On Periodic Solutions of One Class of Systems of Differential Equations
We study the problem of the existence of periodic solutions of two-dimensional linear inhomogeneous periodic systems of differential equations for which the corresponding homogeneous system is Hamiltonian. We propose a new numerical-analytic algorithm for the investigation of the problem of the exis...
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| Дата: | 2005 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2005
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509735609958400 |
|---|---|
| author | Korol', I. I. Король, І. І. |
| author_facet | Korol', I. I. Король, І. І. |
| author_sort | Korol', I. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:00:05Z |
| description | We study the problem of the existence of periodic solutions of two-dimensional linear inhomogeneous periodic systems of differential equations for which the corresponding homogeneous system is Hamiltonian. We propose a new numerical-analytic algorithm for the investigation of the problem of the existence of periodic solutions of two-dimensional nonlinear differential systems with Hamiltonian linear part and their construction. The results obtained are generalized to systems of higher orders. |
| first_indexed | 2026-03-24T02:45:50Z |
| format | Article |
| fulltext |
UDK 517.925
I. I. Korol\ (Ky]v. nac. un-t im. T. Íevçenka)
PRO PERIODYÇNI ROZV’QZKY ODNOHO KLASU
SYSTEM DYFERENCIAL|NYX RIVNQN|
We study the problems of the existence of periodic solutions of two-dimensional linear inhomogeneous
periodic systems of differential equations whose corresponding homogeneous system possesses the
Hamiltonian properties. We suggest a new numerical-analytic algorithm that enables one to investigate
the existence and to construct periodic solutions of two-dimensional nonlinear differential systems with
the Hamiltonian linear part. The results obtained are generalized to systems of higher orders.
Vyvçagt\sq pytannq isnuvannq periodyçnyx rozv’qzkiv dvovymirnyx linijnyx neodnoridnyx pe-
riodyçnyx system dyferencial\nyx rivnqn\, u qkyx vidpovidna odnoridna systema [ hamil\tono-
vog. Zaproponovano novyj çysel\no-analityçnyj alhorytm doslidΩennq isnuvannq i pobudovy
periodyçnyx rozv’qzkiv dvovymirnyx nelinijnyx dyferencial\nyx system iz hamil\tonovog li-
nijnog çastynog. OderΩani rezul\taty uzahal\neno na systemy vywyx porqdkiv.
Analiz riznomanitnyx procesiv u mexanici, fizyci, biolohi], nebesnij mexanici ta
inßyx haluzqx nauky i texniky pryvodyt\ do neobxidnosti vyvçennq periodyçnyx
rozv’qzkiv riznyx typiv dyferencial\nyx rivnqn\ ta ]x system. Zokrema, velyku
kil\kist\ robit prysvqçeno pytannqm isnuvannq ta pobudovy periodyçnyx roz-
v’qzkiv, rozrobleno ßyrokyj spektr zasobiv dlq ]x doslidΩennq [1 – 4]. Sered
nyx moΩna vydilyty çysel\no-analityçnyj metod poslidovnyx periodyçnyx
nablyΩen\ [5, 6], ideg qkoho zhodom bulo pereneseno na ßyrokyj klas zadaç
[7 – 10].
Dana stattq [ prodovΩennqm doslidΩen\ z dano] tematyky. U nij rozhlqda-
gt\sq periodyçni rozv’qzky linijnyx neodnoridnyx system, a takoΩ dlq system
dyferencial\nyx rivnqn\ druhoho porqdku, linijna çastyna qkyx hamil\tonova,
na bazi çysel\no-analityçnoho metodu A. M. Samojlenka rozrobleno novyj al-
horytm doslidΩennq periodyçnyx rozv’qzkiv.
1. Linijni dvovymirni systemy. Rozhlqnemo linijnu neodnoridnu dvovymir-
nu ω-periodyçnu systemu
dx
dt
= P ( t ) x + f ( t ) , (1)
koly vidpovidna odnoridna systema [ hamil\tonovog:
P ( t ) =
0
0
p t
p t
( )
( )−
, (2)
x, f ∈ R2, t ∈ R, f ( t ) , p ( t ) — neperervni ω-periodyçni funkci].
Vidomo [11, c. 77], wo rozv’qzok systemy (1), qkyj pry t = 0 proxodyt\ çerez
toçku ξ = ( ξ1, ξ2 ) , ma[ vyhlqd
x ( t ) = X ( t ) ξ + X t s f s ds
t
( , ) ( )
0
∫ ,
de X ( t ) — matrycant vidpovidno] (1) odnoridno] systemy,
X ( t ) =
cos ( , ) sin ( , )
sin ( , ) cos ( , )
ϕ ϕ
ϕ ϕ
0 0
0 0
t t
t t−
,
X ( t, s ) = X t X s( ) ( )−1 =
cos ( , ) sin ( , )
sin ( , ) cos ( , )
ϕ ϕ
ϕ ϕ
s t s t
s t s t−
,
© I. I. KOROL|, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4 483
484 I. I. KOROL|
ϕ ( s, t ) = p d
s
t
( )τ τ∫ .
