IAOe?IIAEUeIA AEAAeAI?ss IAOE OE?A?IE

?INOEOOO IAOAIAOEEE

IIIIAE* Ieaenaiae? Aaneeueiae/

OAeE 517.9

ANEIIOIOE*IA IIAAAe?IEA OA A?OO?EAOe??

?ICA’ssCE?A EAIOeTHA?A CA’ssCAIEO INOeEEssOI??A

01.01.02 — aeeoa?aioe?aeuei? ??aiyiiy

A a o i ? a o a ? a o

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa o?ceei-iaoaiaoe/ieo iaoe

Ee?a — 1999

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia o a?aeae?e? cae/aeieo aeeoa?aioe?aeueieo ??aiyiue

?inoeoooo iaoaiaoeee IAI Oe?a?ie

Iaoeiaee ea??aiee: aeaaeai?e IAI Oe?a?ie,

aeieoi? oiceei-iaoaiaoe/ieo iaoe, i?ioani?

NAIIEEAIEI Aiaoie?e Ieoaeeiae/

aee?aeoi? ?inoeoooo iaoaiaoeee IAI Oe?a?ie

Ioioeieii iiiiaioe:

aeieoi? oiceei-iaoaiaoe/ieo iaoe

NAIIEEAIEI Aaea??e A?eai?iae/

Ee?anueeee oi?aa?neoao ?i.

Oa?ana Oaa/aiea,

caa?aeoth/ee eaoaae?ith iaoaiaoe/ii? o?ceee

iaoai?ei-iaoaiaoe/iiai oaeoeueoaoo

eaiaeeaeao oiceei-iaoaiaoe/ieo iaoe, aeioeaio

I?AOeUeIAEOEE Ieeiea A?eoi?iae/

Iaoe?iiaeueiee iaaeaaia?/iee oi?aa?neoao ?i. I.I. Ae?aaiiaiiaa,

caa?aeoth/ee eaoaae?ith aeui? iaoaiaoeee

I?iaiaeia onoaiiaa:

Iaeanueeee aea?aeaaiee oi?aa?neoao ?iai? ?.?. Ia/ieeiaa,

eaoaae?a iioeiaeueiiai oi?aae?iiy oa aeiiii?/ii? e?aa?iaoeee,

i. Iaeana

Caoeno aiaeaoaeaoueny 16 ethoiai 1999 ?ieo i 15 aiaeeii ia caniaeaiii
niaoeiaeiciaaii? ?aaee Ae. 26.206.02 i?e Iinoeoooi iaoaiaoeee IAI
Oe?a?ie

ca aae?anith: 252601 Ee?a — 4, aoe. Oa?auaieianueea, 3.

C aeena?oaoei?th iiaeia iciaeiieoenue o aiaeiioaoei ?inoeoooo.

Aaoi?aoa?ao ?icineaiee 14 n?/iy 1999 ?.

A/aiee nae?aoa?

niaoeiaeiciaaii? ?aaee EO*EA A.TH.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeueiinoue oaie. ?iaioa i?enay/aia aeine?aeaeaiith aneiioioe/ii?
iiaaae?iee oa a?oo?eaoe?e ?ica’yce?a eaioetha?a ca’ycaieo inoeeeyoi??a
(ECI), ui yaeythoue niaith nenoaie ne?i/aii? aai ce?/aii? e?eueeino?
ca’ycaieo iaaiei /eiii aeeiai?/ieo nenoai iaioi? ?ici??iino? — oae
caaieo aaciaeo inoeeeyoi??a. ?icaeyaea?oueny aeiaaeie, eiee aaciaei
inoeeeyoi?ii ? iaeiiaei??ia iae?i?eia a?aeia?aaeaiiy a?ae??cea i?yii? a
naaa c ?aaoey?iith aai oaioe/iith aeeiai?eith. Iniiaia oaaaa
i?eae?ey?oueny aeine?aeaeaiith ?nioaaiiy oa no?eeino? ia??iaee/ieo
?ica’yce?a oaeiai ?iaeo nenoai o aeiaaeeo neaaeiai ca’yceo, a oaeiae
aea/aiith no?eeino? ?aaeeio iiaii? oa /anoeiai? oaioe/ii? neio?ii?caoe??
(oiaoi neio?ii?caoe?? o aeiaaeeo, eiee aeeiai?ea aaciaiai inoeeeyoi?a ?
oaioe/iith).

