeaee?noue, a
ieie? yei? ??aiii??i? iaa?oaiiy nenoaie c aeaaaeainii ? iano?eeeie.
Nenoaia, ui aea/a?oueny, ia? iaeio ?aciiainio oaeaee?noue ia?oiai oeio:

,

aea

.

Aeine?aeaeaiiy, ye? aeeeaaeaii o oeueiio i?ae?icae?e?, aeicaieythoue
ciaeoe iaeanoue iano?eeino? ?aaeeio, ui aea/a?oueny:

.

O i’yoiio ?icae?e? aea/a?oueny ieoaiiy i?i no?ee?noue i?aoean?eieo ?oo?a
nenoaie a??ineii?a, ui ?iaeyaea?oueny. Aeey noaoe?iia?iiai ?ooo, a yeiio
ia?oa o?ei ??aiii??ii iaa?oa?oueny iaaeiei aa?oeeae? (a?nue neiao???
o?ea ia ca?aa?oueny c iath), a ae?oaa o?ei aeeiio? ?aaoey?io i?aoean?th,
io?eiaii oiiae no?eeino? ca Eyioiiaei. I?e oeueiio e?iaa??ciaaia nenoaia
??aiyiue cao?aiiai ?ooo ? ia??iaee/iith ca /anii, ? aeey ia? iaea?aeai?
aeinoaoi? oiiae no?eeino? ioeueiaiai ?ica’yceo.

A i?ae?icae?e? 5.1 ?icaeyiooi ieoaiiy i?i ?nioaaiiy o nenoaie ?aaeeia,
i?e yeiio a??ineiie aeeiiothoue ?aaoey?i? i?aoean?? ? yeiio a?aeiia?aea?
?ica’ycie

. (8)

Anoaiiaeaii, ui ??aiino? (8) caaeiaieueiythoue nenoai? ??aiyiue ?ooo o
aeaio aeiaaeeao:

,

,

i?e oiia? aeeiiaiiy ??aiino?

,

iaaeiei aa?oeeae?, a a ae?oaiio aeiaaeeo iaaiaee: ia?oa o?ei aeeiio?
i?aoean?th, a ae?oaa — ??aiii??ia iaa?oaiiy. Ine?eueee iaeaeaa ?ooe
iaeiioeii?, ieoaiiy no?eeino? ?icaeyaea?oueny aeae? eeoa aeey ia?oiai
aeiaaeeo, yeiio a?aeiia?aea? ?ica’ycie

.

, oi nenoaia i?eionea? eeoa ??aiii??ia iaa?oaiiy eiaeiiai c o?e iaaeiei
aa?oeeae? (in? o?e ia ca?aathoueny c iath). ?ica’ycie, ui a?aeiia?aea?
oaeiio ?ooia?, ia? aeaeyae

? ?nio? i?e oiia?

.

A i?ae?icae?e? 5.2 caienaii e?iaa??ciaai? ??aiyiiy cao?aiiai ?ooo.
Iieeaaeath/e a iae?i?eieo ??aiyiiyo ?ooo nenoaie o?e, ui aea/a?oueny,

,

iaoeiaii ??aiyiiy a aa??aoe?yo:

(9)

? aeeiioth/e cai?io

i?eoiaeeii aei ??aiyiiy ae?oaiai ii?yaeeo a?aeiinii x, yea a naith /a?ao
caiaeeii aei ??aiyiiy O?eea

(10)

aea

ia? aeaeyae:

.

aecia/athoueny ca oi?ioeaie:

, oi no?ee?noue aai iano?ee?noue ioeueiaiai ?ica’yceo ??aiyiiy (10)
icia/a? a?aeiia?aeii no?ee?noue aai iano?ee?noue nenoaie (9).

A i?ae?icae?e? 5.3 aeine?aeaeo?oueny ieoaiiy i?i no?ee?noue ioeueiaiai
?ica’yceo ??aiyiiy (10). Ca aeiiiiiaith a?aeiieo e?eoa???a no?eeino?
aeey ??aiyiiy O?eea io?eiaii aeinoaoi? oiiae no?eeino?, ye? iiaeaii o
aeaeyae? ia’?aeiaiiy /ioe?ueio a?oi ia??aiinoae:

(11)

(12)

(13)

(14)

Ooo

, yea aei??aith?

