Investigation of the periodic solutions of nonlinear autonomous systems in the critical case
We analyze the conditions of existence and the numerical-analytic method for the approximate construction of periodic solutions of nonlinear autonomous systems of differential equations in the critical case.
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| Date: | 2008 |
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Institute of Mathematics, NAS of Ukraine
2008
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509202471976960 |
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| author | Korol', I. I. Король, І. І. |
| author_facet | Korol', I. I. Король, І. І. |
| author_sort | Korol', I. I. |
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| description | We analyze the conditions of existence and the numerical-analytic method for the approximate construction of periodic solutions of nonlinear autonomous systems of differential equations in the critical case. |
| first_indexed | 2026-03-24T02:37:21Z |
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UDK 517.925
I. I. Korol\ (UΩhorod. nac. un-t)
DOSLIDÛENNQ PERIODYÇNYX ROZV’QZKIV NELINIJNYX
AVTONOMNYX SYSTEM U KRYTYÇNOMU VYPADKU
We investigate existence conditions and a numerical-analytic method of the approximate construction of
the periodic solutions of nonlinear autonomous differential systems in a critical case.
Yssledugtsq uslovyq suwestvovanyq y çyslenno-analytyçeskyj metod pryblyΩennoho po-
stroenyq peryodyçeskyx reßenyj nelynejn¥x avtonomn¥x dyfferencyal\n¥x system v kryty-
çeskom sluçae.
Teoriq periodyçnyx krajovyx zadaç ma[ ßyroke zastosuvannq pry doslidΩenni
riznomanitnyx texniçnyx ta pryrodnyçyx procesiv. Same tomu pytannqm isnu-
vannq i nablyΩeno] pobudovy periodyçnyx rozv’qzkiv dyferencial\nyx rivnqn\
prydilqlasq znaçna uvaha u pracqx bahat\ox matematykiv, zokrema v [1 – 4]. Pry
c\omu vaΩlyve misce zajmagt\ doslidΩennq avtonomnyx system dyferencial\-
nyx rivnqn\ [5 – 7].
Dana robota [ prodovΩennqm doslidΩen\, rozpoçatyx u robotax [8, 9]. U nij
zaproponovano modyfikacig çysel\no-analityçnoho metodu A. M. Samojlenka
dlq doslidΩennq isnuvannq i nablyΩeno] pobudovy periodyçnyx rozv’qzkiv avto-
nomnyx nelinijnyx system
dx
dt
= Px + g x( ) (1)
u krytyçnomu vypadku. Pry c\omu obmeΩennq na matrycg Lipßycq stosugt\sq
ne vsi[] pravo] çastyny, a lyße nelinijnosti.
1. Pobudova periodyçnyx rozv’qzkiv nelinijnyx avtonomnyx system.
Rozhlqnemo nelinijnu avtonomnu systemu zvyçajnyx dyferencial\nyx rivnqn\
(1), de x, g n∈R , P — stala ( )n n× -vymirna dijsna matrycq, pryçomu
A) vidpovidna (1) linijna odnoridna systema
dx
dt
= Px (2)
ma[ k, 1 ≤ k ≤ n, periodyçnyx rozv’qzkiv, qki magt\ spil\nyj period T = 2Π ν .
Bez obmeΩennq zahal\nosti budemo vvaΩaty, wo matrycq P ma[ vyhlqd P =
= diag( , )A B , de A — ( )k k× -vymirna Ωordanova kanoniçna kososymetryçna
matrycq:
A = diag
0
0
0
0
0 0
1
1
ν
ν
ν
ν–
, ,
–
, , , ,
…
… …
q
q
, (3)
taka, wo vsi rozv’qzky linijno] dyferencial\no] systemy
dx
dt
= Ax
[ periodyçnymy zi spil\nym periodom T = 2Π ν .