Eaioethae ca’ycaieo inoeeeyoi??a oe?iei aeei?enoiaothoueny i?e
iiaeaethaaii? neeaaeieo iae?i?eieo i?ioean?a a ??cieo aaeocyo iaoee ?
oaoi?ee. ?o aeeiai?ea ? iaaecae/aeii aaaaoith ? neeaaeiith, ?acii ?c
oei, ?i i?eoaiaii? aeanoeaino? ie?aieo aeaiaio?a eaioethaa, ye?, a naith
/a?ao, ? iaaaaaoi i?ino?oeie aeeiai?/ieie nenoaiaie. Oea aea? ciiao a
aeayeeo aeiaaeeao aeine?aeeoe aneiioioe/io iiaaae?ieo ?ica’yce?a oaeeo
nenoai, aeei?enoiaoth/e ?ioi?iaoe?th i?i nenoaie iaioi? ?ici??iino?.
Eaioethae ca’ycaieo inoeeeyoi??a oe?eaaeyoue iaoaiaoee?a, a oaeiae
aeine?aeiee?a ?c ??cieo aaeocae o?ceee, a?ieia??, o?i??, aeiiii?ee oa
?i., a ia?oo /a?ao, i?e aea/aii? i?iaeaie ooai?aiiy oii?yaeeiaaieo
no?oeoo?, aeieeiaiiy oa aaiethoe?? i?inoi?iai-/aniaiai oaino, yaeua
neio?ii?caoe?? nenoai iaioi? ?ici??iino? (ye ia??iaee/ii? oae ?
oaioe/ii?). Ii/eiath/e ua c ?ia?o A.A. Aiae?iiiaa oa A.A. A?ooa, aaeeea
oaaaa aeine?aeaeaiith oeeo ieoaiue i?eae?ey?oueny aaaaoueia
iaoaiaoeeaie, a?aei?oeii o oe?e iaeano? i?aoe? A.N. Ao?aeiiae/a,
E.A. Aoi?iiae/a, I.I. A?eeo, ss.A. N?iay, E. A?aaiae?, I. Aoa?ia,
Aeae.A. Ei?ea, A. E?ooaiaa?aa, ?. Notha?oa, A.?. Oaioa oa ?i.

Aeine?aeaeaiiy no?eeino? ?aaeeio oaioe/ii? neio?ii?caoe?? a eaioethaao
ca’ycaieo inoeeeyoi??a iaaoei ao?oeeaiai ?icaeoeo ca inoaii? 5 — 10
?ie?a. A ia?oo /a?ao, oea iia’ycaii ?c canoinoaaiiyi oeueiai yaeua

aeey ?ic?iaee iia?oi?o oaoiieia?e o ?aae?i?iaeaia??? c aeei?enoaiiyi
oaioe/ii? iano/i? aeey eiaeoaaiiy neaiaeo (E. Iaei?a, O. Ea??iee,
I. Oanea? oa ?i.), a oaeiae aiii a?aee?eaa? iia? ia?niaeoeae
aeine?aeaeaiue a a?ieia?? oa iaaeeoeei? (E. Eaiaei, A. Iicae?eaea oa
?i.). ?c ?ioiai aieo oea oe?eaaa ? ia?niaeoeaia iaoaiaoe/ia caaea/a, yea
i?eaa?oa? oaaao aaaaoueio a/aieo ?c ??cieo e?a?i na?oo ? aeoeaii
iaaiai?th?oueny ia i?aeia?iaeieo iaoeiaeo eiioa?aioe?yo.

Iacaaaeath/e ia aaeeeo e?euee?noue ?ia?o ii aeai?e oaiaoeoe?, aaaaoi
ieoaiue caeeoa?oueny iac’yniaaieie. Cie?aia oea noino?oueny i?iaeaie
no?eeino? neio?ii?coth/iai aoo?aeoi?a oa eiai a?oo?eaoe?e i?e iiai?e
oaioe/i?e neio?ii?caoe?? (I. Aoa?i, A. Ioo oa ?i.). Ia nueiaiaei?oi?e
aeaiue i?aeoe/ii ia ?nio? iaoaiaoe/ii? oai??? yaeua /anoeiai? oaioe/ii?
neio?ii?caoe??. Inue /iio iaoaiaoe/i? aeine?aeaeaiiy oeeo ieoaiue ?
aeooaeueieie oa ia?niaeoeaieie.

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. ?iaioa
i?iaiaeeeanue ca?aeii c caaaeueiei ieaiii aeine?aeaeaiue a?aeae?eo
cae/aeieo aeeoa?aioe?aeueieo ??aiyiue ?inoeoooo iaoaiaoeee IAI Oe?a?ie.