;

n — aeia?eueia iaoo?aeueia /enei, yea aecia/a? iiia? iaeano? no?eeino?
aeey oiia (14), ? ia iaeeieoeth a?eueoa ca oeae iiia? aeey oiia (11) —
(13).

Ne?ae aeiaeaoe, ui ia??aiino? (14) ia aeeth/athoue aei naaa iieno
ioeueiai? iaeano? no?eeino?, ioe?iea aeey yei? caieno?oueny ie?aiith
oi?ioeith

. (15)

. Iaeano? no?eeino? iiaoaeiaaii ia iaethieao. Oae, iaeano?, ye?
aecia/athoueny oiiaaie (14), ? e?eaie?i?eieie o?eeooieeaie, ui iaiaaeai?
a?eeaie ia?aaie

.

Caoaaaeeii, ui aeey ioeueiai? iaeano?, yea aecia/a?oueny ia??aiinoyie
(15), ia?ii nioao i?ae ia?aaieaie

ia a?aei?io, iai?eeeaae, a?ae a?aeiiiai e?eoa??th Eyioiiaa

yeee a aeaiiio aeiaaeeo aea? ionoo iiiaeeio.

AENIIAEE

A aeena?oaoe?ei?e ?iaio? ca aeiiiiiaith o?aaeeoe?eieo niinia?a ?
iaoiae?a aiae?oe/ii? iaoai?ee oa oai??? no?eeino? ?ooo aea/aii
aeeiai?/i? aeanoeaino? nenoaie aeaio aaaeeeo a??ineii?a Eaa?aiaea, ye?
ooai?ththoue iai?acaieiaiee eaioetha. Io?eiai? ?acoeueoaoe iiaeooue aooe
aeei?enoai? aeey iiaeaeueoeo aeineiaeaeaiue caaea/ i?i ?oo nenoai
ca’ycaieo oaa?aeeo oie iai?acaieiaiiai oeio, a oaeiae ye ia?oa
iaaeeaeaiiy i?e aea/aii? ?iai/eo ?aaeei?a i?oaeii? aaeee c aeaiia
iii?aie ia e?ioeyo.

A i?ioean? aeine?aeaeaiiy iaea?aeaii ianooii? iniiai? ?acoeueoaoe:

iaeiaeia?. Aeacaii ?ioaa?aee oeeo ??aiyiue.

Anoaiiaeaii, ui nenoaia, yea aea/a?oueny, i?eionea? ??aiii??ia iaa?oaiiy
ia’?eoo ye oe?eiai iaaeiei aa?oeeae?. Iaea?aeaii iaiao?aei? ? aeinoaoi?
oiiae no?eeino? oaeiai ?ooo. ?o aiae?c iieacaa, ui: 1) iaiao?aei? oiiae
no?eeino? ??aiii??iiai iaa?oaiiy iaaeiei aa?oeeae? nenoaie aeaio aaaeeeo
a??ineii?a Eaa?aiaea, ye? ooai?ththoue iai?acaieiaiee eaioetha,
aeeiiothoueny i?e iaioeo oaeaeeinoyo iaa?oaiiy, i?ae oiiae no?eeino?
oeueiai ?ooo aeey iaeiiai a??ineiia Eaa?aiaea c ia?ooiiith oi/eith; 2)
aeey niethuaieo o??aiiaaaeaieo o?e, a oaeiae o aeiaaeeo aaeeei?
aei?noeino? aeinoaoi? oiiae aeeiiothoueny i?e aeia?euei?e oaeaeeino?
iaa?oaiiy ? ca?aathoueny c iaiao?aeieie.

Aeine?aeaeaii aieea aeaaaeaino ia no?ee?noue ??aiii??ieo iaa?oaiue.
Ciaeaeaii ?aciiainio /anoioo ia?oiai oeio, yea a?aeiia?aea? aeiaaeeo
ca?ao /anoioe e?i?eieo eieeaaiue c eooiaith oaeaee?noth ??aiii??ieo
iaa?oaiue. Aeiaaaeaii iano?ee?noue aeacaieo ?oo?a a ieie? oe??? /anoioe
i?e iiya? aeaaaeaino a nenoai?.

iaaeiei aa?oeeae?; a o ae?oaiio aeiaaeeo iaaiaee: ia?oa o?ei aeeiio?
i?aoean?th, a ae?oaa — ??aiii??ia iaa?oaiiy.