Takym çynom, systemu (1) moΩemo zapysaty u vyhlqdi
dx
dt
= Ax + g x( ),
(4)
dx
dt
˜
= Bx̃ + ˜( )g x ,
© I. I. KOROL|, 2008
332 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
DOSLIDÛENNQ PERIODYÇNYX ROZV’QZKIV NELINIJNYX AVTONOMNYX … 333
de t ∈R, x : R → D1 � Rk
, x̃ : R → D2 � Rn k–
, g : D k→ R , g̃ : D n k→ R
–
,
x = col( , ˜)x x , g = col( , ˜)g g , D = D
1
× D
2
— zamknena obmeΩena oblast\ v Rn
.
Z budovy matryci P vyplyva[, wo vidpovidna (4) linijna odnoridna systema
dx
dt
= Ax ,
dx
dt
˜
= Bx̃
ma[ k-parametryçnu sim’g T-periodyçnyx rozv’qzkiv vyhlqdu
x t( ) = eAtξ ,
˜( )x t = 0.
Naßym zavdannqm [ doslidyty naqvnist\ ta zaproponuvaty metod vidßukannq
periodyçnyx po t rozv’qzkiv nelinijno] avtonomno] systemy (1), period qkyx zbi-
ha[t\sq z periodom rozv’qzkiv vidpovidno] linijno] avtonomno] systemy (2).
U prostori T-periodyçnyx funkcij T D( , )R � C D( , )R rozhlqnemo sim’g k-
parametryçnyx vidobraΩen\ L xξ : T n( , )R R → T n( , )R R i funkcional µ( )x :
C n( )R → D :
( )( )L x tξ = e xPt
0 + U t s g x s ds
t
( , ) ( )( )∫
0
– V t s g x s ds
t
T
( , ) ( )( )∫ ,
U t s( , ) =
1 0
0
1
–
–
( – )
– –
– – ( – )
t
T
e
I I e e
A t s
n k n k
BT B t s
− ( )( )
,
V t s( , ) =
t
T
e
I e e
A t s
n k
BT B t s
( – )
–
– – ( – )–
0
0
1( )
,
µ( )x = e g x s dsAs
T
– ( )( )∫
0
,
x0 = col( , ˜ )x x0 0 , x 0 = ξ, x̃0 = 0, x
k
0 ∈R , ̃
–x n k
0 ∈R .
Çerez I j budemo poznaçaty odynyçnu matrycg porqdku j. Do systemy (1) za-
stosu[mo çysel\no-analityçnyj metod [8, 9], i ]] T-periodyçnyj rozv’qzok bude-
mo ßukaty qk hranycg poslidovnosti T-periodyçnyx funkcij
x tm +1( , )ξ = x t0( , )ξ + U t s g x s ds
t
m( , ) ( , )
0
∫ ( )ξ – V t s g x s ds
t
T
m( , ) ( , )∫ ( )ξ ,
(5)
x t0( , )ξ = e xPt
0 , m = 0{ } ∪ N .
ZauvaΩennq 1. Operator L xξ moΩemo zapysaty takym çynom:
( )( )L x tξ = e I e e e g x s ds
Pt
n k
BT BT Bs
T
ξ
–
– –– ˜ ( )( ) ( )
∫
1
0
+
+
0
t
P t se g x ds∫ ( – ) ( ) –
t
T
e xAtµ( )
0
. (6)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
334 I. I. KOROL|
ZauvaΩennq 2. Pry praktyçnij pobudovi nablyΩenyx rozv’qzkiv çleny pos-
lidovnosti (5) zruçno podaty u vyhlqdi
x tm +1( , )ξ = x t0( , )ξ + e g x s dsA t s
m
t
( – ) ( , )ξ( )∫
0
– t
T
e g x s dsA t s
m
T
( – ) ( , )ξ( )∫
0
,
˜ ( , )x tm +1 ξ = e g x s dsB t s
m
t
( – ) ˜ ( , )ξ( )∫
0
+
+ I e e e g x s dsn k
BT BT
T
B t s
m–
– ( – )– ˜ ( , )( ) ( )∫
1
0
ξ ,
x t0( , )ξ = eAtξ , ˜ ( , )x t0 ξ = 0, xm = ( , ˜ )x xm m .