Iaoa ? caaea/? aeine?aeaeaiiy. Iaoith aeaii? ?iaioe ? aeine?aeaeaiiy
aneiioioe/ii? iiaaae?iee ?ica’yce?a eaioetha?a ca’ycaieo iae?i?eieo
a?aeia?aaeaiue, a naia: aeieeiaiiy oa caa?aaeaiiy no?eeeo ia??iaee/ieo
i?inoi?iai-/aniaeo no?oeoo?, no?ee?noue ?aaeeio iiaii? oa /anoeiai?
oaioe/ii? neio?ii?caoe??, o?ainaa?naeueieo a?oo?eaoe?e oeeee?a, ye?
iaeaaeaoue oaioe/iiio neio?ii?coth/iio aoo?aeoi?o ? i?eaiaeyoue aei
ao?aoe iei aneiioioe/ii? no?eeino?.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a. Iniiaieie ?acoeueoaoaie, ye?
aecia/athoue iaoeiao iiaecio oa aeiinyoueny ia caoeno ? oae?:

Aeiaaaeaii no?ee?noue ia??iaee/ieo ?ica’yce?a eaioetha?a ca’ycaieo
inoeeeyoi??a o aeiaaeeo neaaeiai ca’yceo. Iieacaii, ui ?aaoey?i?
i?inoi?iai-/ania? no?oeoo?e caa??aathoueny a nenoai? ? ci?iththoueny
iaia?a?aii i?e iaeeo cao?aiiyo ii/aoeiaeo aeaieo oa ia?aiao?a ca’yceo.

Cai?iiiiiaaii ? iaa?oioiaaii canoinoaaiiy iaoiaeo ii?iaeue-

ieo oi?i aeey aea/aiiy o?ainaa?naeueieo a?oo?eaoe?e oeeee?a, ui
iaeaaeaoue oaioe/iiio neio?ii?coth/iio aoo?aeoi?o ? aecia/athoue iiiaio
ao?aoe no?eeino? ?aaeeio oaioe/ii? neio?ii?caoe??.

Aeey nenoaie aeaio neiao?e/ii ca’ycaieo eaaae?aoe/ieo a?aeia?aaeaiue
io?eiaii oi/i? oi?ioee aeey aecia/aiiy oeio o?ainaa?naeueieo a?oo?eaoe?e
ia?ooiii? oi/ee oa oeeeeo ia??iaeo 2. Aeey oeeee?a a?eueoeo ia??iae?a
oae? oi?ioee io?eiaii o aeaeyae? ?aeo?aioieo ni?aa?aeiioaiue.

Iaea?aeaii iaiao?aei? oiiae neeueii? /anoeiai? neio?ii?caoe?? aeey
nenoaie o?ueio ca’ycaieo iaeiiaei??ieo oaioe/ieo a?aeia?aaeaiue.
?acoeueoao ocaaaeueiaii aeey nenoaie aeia?eueii? ne?i/aiii? e?eueeino?
ca’ycaieo oaioe/ieo a?aeia?aaeaiue.

Aeine?aeaeaii neaaeo oa neeueio /anoeiao neio?ii?caoe?th a iaei?e
nenoai? o?ueio ca’ycaieo oaioe/ieo a?aeia?aaeaiue ?c neiao?e/iei ?
ianeiao?e/iei ca’yceii.

I?aeoe/ia cia/aiiy io?eiaieo ?acoeueoao?a. Io?eiai? ?acoeueoaoe
ocaaaeueiththoue oa aeiiiaiththoue a?aeiia?aei? aeine?aeaeaiiy nenoai
ca’ycaieo iae?i?eieo a?aeia?aaeaiue. I?e aecia/aii? i?inoi?iai? oi?ie
ia??iaee/ieo ?ica’yce?a o aeiaaeeo neaaeiai ca’yceo iiaeooue aooe
aeei?enoai? ?acoeueoaoe ia?oiai ?icae?eo. Cai?iiiiiaaiee iaoiae
aeine?aeaeaiiy o?ainaa?naeueieo a?oo?eaoe?e oeeee?a, ye? iaeaaeaoue
neio?ii?coth/iio oaioe/iiio aoo?aeoi?o, iiaea aooe canoiniaoaaiee i?e
aeine?aeaeaii? a?oo?eaoe?? ao?aoe no?eeino? oeei aoo?aeoi?ii.
?acoeueoaoe o?aoueiai ?icae?eo aoaeooue ei?enieie i?e ?ica’ycaii?
eiie?aoieo i?eeeaaeieo caaea/ ?c aeei?enoaiiyi yaeua neeueii? aai
neaaei? /anoeiai? oaioe/ii? neio?ii?caoe??.

Iniaenoee aianie caeiaoaa/a. Aecia/aiiy caaaeueiiai ieaio oa iai?yiie
aeine?aeaeaiue, iinoaiiaea caaea/ iaeaaeaoue iaoeiaiio ea??aieeo — A.I.
Naiieeaieo oa ni?aaaoi?o iaoeiaeo i?aoeue — TH.E. Iaeno?aieo. An?
?acoeueoaoe aeena?oaoe??, ye? aeiinyoueny ia caoeno, iaeaaeaoue aaoi?o.