Aeey noaoe?iia?iiai ?ooo, a yeiio ia?oa o?ei ??aiii??ii iaa?oa?oueny
iaaeiei aa?oeeae? (a?nue neiao??? o?ea ia ca?aa?oueny c iath), a ae?oaa
o?ei aeeiio? ?aaoey?io i?aoean?th, io?eiaii oiiae no?eeino? ca
Eyioiiaei. I?e oeueiio e?iaa??ciaaia nenoaia ??aiyiue cao?aiiai ?ooo ?
ia??iaee/iith ca /anii, ? aeey ia? iaea?aeai? aeinoaoi? oiiae no?eeino?
ioeueiaiai ?ica’yceo. Oe? oiiae i?iaiae?ciaaii o aeiaaeeo, eiee iaeaeaa
a??ineiie ? neeueii nooeaieie iaeii??aeieie aeeneaie ? iathoue iaeiaeia?
inueia? iiiaioe ?ia?oe??.

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

Aaeanei Y??a?a A. Onoie/eainoue ?aaiiia?iuo a?auaiee nenoaiu aeaoo
ae?ineiiia Eaa?aiaea, ia?acothueo iieocaieioooth oeaiue // Iaoaieea
oaa?aeiai oaea. — 1997. — Aui. 29. — N. 50-54.

Aaeanei Y??a?a A. O/ao iaeie ianeiiao?ee a nenoaia aeaoo oae,
ia?acothuae iieocaieioooth oeaiue // Iaoaieea oaa?aeiai oaea. — 1998. —
Aui. 26(II). — N. 144-147.

Aiea?aaneay E.A., Aaeanei Y??a?a A. ?aaoey?iay i?aoeanney nenoaiu
ae?ineiiia Eaa?aiaea, ia?acothueo iieocaieioooth oeaiue // O?oaeu
Einoeoooa i?eee. iaoaiaoeee e iaoaieee, o.2. — Iiaeaee?iaaiea
iai?a?uaiuo e aeene?aoiuo nenoai. — Aeiiaoee. — 1998. — N. 3-9.

Aaeanei Y??a?a A., Iocu?aa A.A. Ia onoie/eainoe aeaeaeaiey nenoaiu aeaoo
ae?ineiiia Eaa?aiaea, ia?acothueo iieocaieioooth oeaiue // O?oaeu
Einoeoooa i?eee. iaoaiaoeee e iaoaieee, o.2. — Iiaeaee?iaaiea
iai?a?uaiuo e aeene?aoiuo nenoai. — Aeiiaoee. — 1998. — N. 17-21.

Bolgrabskaya I.A. and Velasco Herrera G. Investigation of the solutions
of the difference equations system which describe small oscillation of
elastic rods // Fourth International Conference on difference equations
and applications. — Poznan (Poland). — 1998. — P. 57-60.

Aaeanei Y??a?a A. Noaoeeiia?iua ?aaeeiu nenoaiu aeaoo ae?ineiiia
Eaa?aiaea, ia?acothueo iieocaieioooth oeaiue // O?oaeu Iaaeaeoia?iaeiie
eiioa?aioeee “Iaoaiaoeea a eiaeono?ee”. — Oaaai?ia: Ecae-ai Oaaai?.
ain. iaae. ei-oa. — 1998. — C. 72-74.

Aaeanei Y??a?a A. Enneaaeiaaiea onoie/eainoe noaoeeiia?iuo aeaeaeaiee
iaeiie iieocaieiooie nenoaiu naycaiiuo oaa?aeuo oae // Oacenu aeieeaaeia
Iaaeaeoia?iaeiie eiioa?aioeee “Onoie/eainoue, oi?aaeaiea e aeeiaieea
oaa?aeiai oaea”. — Aeiiaoee. — 1996. — N. 111-112.

AIIOAOe?ss

Aaeanei Y??a?a A. No?ee?noue noaoe?iia?ieo ?aaeei?a nenoaie oaa?aeeo
o?e, ye? ooai?ththoue iai?acaieiaiee eaioetha. — ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa
o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.02.01 — oai?aoe/ia
iaoai?ea. — ?inoeooo i?eeeaaeii? iaoaiaoeee ? iaoai?ee IAI Oe?a?ie,
Aeiiaoeuee, 1999.