Prypuska[mo, wo v oblasti ( , )t x ∈ R × D systema (1) zadovol\nq[ nastupni
umovy:
B) vektor-funkciq g x( ) [ vyznaçenog, neperervnog i zadovol\nq[ umovy
obmeΩenosti i Lipßycq z nevid’[mnymy stalymy vektorom M i matryceg K:
g x( ) ≤ M, g x g x( ) – ( )′ ′′ ≤ K x x′ ′′– , (7)
pryçomu M = ( , , )M Mn1 … , K = Ki j{ }, i, j = 1, n , x = ( , , )x xn1 … i vsi nerivnos-
ti rozumi[mo pokomponentno;
C) isnu[ neporoΩnq mnoΩyna toçok ξ β∈D taka, wo vektor-funkciq
x t0( , )ξ naleΩyt\ oblasti D razom iz svo]m β-okolom, de β = max ( )( )
,t T
SM t
∈[ ]0
,
Sx : C
n( , )R R → C
n( , )R R — linijnyj operator:
( )( )Sx t =
0
t
t
T
U t s x s ds V t s x s ds∫ ∫+( , ) ( ) ( , ) ( ) ;
D) r Q( ) < 1, de r Q( ) — spektral\nyj radius operatora Qx = S Kx( ) , qkyj [
kompozyci[g operatora S iz mnoΩennqm na matrycg K:
( )( )Qx t =
0
t
t
T
U t s K x s ds V t s K x s ds∫ ∫+( , ) ( ) ( , ) ( ) .
Doslidymo umovy isnuvannq i metod nablyΩeno] pobudovy periodyçnoho roz-
v’qzku zadanoho periodu T systemy (1). Nastupne tverdΩennq mistyt\ neobxidni
umovy isnuvannq T-periodyçnoho rozv’qzku systemy (1).
Teorema 1. Nexaj vykonu[t\sq umova A i avtonomna dyferencial\na sys-
tema (1) ma[ T-periodyçnyj rozv’qzok ϕ( )t = ϕ ξ( , )t ∗
. Todi joho poçatkovym
znaçennqm [ ϕ( )0 = ϕ0 = ( , ˜ )ϕ ϕ0 0 , de
ϕ0 = ξ∗,
(8)
ϕ̃0 = I e e e g s dsn k
BT BT
T
Bs
–
– –– ˜ ( )( ) ( )∫
1
0
ϕ
i ξ∗
take, wo
µ ϕ( ) = 0. (9)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
DOSLIDÛENNQ PERIODYÇNYX ROZV’QZKIV NELINIJNYX AVTONOMNYX … 335
Dovedennq. Nexaj ϕ( )t [ rozv’qzkom systemy (1), todi ma[ misce totoΩnist\
ϕ( )t ≡ e e g s dsPt P t s
t
ϕ ϕ( ) ( )( – )0
0
+ ( )∫ . (10)
Z T-periodyçnosti ϕ( )t oderΩu[mo linijnu alhebra]çnu systemu
I en
PT– ( )( )ϕ 0 = e e g s dsPT Ps
T
– ( )ϕ( )∫
0
.
Beruçy do uvahy strukturu matryci P, otrymu[mo systemu dlq znaxodΩennq
ϕ( )0 :
0 0
0
0
I en k
BT
– –
( )
ϕ =
e g s ds
e g s ds
A T s
T
B T s
T
( – )
( – )
( )
˜ ( )
ϕ
ϕ
( )
( )
∫
∫
0
0
. (11)
Zrozumilo, wo systema (11) [ sumisnog todi i til\ky todi, koly vykonu[t\sq
umova (9) i pry c\omu ϕ( )0 = ϕ0.