Ai?iaaoeiy ?acoeueoao?a aeena?oaoe??. ?acoeueoaoe aeena?oaoe?eii? ?iaioe
aeiiia?aeaeenue ? iaaiai?thaaeenue ia nai?ia?ao a?aeae?eo cae-

/aeieo aeeoa?aioe?aeueieo ??aiyiue ?inoeoooo iaoaiaoeee IAI Oe?a?ie; ia
i?aeia?iaeiiio neiiic?oi? «Ergodic theory and dynamical systems»
(/a?aaiue-eeiaiue, 1995 ?., Aa?oaaa, Iieueua); ia i?aeia?iaeieo iaoeiaeo
eiioa?aioe?yo «Nonlinear dynamics, chaotic and complex systems»
(eenoiiaae, 1995 ?., Caeiiaia, Iieueua); «Contemporary problems in
theory of dynamical systems» (eeiaiue, 1996 ?., Ieaei?e Iiaai?iae,
?in?y); «Applied chaotic systems» (aa?anaiue, 1996 ?., Eiaecue,
Iieueua); ia i?aeia?iaei?e oeie? «Chaotic synchronization and
two-dimensional maps » (o?aaaiue, 1998 ?., E?iaa?, Aeai?y); ia
niaoe?ae?ciaaiiio i?aeia?iaeiiio nai?ia?? «Nonlinear dynamics of
electronic systems» (eeiaiue, 1998 ?., Aoaeaiaoo, Oai?ueia); ia
i?aeia?iaei?e iaoeiai-i?aeoe/i?e eiioa?aioe?? ii aeeiai?/iei nenoaiai
(aa?anaiue, 1998 ?., O???no, ?oae?y); ia i?aeia?iaeiiio neiiic?oi?
«Nonlinear theory and its applications» (aa?anaiue, 1998 ?.,
E?ain-Iiioaia, Oaaeoea??y).

Ioaeieaoei?. Ca oaiith aeena?oaoe?? iioae?eiaaii 10 iaoeiaeo i?aoeue, ?c
ieo 4 — o iaoeiaeo aeo?iaeao, 1 — o ca??ieeo iaoeiaeo i?aoeue, 2 — o
ca??ieeao i?aoeue i?aeia?iaeieo iaoeiaeo eiioa?aioe?e, 3 — o ca??ieeao
oac i?aeia?iaeieo iaoeiaeo eiioa?aioe?e.

No?oeoo?a oa ia’?i aeena?oaoe??. Aeena?oaoe?eia ?iaioa neeaaea?oueny ?c
anooio, o?ueio ?icae?e?a, ?icaeoeo ia 6 ia?aa?ao?a, oa nieneo oeeoiaaii?
e?oa?aoo?e ?c 67 iaca ? aeeeaaeaia ia 110 noi??ieao.

INIIAIEE CI?NO ?IAIOE

O anooii iaa?oioiaaii aeooaeuei?noue oaie, i?iaiae?ciaaii no/aniee noai
i?iaeaie, noi?ioeueiaaii caaea/? aeine?aeaeaiiy oa ei?ioei aeeeaaeaii
iniiai? ?acoeueoaoe.

aeaeyaeo

(1)

Nenoaia (1) caaea? aeene?aoiee caeii ci?ie o /an? ii/aoeiaiai aae-

.

aecia/a? neeo oeueiai ca’yceo. Ia iao?eoeth ca’yceo G iaeeaaeaii
oiiae:

(2)

. Oaeiae aeiaaa?oueny, uia nenoaia (1) aoea neiao?e/iith a oiio nain?,
ui ca’ycie eiaeiiai inoeeeyoi?a ?c ?ioeie caeaaeeoue eeoa a?ae
a?aeoeeaiiy aeaiaio?a eaioethaa iaeei a?ae iaeiiai ? ia caeaaeeoue a?ae
aeai?o inoeeeyoi?a.

) ia? oaeo ae i?inoi?iao no?oeoo?o ye ? ?ica’ycie iacao?aii? nenoaie.
Oe? oaeoe iathoue i?noea i?e aeeiiaii? oiia:

, aea I — a?ae??cie i?yii?, ? e?ioeoeaaith, oiaoi ?nio? L > 0 oaea, ui

(3)

, ? ne?i/aiith, a ioaea ?nio? iaeiaioa ni?eueia e?aoia ia??iae?a an?o
i?eoyaoth/eo oeeee?a, yea aoaeaii iicia/aoe m.

oae?, ui

(4)

.

oaea, ui

:

(5)

.

OAI?AIA 1.1.1. Iaoae aeeiiothoueny oiiae 1 — 4. Oiae? ?ica’ycie

aeieeaa? ia??ai?noue

? aneiioioe/ii m-ia??iaee/iei, oiaoi ?nio? ia??iaee/iee ?ica’ycie
nenoaie (1)

,

oaeee, ui

. (6)

? aneiioioe/ii i?inoi?iai l-ia??iaee/iei, oiaoi

,

oaeiae ? aneiioioe/ii i?inoi?iai l-ia??iaee/iei, oiaoi

.

, ui ? noaa?aeaeo? ianooiiee iane?aeie.

? ??aiii??ii ca N aneiioioe/ii no?eeei ii a?aeiioaiith aei iaeeo
cao?aiue ii/aoeiaeo aeaieo.