Aeena?oaoe?th i?enay/aii aeine?aeaeaiith aeeiai?/ieo aeanoeainoae
nenoaie aeaio aaaeeeo ai?ineiiia Eaa?aiaea iai?acaieiaiiai oeio o
aeiaaeeo, eiee o?ea c/eaiiaai? i?ae niaith i?oaeiei oi?aa?naeueiei
oa?i??ii. Io?eiaii ??ci? oi?ie iae?i?eieo ??aiyiue ?ooo oaei? nenoaie.
Iieacaii, ui aiia i?eionea? ??aiii??ia iaa?oaiiy ia’?eoo ye oe?eiai
iaaeiei aa?oeeae?. Iaea?aeaii iaiao?aei? ? aeinoaoi? oiiae no?eeino?
oaeiai ?ooo. Aeine?aeaeaii aieea aeaaaeaino ia no?ee?noue ??aiii??ieo
iaa?oaiue. Anoaiiaeaii oiiae ?nioaaiiy o nenoaie, ui aea/a?oueny,
?aaoey?ieo i?aoean?e ? aeacaii aeinoaoi? oiiae ?o no?eeino?.

Eeth/ia? neiaa: nenoaia a??ineii?a Eaa?aiaea iai?acaieiaiiai oeio,
i?oaeiee oi?aa?naeueiee oa?i??, ??aiii??ia iaa?oaiiy, no?ee?noue,
?aaoey?ia i?aoean?y.

Aaeanei Y??a?a A. Onoie/eainoue noaoeeiia?iuo ?aaeeiia nenoaiu oaa?aeuo
oae, ia?acothueo iieocaieioooth oeaiue. — ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.02.01 — oai?aoe/aneay
iaoaieea. — Einoeooo i?eeeaaeiie iaoaiaoeee e iaoaieee IAI Oe?aeiu,
Aeiiaoee, 1999.

Aeenna?oaoeey iinayuaia enneaaeiaaieth aeeiaie/aneeo naienoa nenoaiu
aeaoo oyaeaeuo ae?ineiiia Eaa?aiaea iieocaieiooiai oeia, a neo/aa,
eiaaea oaea niaaeeiaiu iaaeaeo niaie oi?oaei oieaa?naeueiui oa?ie?ii.
Aeaiiay nenoaia iiaeao iineoaeeoue a ea/anoaa ia?aiai i?eaeeaeaiey i?e
eco/aiee aeeiaie/aneeo naienoa oi?oaie aaeee n aeaoiy iii?aie ia
eiioeao.

Iieo/aiu ?acee/iua oi?iu iaeeiaeiuo o?aaiaiee aeaeaeaiey oaeie nenoaiu e
oeacaiu eo ia?aua eioaa?aeu.

Onoaiiaeaii, /oi eco/aaiay nenoaia aeiioneaao ?aaiiia?iia a?auaiea
iauaeoa eae oeaeiai aie?oa aa?oeeaee.

Iieo/aiu iaiaoiaeeiua e aeinoaoi/iua oneiaey onoie/eainoe oaeiai
aeaeaeaiey.

Aeieacaii, /oi: iaiaoiaeeiua oneiaey onoie/eainoe ?aaiiia?iiai a?auaiey
aie?oa aa?oeeaee nenoaiu aeaoo oyaeaeuo ae?ineiiia Eaa?aiaea,
ia?acothueo iieocaieioooth oeaiue, auiieiythony i?e iaiueoeo nei?inoyo
a?auaiey, /ai oneiaey onoie/eainoe oaeiai aeaeaeaiey aeey iaeiiai
ae?ineiia Eaa?aiaea n iaiiaeaeaeiie oi/eie; aeey niethnioouo
o?aaiiaaoaiiuo oae, a oaeaea a neo/aa aieueoie aeanoeinoe a oa?ie?a,
aeinoaoi/iua oneiaey auiieiythony i?e i?iecaieueiie nei?inoe a?auaiey e
niaiaaeatho n iaiaoiaeeiuie.