Oskil\ky p ( t ) — neperervna ω-periodyçna funkciq, to moΩemo zapysaty ]]
u vyhlqdi
p ( t ) = p a
j
t b
j
t
j
j j0
1
2 2+
+
=
∞
∑ cos sin
π
ω
π
ω
,
de
p0 =
1
0ω
ω
p s ds( )∫ , aj =
2 2
0
ω
π
ω
ω
p s
js
ds( )cos
∫ , bj = 2 2
0
ω
π
ω
ω
p s
js
ds( )sin
∫ .
Rozhlqnemo pytannq isnuvannq ω-periodyçnyx rozv’qzkiv systemy (1).
Lema,1. 1. Qkwo p ( t ) take, wo
p t dt( )
0
ω
∫ ≠ 2
π
l
, (3)
de l — cile çyslo, to isnu[ [dynyj ω-periodyçnyj rozv’qzok systemy (1).
2. U rezonansnomu vypadku, tobto qkwo
p t dt( )
0
ω
∫ = 2
π
l
, (4)
systema (1) ma[ ω-periodyçni rozv’qzky todi i til\ky todi, koly
X s f s ds−∫ 1
0
( ) ( )
ω
= 0. (5)
Pry c\omu dlq dovil\no] toçky ξ = ( ξ1, ξ 2 ) isnu[ ω -periodyçnyj rozv’qzok
systemy (1) z poçatkovog umovog x ( 0 ) = ξ .
Dovedennq. Qkwo umovu (3) vykonano, to mul\typlikatory systemy (1) vid-
minni vid odynyci. Zhidno z teoremog 23.1 [11] systema (1) pry c\omu ma[ [dynyj
ω-periodyçnyj rozv’qzok vyhlqdu
x ( t ) = G t f d( , ) ( )τ τ τ
ω
0
∫ ,
de G ( t, τ ) — funkciq Hrina,
G ( t, τ ) =
X t E X X t
X t E X X t
( )( ( )) ( ), ,
( )( ( )) ( ), .
− ≤ ≤ ≤
+ − ≤ < ≤
− −
− −
ω τ τ ω
ω ω τ τ ω
1 1
1 1
0
0
Qkwo vykonu[t\sq umova (4), to, vraxovugçy, wo systema (1) [ samosprqΩe-
nog, tverdΩennq lemy bezposeredn\o vyplyva[ z teoremy 23.2 [11].
Lema,2. Nexaj dlq systemy (1) vykonu[t\sq umova (4). Todi zavΩdy isnu[
ω-periodyçna funkciq θ ( t ) taka, wo systema
dx
dt
= P ( t ) x + f ( t ) – θ ( t ) (6)
ma[ dvoparametryçnu sim’g ω-periodyçnyx rozv’qzkiv.
Dovedennq. Umova ortohonal\nosti (5) dlq systemy (6) zapyßet\sq tak:
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
PRO PERIODYÇNI ROZV’QZKY ODNOHO KLASU SYSTEM … 485
X s f s s ds− −∫ 1
0
( ) ( ) ( )( )θ
ω
= 0. (7)
Budemo ßukaty θ ( t ) u vyhlqdi
θ ( t ) = X ( t ) µ , (8)
de µ — deqkyj stalyj dvovymirnyj vektor.
Oçevydno, wo [dynym rozv’qzkom rivnqnnq (7) [
µ0 =
1 1
0ω
ω
X s f s ds−∫ ( ) ( ) .
OtΩe, zhidno z lemogK1 systema (6), (8) pry µ = µ0 ma[ ω-periodyçni roz-
v’qzky, qki utvorggt\ dvoparametryçnu sim’g.
Lema,3. Nexaj poslidovnist\ neperervnyx pry t ∈ [ 0, ω ] funkcij rm ( t ) za-
da[t\sq rekurentnym spivvidnoßennqm
rm ( t ) = 1 1
2
0
1
2−
+ −− −∫ ∫t
t r s ds
t
t r s dsm
t
m
tω ω
ω
ω
( ) ( ) , (9)
r0 ( t ) = 1, m = 1, 2, … .