, ui caaeiaieueiythoue ni?aa?aeiioaiiy (6).

, oiaoi

.

Oey aeanoea?noue iiaea aooe i?i?ioa?i?aoiaaia c oi/ee ci?o aeieeiaiiy,
caa?aaeaiiy oa aaiethoe?? i?inoi?iaeo no?oeoo? i?ae ae??th cao?aiue, ui
ae?thoue a nenoai? (1).

caaea? i?inoi?iao no?oeoo?o aeaeo

, (7)

aecia/athoueny ye

.

) nenoaie (1) aoaeue-yea no?oeoo?a (7), yeui ?? acyoe ca ii/aoeia?
aeai? caaea/?, ii?iaeaeo? m-ia??iaee/iee ?ica’ycie, a oiio oaea
no?oeoo?a iiaea aooe iacaaia ia??iaee/iith ia??iaeo m.

, oaeeo ui

,

.

Oai?aia 1.1.1 aeicaiey? c?iaeoe aeniiaie, ui i?e aeeiiaii? oiia 1 — 4
ia??iaee/i? no?oeoo?e, ye? ii?iaeaeothoueny ?ica’yceaie iacao?aii?
nenoaie (1), caa??aathoueny i?e iaeeo cao?aiiyo ii/aoeiaeo aeaieo, iaei
ci?ithth/enue i?e aaaaeaii? neaaeiai ca’yceo.

aeaeyaeo

, (8)

— ia?aiao? ca’yceo.

.

aaaeeeaa cia/aiiy ia? aea/aiiy o?ainaa?naeueieo (noiniaii ae?aaiiae?)
a?oo?eaoe?e oaeeo oeeee?a.

— ia?ooiia oi/ea a?aeia?aaeaiiy f.

aeani? cia/aiiy ? a?aeiia?aei? ?i aeani? aaeoi?e iao?eoe? sseia?
a?aeia?aaeaiiy F ia/eneththoueny ca oi?ioeaie

— iio?aeia ooieoe?? f.

, ? iaoae

. (9)

, yea io?eio?oueny ?c (8) i?e cai?i?

. (10)

o?ainaa?naeueii. Ia? i?noea oaea oai?aia.

, i?e/iio

. (11)

, i?e/iio

. (12)

). Oaeei /eiii, a oai?ai? 2.1.3 noaa?aeaeo?oueny, ui o?ainaa?naeueia
a?oo?eaoe?y iiaeai?iiy ia??iaeo aoaea i’yeith, a o?ainaa?naeueia
a?oo?eaoe?y «aeeee» aoaea aei?noeith.

.

.

:

.

, i?e/iio

(13)

ciaoiaeeoueny ?c ianooiiiai ?aeo?aioiiai ni?aa?aeiioaiiy:

.

, i?e/iio

, (14)

ciaoiaeyoueny ?c ni?aa?aeiioaiue

,

.

aeey a?aeia?aaeaiiy F.

O o?aoueiio ?icae?e? aeine?aeaeo?oueny yaeua /anoeiai? neio?ii?caoe??
aeey aeayeeo eaioetha?a ca’ycaieo iaeiiaei??ieo oaioe/ieo
a?aeia?aaeaiue.

Aeey N-aei??ii? aeeiai?/ii? nenoaie yaeua /anoeiai? neio?ii?caoe??
iieyaa? a oiio, ui aeayea e?euee?noue ?c 1 < N' < N oaciaeo ci?iieo c /anii ii/eia? eieeaaoeny neio?iiii, a oie /an ye aeey an?o ?ioeo ci?iieo oey aeanoea?noue ia aeeiio?oueny. sseui aei oiai ae neo?ii?ciaai? ci?ii? eieeaathoueny oaioe/ii, oi aoaeaii aiai?eoe, ui ia? i?noea /anoeiaa oaioe/ia neio?ii?caoe?y. caaaeueiiai aeaeyaeo , (15) — ia?aiao?e ca'yceo. I?e /anoeia?e neio?ii?caoe?? aneiioioe/ii c /anii aeeiai?ea nenoaie ?aae?co?oueny a iaei?e ?c ae?aaiiaeueieo ieiuei , . (16) ye?, i/aaeaeii, ? ?iaa??aioieie aeey a?aeia?aaeaiiy (15). Ae?aaiiaeue iaeaaeeoue eiaei?e ?c ieiuei (16) ? oaeiae ? ?iaa??aioiith. , yeee aoaeaii iaceaaoe neio?ii?coth/ei aoo?aeoi?ii; 2) neio?ii?coth/ee aoo?aeoi? A ? oaeiae aoo?aeoi?ii ? aeey a?aeia?aaeaiiy F. I?e aeeiiaii? oiia 1 — 2 /anoeiaa oaioe/ia neio?ii?caoe?y aoaea iaoe i?noea aeey iiiaeeie aeiaeaoii? i??e Eaaaaa ii/aoeiaeo aeaieo. Aoaeaii aiai?eoe, ui /anoeiaa neio?ii?caoe?y ? neeueiith, yeui a?aeiia?aeiee neio?ii?coth/ee aoo?aeoi? ? aneiioi- oe/ii no?eeei ca Eyioiiaei aeey a?aeia?aaeaiiy F, ? neaaeith — a i?ioeeaaeiiio aeiaaeeo. Iaiao?aei? oiiae neeueii? oaioe/ii? /anoeiai? neio?ii?caoe?? aeey nenoaie (15) oi?ioeththoueny o ianooii?e oai?ai?. , ? iano?eeei ca Eyioiiaei. O § 3.1.2 iaiao?aei? oiiae neeueii? oaioe/ii? /anoeiai? neio?ii?caoe?? ocaaaeueiththoueny aeey nenoaie aeia?eueii? ne?i/aiii? e?eueeino? N > 3
ca’ycaieo iaeiiaei??ieo oaioe/ieo a?aeia?aaeaiue.