Teoremu dovedeno.
VkaΩemo dostatni umovy isnuvannq T-periodyçnoho rozv’qzku systemy (1).
Teorema 2. Nexaj vykonu[t\sq umova A . Qkwo pry c\omu ξ = ξ∗
i funkciq
ϕ( )t = ϕ ξ( , )t ∗
taki, wo vykonu[t\sq systema rivnqn\
ϕ = Lξϕ , (12)
µ ϕ( ) = 0, (13)
to ϕ( )t [ T -periodyçnym rozv’qzkom avtonomno] dyferencial\no] syste-
myN(1), a joho poçatkove znaçennq vyznaça[t\sq zhidno z (8).
Dovedennq. Nexaj funkciq ϕ( )t i vektornyj parametr ξ = ξ∗ ∈Rk
zado-
vol\nqgt\ rivnqnnq (12). Z (6) vyplyva[
ϕ ξ( , )t ≡ e I e e g s ds
Pt
n k
BT B T s
T
ξ
ϕ ξ–
– ( – )– ˜ ( , )( ) ( )
∫
1
0
+
+ e g s dsP t s
t
( – ) ( , )ϕ ξ( )∫
0
–
t
T
e sAtµ ϕ ξ( , )( )
0
.
Oskil\ky ξ = ξ∗
i ϕ( )t = ϕ ξ( , )t ∗
zadovol\nqgt\ rivnqnnq (13), z ostann\o] to-
toΩnosti vyplyva[
ϕ( )t ≡ ePtϕ0 + e g s dsP t s
t
( – ) ( )ϕ( )∫
0
,
a tomu ϕ( )t [ rozv’qzkom systemy (1) z poçatkovym znaçennqm (8). PokaΩemo,
wo ϕ( )t [ T-periodyçnog funkci[g:
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
336 I. I. KOROL|
ϕ( )T ≡ ePTϕ0 + e e g s dsPT Ps
T
– ( )ϕ( )∫
0
=
=
e
e I e e e g s ds
AT
BT
n k
BT BT Bs
T
ξ
ϕ
∗
( ) ( )
∫–
– –– ˜ ( )
1
0
+
e
e e g s ds
AT
BT Bs
T
µ ϕ
ϕ
( )
˜ ( )– ( )
∫
0
=
=
ξ
ϕ
∗
( ) +[ ] ( )
∫I e e I e e g s dsn k
BT BT
n k
BT Bs
T
–
–
–
–– ˜ ( )
1
0
.
Oskil\ky
I e en k
BT BT
–
–
–( ) 1
+ In k– = I en k
BT
–
–
–( ) 1
,
to ϕ( )T = ϕ0 .
Teoremu dovedeno.
Vraxovugçy umovy B i C, otrymu[mo ocinku
( )( ) – ( , )L x t x tξ ξ0 ≤
0
t
U t s M ds∫ ( , ) +
t
T
V t s M ds∫ ( , ) = ( )( )SM t ≤ β. (14)
OtΩe, pry vsix t ∈R, ξ β∈D vsi çleny poslidovnosti (5) naleΩat\ oblasti D.