Enneaaeiaaii aeeyiea aeaaaeaina a nenoaia ia onoie/eainoue ?aaiiia?iuo
a?auaiee. Iaeaeaia ?aciiainiay /anoioa ia?aiai oeia, eioi?ay
niioaaonoaoao neo/ath niaiaaeaiey /anoiou eeiaeiuo eieaaaiee n oaeiaie
nei?inoueth ?aaiiia?iuo a?auaiee. Aeieacaia iaonoie/eainoue oeacaiiuo
aeaeaeaiee a ie?anoiinoe yoie /anoiou i?e iiyaeaiee aeaaaeaina a
nenoaia.

Onoaiiaeaii, /oi ?aaoey?iua i?aoeannee nenoaiu aeaoo ae?ineiiia
Eaa?aiaea iieocaieiooiai oeia iiaoo nouanoaiaaoue eeoue a aeaoo neo/ayo:
a ia?aii neo/aa aoi?ia oaei niaa?oaao ?aaoey?ioth i?aoeanneth, a ia?aia
— ?aaiiia?ii a?auaaony aie?oa aa?oeeaee; a ai aoi?ii neo/aa, iaiai?io, —
ia?aia oaei niaa?oaao ?aaoey?ioth i?aoeanneth, a aoi?ia — ?aaiiia?iia
a?auaiea.

Aeey noaoeeiia?iiai aeaeaeaiey, i?e eioi?ii ia?aia oaei ?aaiiia?ii
a?auaaony aie?oa aa?oeeaee (inue neiiao?ee oaea ia niaiaaeaao n iath), a
aoi?ia oaei niaa?oaao ?aaoey?ioth i?aoeanneth, iieo/aiu oneiaey
onoie/eainoe ii Eyioiiao. I?e yoii eeiaa?eciaaiiay nenoaia o?aaiaiee
aiciouaiiiai aeaeaeaiey eiaao eiyooeoeeaiou, niaea?aeauea ia?eiaee/aneea
ooieoeee, caaenyuea io a?aiaie. Aaaaeaia niaoeeaeueiay caiaia
ia?aiaiiuo, iicaieythuay naanoe eco/aiea onoie/eainoe o?eaeaeueiiai
?aoaiey yoie nenoaiu e enneaaeiaaieth onoie/eainoe ioeaaiai ?aoaiey
o?aaiaiey Oeeea n ecaanoiuie eiyooeoeeaioaie. Aeey iaai iieo/aiu
aeinoaoi/iua oneiaey onoie/eainoe ioeaaiai ?aoaiey. N iiiiuueth
/eneaiii-aiaeeoe/aneiai iaoiaea iino?iaiu iaeanoe onoie/eainoe ia
ieineinoe ia?aiao?ia, oa?aeoa?ecothueo niioiioaiey iaaeaeo oaeiauie
nei?inoyie i?aoeannee, ?aaiiia?iiai a?auaiey e eiyooeoeeaioii aeanoeinoe
oa?ie?a. Yoe oneiaey i?iaiaeece?iaaiu a neo/aa, eiaaea iaa ae?ineiia
yaeythony neeueii niethnioouie iaeii?iaeiuie aeeneaie e eiatho
iaeeiaeiaua inaaua iiiaiou eia?oeee.

Eeth/aaua neiaa: nenoaia ae?ineiiia Eaa?aiaea iieocaieiooiai oeia,
oi?oaee oieaa?naeueiue oa?ie?, ?aaiiia?iia a?auaiea, onoie/eainoue,
?aaoey?iay i?aoeanney.

Velasco Herrera G. Stability of stationary conditions of a system of
rigid bodies formed in half-closed chain. — Manuscript.

Thesis for a degree of Candidate of Sciences (Physics and Mathematics)
by speciality 01.02.01 — theoretical mechanics. — Institute of Applied
Mathematics and Mechanics of National Academy of Sciences of Ukraine,
Donetsk, 1999.

The dissertation is devoted to investigation of dynamical properties of
a system of two heavy Lagrange’s gyroscopes formed in half-closed chain
in the case when the bodies are coupled by elastic universal hinge.
Different forms of nonlinear equations of motion of this system are
found. It is shown that the system allows the permanent rotation of
object around vertical. Necessary and sufficient conditions of stability
of this motion are found. The influence of asymmetry on stability of
permanent rotations is investigated. Conditions of existence of regular
precession of the system are received. Sufficient conditions of
stability of their motions are obtained.

Key words: system of Lagrange’s gyroscopes formed in half-closed chain,
elastic universal hinge, permanent rotation, stability, regular
precession.

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