, yea i?noeoue i?eiaeii? aeaa aeaiaioe. I?ae?ino??

(17)

iaceaa?oueny neio?ii?coth/ei i?aei?inoi?ii aai, aiane?aeie eiai
aaiiao?e/ii? iaoo?e, ae?aaiiaeueiei i?aei?inoi?ii.

ca’ycaieo iaeiiaei??ieo a?aeia?aaeaiue, aeey yei? an? ae?aaiiaeuei?
i?aei?inoi?e (17) ? ?iaa??aioieie:

(18)

— ia?aiao?e ca’yceo.

Ii?o/ ?c nenoaiith (18) ?icaeyaea?oueny nenoaia ?c ciai?oi?i ca’yceii:

(19)

A oai?ai? 3.1.2 noi?ioeueiaaii oiiae ia ia?aiao?e ca’yceo, ui
caaacia/othoue oiiieia?/io ni?yaeai?noue nenoai (18) oa (19).

Anoaiiaeaii oaeiae oaeo oai?aio.

a nenoaiao (18) oa (19) aneiioioe/ia no?ee?noue iaeiiai ?c
ae?aaiiaeueieo i?aei?inoi??a (17) icia/a? aneiioioe/io no?ee?noue
ae?aaiiae?.

?c oai?aie 3.1.3 aeieeaa? ocaaaeueiaiiy oai?aie 3.1.1.

, ? iano?eeei ca Eyioiiaei.

?ioeie neiaaie, /anoeiaa oaioe/ia neio?ii?caoe?y ?c neio?ii?coth/ei

, iiaea aooe o?eueee neaaeith. Oaeei /eiii, aeey neeueii? /anoeiai?
oaioe/ii? neio?ii?caoe?? a eaioethaao ca’ycaieo a?aeia?aaeaiue
iaiao?aeii ?icaeyaeaoe nenoaie, aeey yeeo aai ia an? ae?aaiiaeuei?
i?aei?inoi?e ? ?iaa??aioieie, aai neio?ii?coth/ee aoo?aeoi? ia i?noeoue
/anoeio ae?aaiiae?. sse i?eeeaaee canoinoaaiiy oeueiai e?eoa??th o § 3.2
aeine?aeaeaii eiie?aoio nenoaio o?ueio ca’ycaieo oaioe/ieo
a?aeia?aaeaiue, aeey yei? o?euee? aea? ae?aaiiaeuei? ieiueie (16) ?
?iaa??aioieie. I?e ii?aeiaii? aiae?oe/ieo oa /enaeueieo iaoiae?a
ciaeaeaii iaeano? a i?inoi?? ia?aiao??a aeey neeueii? oa neaaei?
/anoeiai? neio?ii?caoe??, a oaeiae iaeano? ni?a?nioaaiiy neeueii?
/anoeiai? oa neaaei? iiaii? neio?ii?caoe??. O § 3.3 ?icaeyiooi
aiaeia?/io nenoaio ?c neiao?e/iei ca’yceii ? aeine?aeaeaii neeueio
/anoeiao neio?ii?caoe?th aeey aeiaaeeo neio?ii?coth/iai aoo?aeoi?a, yeee
ia ia?aoeia?oueny ?c ae?aaiiaeeth.

AENIIAEE

Aeine?aeaeaii no?ee?noue ia??iaee/ieo ?ica’yce?a a nenoai? caaaeueiiai
aeaeo N ca’ycaieo iae?i?eieo a?aeia?aaeaiue o aeiaaeeo neaaeiai ca’yceo.
Iieacaii, ui ?aaoey?i? i?inoi?iai-/ania? no?oeoo?e caa??aathoueny i?e
iaeeo cao?aiiyo ii/aoeiaeo aeaieo ? ? no?eeeie noiniaii iaeeo cao?aiue
ia?aiao?a ca’yceo.