Krim toho, z (5), (7) oderΩu[mo
x t x tm m+1( , ) – ( , )ξ ξ = L x x tm mξ ξ ξ( , ) – ( , ) ( )–⋅ ⋅( )( )1 ≤
≤ Q x x tm m( , ) – ( , ) ( )–⋅ ⋅( )ξ ξ1 ≤ Q x x tm m
2
1 2– –( , ) – ( , ) ( )⋅ ⋅( )ξ ξ ≤ …
… ≤ Q x x tm
1 0( , ) – ( , ) ( )⋅ ⋅( )ξ ξ ≤ Qmβ ,
z çoho vyplyva[, wo pry vsix m, j ∈N
x t x tm j m+ ( , ) – ( , )ξ ξ ≤
i
j
m i m ix t x t
=
+ + +∑
0
1
1
–
( , ) – ( , )ξ ξ ≤
≤
i
j
m iQ x x t
=
+∑ ⋅ ⋅( )
0
1
1 0
–
( , ) – ( , ) ( )ξ ξ ≤
i
j
m iQ
=
+∑
0
1–
β. (15)
Z (14) i umovy D vyplyva[, wo, zhidno z pryncypom stysnutyx vidobraΩen\,
rivnqnnq x = L xξ ma[ [dynyj rozv’qzok v T D( , )R , qkyj pry vsix ξ β∈D zbi-
ha[t\sq z hranyçnog funkci[g x t∗( , )ξ poslidovnosti (5). Pry c\omu parametr
ξ = ξ∗
budemo vybyraty tak, wob µ ξx∗ ⋅( )( , ) = 0. Todi za teoremogN2 funkciq
x t∗( ) = x t∗ ∗( , )ξ , x∗( )0 = ϕ0 , [ T-periodyçnym rozv’qzkom systemyN(1). Perexo-
dqçy v (15) do hranyci pry j → ∞ , oderΩu[mo ocinku zbiΩnosti poslidovnos-
tiN(5):
x t x tm
∗( , ) – ( , )ξ ξ ≤ I Q Qn
m– –( ) 1 β. (16)
Takym çynom, moΩna sformulgvaty nastupnyj rezul\tat.
Teorema 3. Nexaj systema (1) zadovol\nq[ umovy A – D. Todi:
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
DOSLIDÛENNQ PERIODYÇNYX ROZV’QZKIV NELINIJNYX AVTONOMNYX … 337
1) poslidovnist\ funkcij x tm( , )ξ vyhlqdu (5) pry m → ∞ rivnomirno zbi-
ha[t\sq vidnosno ( , )t ξ ∈ R × Dβ do hranyçno] funkci] x t∗( , )ξ i pry vsix na-
tural\nyx m spravdΩugt\sq ocinky zbiΩnosti (16);
2) dlq toho wob funkciq x t∗( ) = x t∗( , )ξ bula T-periodyçnym rozv’qzkom
dyferencial\no] systemy (1), neobxidno i dostatn\o, wob toçka ξ = ξ∗
bula
rozv’qzkom rivnqnnq
∆ ( )ξ =df
e g x s dsAs
T
– ( , )∗( )∫ ξ
0
= 0;
3) poçatkove znaçennq x∗( )0 = x x∗ ∗( )( ), ˜ ( )0 0 T -periodyçnoho rozv’qzku
x t∗( ) vyznaça[t\sq za formulog
x∗( )0 = ξ∗,
(17)
˜ ( )x∗ 0 = I e e e g x s dsn k
BT BT Bs
T
–
– –– ˜ ( )( ) ( )∗∫
1
0
.
Sformulg[mo konstruktyvni dostatni umovy isnuvannq periodyçnyx roz-
v’qzkiv, dlq perevirky qkyx ne potribno ßukaty hranyçnu funkcig x t∗( , )ξ , a
dosyt\ znajty poslidovni nablyΩennq x tm( , )ξ .
Teorema 4. Nexaj systema (1) zadovol\nq[ umovy A – D i, krim toho:
1) isnu[ opukla zamknena oblast\ ′Dβ � Dβ � Rk
taka, wo pry deqkomu
fiksovanomu natural\nomu m vidobraΩennq ∆m( )ξ : Dβ → Rk :
∆m( )ξ =df
µ ξxm( , )⋅( ) = e g x s dsAs
m
T
– ( , )ξ( )∫
0
,
mistyt\ v oblasti ′Dβ [dynu osoblyvu toçku ξ0m nenul\ovoho indeksu;
2) na hranyci ∂ β′D oblasti ′Dβ vykonu[t\sq nerivnist\
inf ( )
ξ ∂ β
ξ
∈ ′D
m∆ > H I Q Qn
m( – )–1 β ,
de H =
0
T Ase ds K∫ –
.
Todi systema (1) ma[ T-periodyçnyj rozv’qzok x = x t∗( ) = x t∗ ∗( , )ξ , ξ∗
∈
∈ ′Dβ , z poçatkovym znaçennqm x∗( )0 vyhlqdu (17).
Dovedennq. Z oznaçennq vidobraΩen\ ∆ ( )ξ , ∆m( )ξ , beruçy do uvahy umovu
Lipßycq (7) i ocinku (16), otrymu[mo
∆ ∆( ) – ( )ξ ξm ≤ e g x s g x s dsAs
T
m
– ( , ) – ( , )
0
∫ ∗( ) ( )ξ ξ ≤
≤ e K x s x s dsAs
T
m
– ( , ) – ( , )
0
∫ ∗ ξ ξ ≤ H I Q Qn
m– –( ) 1 β.
Z ostann\o] ocinky, qk i pry dovedenni teoremyN3 [8], moΩna pokazaty homotop-
nist\ poliv ∆ ( )ξ i ∆m( )ξ , wo zaverßu[ dovedennq teoremy.
Takym çynom, qkwo linijna avtonomna systema (2) ma[ p periodyçnyx po t
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
338 I. I. KOROL|
rozv’qzkiv z periodamy vidpovidno T1, … , Tp , to, zastosovugçy navedeni vywe
mirkuvannq pry T = Ti , i = 1, p, moΩemo doslidyty naqvnist\ i pobuduvaty roz-
v’qzky nelinijno] systemy dlq koΩnoho z periodiv Ti .
2. Kvazilinijni avtonomni systemy. Doslidymo pytannq isnuvannq i nably-
Ωeno] pobudovy T-periodyçnoho rozv’qzku kvazilinijno] avtonomno] systemy
zvyçajnyx dyferencial\nyx rivnqn\
dy
dt
= Py + ε εg y( , ), (18)
de ε ∈ 0 0, ε[ ] — malyj dodatnyj parametr, y = ( , ˜)y y , y D∈ 1 � R
k
, ỹ D∈ 2 �
� R
n k–
, D = D1 × D2, y, g n∈R , P = diag( , )A B — stala ( )n n× -vymirna mat-
rycq, A — ( )k k× -matrycq vyhlqdu (3), pryçomu
A1) porodΩugça dlq (18) linijna systema
dy
dt
= Py , (19)
qka oderΩu[t\sq z (18) pry ε = 0, ma[ k-parametryçnu sim’g T-periodyçnyx
rozv’qzkiv;
B1) v oblasti D × 0 0, ε[ ] funkciq g y( , )ε [ vyznaçenog, neperervnog i vy-
konugt\sq umovy obmeΩenosti i Lipßycq
g y( , )ε ≤ M, g y g y( , ) – ( , )′ ′′ε ε ≤ K y y′ ′′– .
Rozhlqnemo zadaçu znaxodΩennq periodyçnoho z zadanym fiksovanym perio-
dom T rozv’qzku avtonomno] systemy (18), qkyj pry ε = 0 peretvorg[t\sq na
odyn iz T-periodyçnyx rozv’qzkiv
y t( ) = eAtξ ,
(20)
˜( )y t = 0
porodΩugço] systemy (19).
Z metog vidßukannq T-periodyçnoho rozv’qzku systemy (18) pobudu[mo re-
kurentnu poslidovnist\
y tm +1( , , )ξ ε =
= y t0( , , )ξ ε + ε ξ ε ε ξ ε εe g y s ds t
T
e g y s dsA t s
m
Tt
A t s
m
( – ) ( – )( , , ), – ( , , ),( ) ( )
∫∫
00
,
˜ ( , , )y tm +1 ξ ε = ε ξ ε ε
0
t
B t s
me g y s ds∫
( )( – ) ˜ ( , , ), +
(21)
+ I e e e g y s dsn k
BT BT
T
B t s
m–
– ( – )– ˜ ( , , ),( ) ( )
∫
1
0
ξ ε ε ,
y t0( , , )ξ ε = eAtξ , ˜ ( , )y t0 ξ = 0, y y ym m m= ( , ˜ ) .