Cai?iiiiiaaii ? iaa?oioiaaii iaoiae aecia/aiiy oeio o?ainaa?naeueieo
a?oo?eaoe?e oeeee?a, ye? iaeaaeaoue neio?ii?coth/iio

aoo?aeoi?o ? aecia/athoue iiiaio ao?aoe no?eeino? ?aaeeio oaioe/ii?
neio?ii?caoe??.

Iaea?aeaii oi/i? oi?ioee aeey aecia/aiiy oeio o?ainaa?naeueieo
a?oo?eaoe?e ia?ooiii? oi/ee ? oeeeeo ia??iaeo 2 a nenoai? aeaio
neiao?e/ii ca’ycaieo eaaae?aoe/ieo a?aeia?aaeaiue. Aeey oeeee?a a?eueoeo
ia??iae?a oae? oi?ioee io?eiaii o aeaeyae? ?aeo?aioieo ni?aa?aeiioaiue.

Anoaiiaeaii iaiao?aei? oiiae neeueii? /anoeiai? oaioe/ii?
neio?ii?caoe?? a eaioethaao ca’ycaieo a?aeia?aaeaiue. I?iaaaeaii
aeine?aeaeaiiy neeueii? oa neaaei? /anoeiai? neio?ii?caoe?? a nenoaiao
o?ueio ca’ycaieo oaioe/ieo a?aeia?aaeaiue ?c neiao?e/iei ? ianeiao?e/iei
ca’yceii.

Iniiai? ?acoeueoaoe aeena?oaoe?? iioae?eiaaii a ianooiieo ?iaioao:

Iaeno?aiei TH.E., Iiiiae/ I.A. ?nioaaiiy oa no?ee?noue ia??iaee/ieo
?ica’yce?a eaioetha?a ca’ycaieo inoeeeyoi??a // Oe?. iao. aeo?.— 1997.
— 47. —? 7. — N. 943-950.

Maistrenko Yu.L., Maistrenko V.L., Popovich A., Mosekilde E. Role of
the Absorbing Area in Chaotic Synchronization // Phys. Rev. Let. — 1998.
— 80. — ? 8. — P. 1638-1641.

Maistrenko Yu.L., Maistrenko V.L., Popovich A., Mosekilde E.
Transverse Instability and Riddled Basins on a System of Two Coupled
Logistic Maps // Phys. Rev. E. — 1998. — 57. — ? 3. — P. 2713-2724.

Hasler M., Maistrenko Yu., Popovych O. Simple Example of Partial
Synchronization of Chaotic Systems // Phys. Rev. E. — 1998. — 58. — ?
5. — P. 6843-6846.

Maistrenko Yu., Popovych O. Necessary Conditions for Chaotic Partial
Synchronization // Iaeeiaeiua e?aaaua caaea/e iaoaiaoe/aneie oeceee e eo
i?eeiaeaiey. — Eeaa: Ei-o iaoaiaoeee IAI Oe?aeiu, 1998. — N. 144-147.

Hasler M., Maistrenko Yu., Popovych O. An Example of Partial
Synchronization // Proceedings of NDES’98. — Budapest (Hungary). — 1998.
— P. 241-244.

Hasler M., Maistrenko Yu., Popovych O. Partial Synchronization in
Coupled Map Systems // Proceedings of NOLTA’98. — Crans-Montana
(Switzerland). — 1998. — P. 327-330.

Maistrenko Yu.L., Popovich A.V. Stability in Coupled Map Lattices //
Abstract of NDCCS’95. — Zacopane(Poland). — 1995. — P. 115.

Popovich A.V. Periodic Solutions of Coupled Map Lattices with Small
Coupling // Abstracts of CPTDS’96. — Nizhny Novgorod (Russia). — 1996.
— P. 43-44.

Maistrenko Yu.L., Popovich A.V. Preservation of Periodic
Spatio-temporal Sturtures in Map Lattices with a Small Coupling //
Abstracts of International Conference on Applied Chaotic Systems. —
Inowlodz (Poland). — 1996. — P. 17-18.

Iiiiae/ I.A. Aneiioioe/ia iiaaae?iea oa a?oo?eaoe?? ?ica’yce?a

eaioetha?a ca’ycaieo inoeeeyoi??a. — ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa
o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.01.02 —
aeeoa?aioe?aeuei? ??aiyiiy. — ?inoeooo iaoaiaoeee IAI Oe?a?ie, Ee?a,
1998.