Qkwo ε0 [ dostatn\o malym, to navedeni vywe tverdΩennq [ spravedlyvymy
dlq systemy (18), poslidovnyx nablyΩen\ (21) i vykonugt\sq ocinky
y t y t∗( , , ) – ( , , )ξ ε ξ ε0 ≤ εβ , (22)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
DOSLIDÛENNQ PERIODYÇNYX ROZV’QZKIV NELINIJNYX AVTONOMNYX … 339
y t y tm
∗( , , ) – ( , , )ξ ε ξ ε ≤ ε ε ε βI Q Qn
m– –( ) ( )1
,
inf ( , )
ξ ∂ β
ξ ε
∈ ′D
mΓ > ε ε ε βH I Q Qn
m– –( ) ( )1 , m ∈N , (23)
de y t∗( , , )ξ ε = lim ( , , )
m
my t
→∞
ξ ε , Γm( , )ξ ε =df
0
T As
me g y s ds∫ ( )– ( , , ),ξ ε ε . Pry m = 0
dlq systemy (18) magt\ misce nastupni dostatni umovy isnuvannq rozv’qzku.
Teorema 5. Nexaj dlq systemy (18) vykonugt\sq umovy A1, B1 i pry vsix
ε ∈ 0 1, ε[ ] vidobraΩennq Γ0( , )ξ ε ma[ v oblasti ′Dβ � Dβ izol\ovanu osobly-
vu toçku ξ = ξ0 nenul\ovoho indeksu.
Todi isnu[ take ε0, wo pry vsix 0 < ε < ε0 systema (18) ma[ T -periodyç-
nyj rozv’qzok, qkyj pry ε = 0 peretvorg[t\sq v T-periodyçnyj rozv’q-
zokN(20) porodΩugço] linijno] systemy (18).
Dovedennq. Z (22) vyplyva[, wo pry m = 0 nerivnist\ (23) nabyra[ vyhlqdu
inf ( , )
ξ β
ξ ε
∈∂ ′D
Γ0 > ε βH . (24)
V qkosti oblasti ′Dβ viz\memo kolo radiusa δ z centrom u toçci ξ0. Os-
kil\ky ξ0 [ izol\ovanog osoblyvog toçkog, to pry dostatn\o malyx δ oblast\
′Dβ ne bude mistyty inßyx osoblyvyx toçok vidobraΩennq Γ0( , )ξ ε , a tomu
inf ( , )
–ξ ξ δ
ξ ε
0
0=
Γ = η > 0. (25)
Z (24) i (25) vyplyva[, wo η > ε βH , otΩe, pry vsix 0 < ε < ε0, de ε0 =
=N min( , )ε ε1 2 i ε β2 H < η, isnu[ T -periodyçnyj rozv’qzok y = y t∗( , )ε systemy
(18) takyj, wo
y∗( , ) –0 0ε ϕ < δ,
y t y t∗( , ) – ( , )ε ε0 < εβ .
Z (21) vyplyva[, wo pry vsix ξ
∗ ∈Rk
, m ∈ 0 ∪ N funkci] y tm( , , )ξ 0 [ roz-
v’qzkamy porodΩugço] linijno] systemy (19), a otΩe, y t∗( , )0 teΩ [ rozv’qzkom
systemy (19).
1. Samojlenko A. M., Ronto N. Y. Çyslenno-analytyçeskye metod¥ yssledovanyq peryody-
çeskyx reßenyj. – Kyev: Vywa ßk., 1976. – 180 s.