A aeena?oaoe?? aeine?aeaeaii no?ee?noue ia??iaee/ieo ?ica’yce?a
eaioetha?a ca’ycaieo inoeeeyoi??a o aeiaaeeo, eiee aaciaei inoeeeyoi?ii
? iaeiiaei??ia iae?i?eia a?aeia?aaeaiiy a?ae??ceo a naaa. Cai?iiiiiaaii
? iaa?oioiaaii iaoiae aeine?aeaeaiiy o?ainaa?naeueieo a?oo?eaoe?e
neiao?e/ieo oeeee?a, ye? iaeaaeaoue neio?ii?coth/iio iiiaiaeaeo.
Iaea?aeaii oiiae aeey neeueii? oa neaaei? /anoeiai? oaioe/ii?
neio?ii?caoe??, ye? canoiniaaii i?e aeine?aeaeai? /anoeiai?
neio?ii?caoe?? a eiie?aoieo nenoaiao ?c neiao?e/iei ? ianeiao?e/iei
ca’yceii.

Eeth/ia? neiaa: eaioethae ca’ycaieo inoeeeyoi??a, iae?i?eia
a?aeia?aaeaiiy, no?ee?noue ca Eyioiiaei, a?oo?eaoe?y, neio?ii?caoe?y.

Iiiiae/ A.A. Aneiioioe/aneia iiaaaeaiea e aeoo?eaoeee ?aoaiee oeaii/ae
naycaiiuo inoeeeeyoi?ia. — ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.01.02 —
aeeooa?aioeeaeueiua o?aaiaiey.— Einoeooo iaoaiaoeee IAI Oe?aeiu, Eeaa,
1998.

A aeenna?oaoeee enneaaeiaaii onoie/eainoue ia?eiaee/aneeo ?aoaiee
oeaii/ae naycaiiuo inoeeeeyoi?ia a neo/aa, eiaaea aaciaui inoeeeyoi?ii
yaeyaony iaeiiia?iia iaeeiaeiia ioia?aaeaiea io?acea a naay.
I?aaeeiaeai e iainiiaai iaoiae enneaaeiaaiey oeia o?ainaa?naeueiuo
aeoo?eaoeee neiiao?e/iuo oeeeeia, eioi?ua i?eiaaeeaaeao neio?iiecothuaio
iiiaiia?aceth. Iieo/aiu oneiaey aeey neeueiie e neaaie /anoe/iie
oaioe/aneie neio?iiecaoeee, eioi?ua i?eiaiaiu i?e enneaaeiaaiee
/anoe/iie neio?iiecaoeee a eiie?aoiuo nenoaiao n neiiao?e/iie e
ianeiiao?e/iie naycueth.

Eeth/aaua neiaa: oeaii/ee naycaiiuo inoeeeeyoi?ia, iaeeiaeiia
ioia?aaeaiea, onoie/eainoue ii Eyioiiao, aeoo?eaoeey, neio?iiecaoeey.

Popovych O.V. Asymptotic behavior and bifurcations of solutions of
coupled oscillator lattices.- Manuscript.

Thesis for candidate degree by speciality 01.01.02 — differential
equations.- The Institute of Mathematics, National Academy of Sciences
of Ukraine, Kyiv, 1998.

The thesis is devoted to the investigations of asymptotic behavior and
bifurcations of solutions of coupled oscillator lattices. As a base
oscillator a one-dimensional nonlinear map of an interval into itself
with regular or chaotic dynamics is considered. The main attention is
paid to the problem of stability of periodic solutions and the
phenomenon of total and partial chaotic synchronization. The thesis
consists of introduction, three chapters and references.

In the introduction the importance of the topic is justified, the
current stage of the investigations in the field is analyzed and the
main results

are briefly described.

In the first chapter a system of N coupled one-dimensional maps with
a small coupling is considered. Under general conditions on the base
map and initial data it is proved that periodic solutions are stable
with respect to small perturbations of initial data and coupling
parameter. It is also stated that regular spatio-temporal structures
are preserving under small perturbations of initial data and change a
little when a weak coupling is introduced.

The second chapter is devoted to the investigation of transversal
bifurcations of symmetrical point cycles which belong to the
synchronous state. The method of using the theory of normal form to the
problem is proposed and justified. For the system of two coupled
logistic maps explicit formulas to determine the type of the
bifurcations are obtained for fixed point and period-2 cycle. For the
cycles of higher period these formulas are obtained in the form of
recurrent relations. Obtained formulas allow us to conclude whether
transversal bifurcation of the cycle considered is supercritical or
subcritical.

In the third chapter the problem of weak and strong partial chaotic
synchronization is examined. For a system of coupled chaotic maps in
general form necessary conditions for strong partial synchronization
are stated. In particular it was proved that strong partial
synchronization is impossible in the system having all invariant
diagonal subspaces and with synchronizing attractor that contains a
part of the diagonal. As an application of this result two systems of
three coupled one-dimensional chaotic maps with symmetrical and
non-symmetrical coupling are considered. With using analytical and
numerical methods regions in parameter space of weak and strong partial
synchronization as well as coexistence of strong partial and weak total
synchronization are found.

The main results of the thesis have been published in 10 scientific
publications and have been reported at a number of international
scientific conferences and symposiums.

Key words: coupled oscillator lattices, nonlinear map, stability by
Lyapunov, bifurcation, synchronization.

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