2. Boholgbov N. N., Mytropol\skyj G. A. Asymptotyçeskye metod¥ v teoryy nelynejn¥x ko-
lebanyj. – M.: Nauka, 1974. – 503 s.
3. Hrebenykov E. A., Rqbov G. A. Konstruktyvn¥e metod¥ analyza nelynejn¥x system. – M.:
Nauka, 1979. – 432 s.
4. Malkyn Y. H. Nekotor¥e zadaçy teoryy nelynejn¥x kolebanyj. – M.: Hostexyzdat, 1956. –
491 s.
5. Samojlenko A. M., Ronto N. Y. Çyslenno-analytyçeskye metod¥ yssledovanyq reßenyj
kraev¥x zadaç. – Kyev: Nauk. dumka, 1985. – 224 s.
6. Boichuk A. A., Samoilenko A. M. Generalized inverse operators and Fredholm boundary value
problems. – Utrecht; Boston: VSP, 2004. – 320 p.
7. Çujko S. M. Oblast\ sxodymosty yteracyonnoj procedur¥ dlq avtonomnoj kraevoj zadaçy
// Nelinijni kolyvannq. – 2006. – 9, # 3. – S. 416 – 432.
8. Korol\ I. I., Perestgk M. O. We raz pro çysel\no-analityçnyj metod poslidovnyx perio-
dyçnyx nablyΩen\ A. M. Samojlenka // Ukr. mat. Ωurn. – 2006. – 58, # 4. – S. 472 – 489.
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OderΩano 23.10.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
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| id | umjimathkievua-article-3159 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:37:21Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fc/4aff0121343d3012818435b4b5190dfc.pdf |
| spelling | umjimathkievua-article-31592020-03-18T19:47:10Z Investigation of the periodic solutions of nonlinear autonomous systems in the critical case Дослідження періодичних розв'язків нелінійних автономних систем у критичному випадку Korol', I. I. Король, І. І. We analyze the conditions of existence and the numerical-analytic method for the approximate construction of periodic solutions of nonlinear autonomous systems of differential equations in the critical case. Исследуются условия существования и численно-аналитический метод приближенного построения периодических решений нелинейных автономных дифференциальных систем в критическом случае. Institute of Mathematics, NAS of Ukraine 2008-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3159 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 3 (2008); 332–339 Український математичний журнал; Том 60 № 3 (2008); 332–339 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3159/3066 https://umj.imath.kiev.ua/index.php/umj/article/view/3159/3067 Copyright (c) 2008 Korol' I. I. |
| spellingShingle | Korol', I. I. Король, І. І. Investigation of the periodic solutions of nonlinear autonomous systems in the critical case |
| title | Investigation of the periodic solutions of nonlinear autonomous systems in the critical case |
| title_alt | Дослідження періодичних розв'язків нелінійних автономних систем у критичному випадку |
| title_full | Investigation of the periodic solutions of nonlinear autonomous systems in the critical case |
| title_fullStr | Investigation of the periodic solutions of nonlinear autonomous systems in the critical case |
| title_full_unstemmed | Investigation of the periodic solutions of nonlinear autonomous systems in the critical case |
| title_short | Investigation of the periodic solutions of nonlinear autonomous systems in the critical case |
| title_sort | investigation of the periodic solutions of nonlinear autonomous systems in the critical case |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3159 |
| work_keys_str_mv | AT korol039ii investigationoftheperiodicsolutionsofnonlinearautonomoussystemsinthecriticalcase AT korolʹíí investigationoftheperiodicsolutionsofnonlinearautonomoussystemsinthecriticalcase AT korol039ii doslídžennâperíodičnihrozv039âzkívnelíníjnihavtonomnihsistemukritičnomuvipadku AT korolʹíí doslídžennâperíodičnihrozv039âzkívnelíníjnihavtonomnihsistemukritičnomuvipadku |