Todi pry vsix cilyx m ≥ 2, t ∈ [ 0, ω ] vykonugt\sq ocinky
rm ( t ) ≤ q r tm−1
1( ), q =
2
15
ω . (10)
Dovedennq. Zhidno z (9) otrymu[mo
r1 ( t ) =
2t t( )ω
ω
−
,
r2 ( t ) = 1 1
2
0
−
∫t
t r s ds
t
ω
( ) +
t
t r s ds
tω
ω
ω
− ∫ 1
2( ) = r1 ( t ) h ( t ),
de
h ( t ) =
1
30
6 15 10 6 94 3 2 4 3 2 2 3 4
ω
ω ω ω ω ω ωt t t t t t t− + + − + + +{ }.
Oskil\ky
max
[ , ]t ∈ 0 ω
h ( t ) = h
ω
2
= q,
to r2 ( t ) ≤ q r1 ( t ) . Za indukci[g moΩemo vstanovyty, wo ocinka (10) vykonu[t\-
sq pry vsix m ≥ 2.
Lemu dovedeno.
2. Periodyçni rozv’qzky nelinijnyx dvovymirnyx system. Rozhlqnemo
nelinijnu systemu dyferencial\nyx rivnqn\
dx
dt
= P ( t ) x + g ( t, x ) , (11)
de P ( t ) — matrycq vyhlqdu (2), dlq qko] vykonu[t\sq rivnist\ (4).
Prypustymo, wo v oblasti
( t, x ) ∈ Ω = R × D, D = x r x R0 ≤ ≤ ≤{ },
x = x x1
2
2
2+ ,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
486 I. I. KOROL|
systema (11) zadovol\nq[ nastupni umovy:
funkciq g ( t, x ) [ vyznaçenog, neperervnog, ω-periodyçnog po t i
sup ( , )
( , )t x
g t x
∈Ω
= M ; (12)
isnu[ nevid’[mna stala K taka, wo pry vsix ( , )t x′ , ( , )t x′′ ∈ Ω
g t x g t x( , ) ( , )′ − ′′ ≤ K x x′ − ′′ ; (13)
pry c\omu
M ≤
R r−
ω
, q K < 1. (14)
Doslidymo pytannq isnuvannq i pobudovy periodyçnyx rozv’qzkiv systemy (11).
Rozhlqnemo poslidovnist\ ω-periodyçnyx funkcij
xm ( t, ξ ) = x t X t s g s x s X s g x d ds
t
m m0
0
1 1
0
1( , ) ( , ) ( , ( , )) ( , ) ( , ( , ))ξ ξ
ω
τ τ τ ξ τ
ω
+ −
∫ ∫− − ,
(15)
x0 ( t, ξ ) = X ( t ) ξ , m = 1, 2, … .
Lema,4. Funkciq x ( t ) [ ω -periodyçnym rozv’qzkom systemy (11) z poçat-
kovog umovog x ( 0 ) = ξ todi i til\ky todi, koly x ( t ) [ rozv’qzkom intehral\-
noho rivnqnnq
x ( t ) = X t X t s g s x s X s g x d ds
t
( ) ( , ) ( , ( )) ( , ) ( , ( ))ξ
ω
τ τ τ τ
ω
+ −
∫ ∫
0 0
1
(16)
i vykonu[t\sq umova ortohonal\nosti
X s g s x s ds−∫ 1
0
( ) ( , ( ))
ω
= 0. (17)
Dovedennq. Nexaj x ( t ) — ω-periodyçna vektor-funkciq taka, wo
dx
dt
≡ P ( t ) x ( t ) + g ( t, x ( t )) , x ( 0 ) = ξ .
Todi
x ( t ) = X t X t s g s x s ds
t
( ) ( , ) ( , ( ))ξ + ∫
0
. (18)
Oskil\ky X ( ω ) = E ,
x ( t + ω ) = x t X t X s g s x s ds( ) ( ) ( ) ( , ( ))+ −∫ 1
0
ω
,
to z ω-periodyçnosti x ( t ) vyplyva[ vykonannq umovy (17). Pry c\omu rivnqnnq
(16) i (18) zbihagt\sq, wo i dovodyt\ neobxidnist\ vykonannq umov (16), (17).
Dostatnq umova [ oçevydnog, oskil\ky v ω-periodyçnosti rozv’qzku rivnqn-
nq (16) moΩna perekonatysq bezposeredn\og perevirkog, a qkwo spravdΩu[t\sq
rivnist\ (17), to rivnqnnq (16) peretvorg[t\sq na (18).
Teorema,1. Nexaj systema (11) zadovol\nq[ umovy (12) – (14). Todi:
1) poslidovnist\ funkcij xm ( t, ξ ) vyhlqdu (15) pry m → ∞ rivnomirno
zbiha[t\sq vidnosno ( t, ξ ) ∈ R × D0 ,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
PRO PERIODYÇNI ROZV’QZKY ODNOHO KLASU SYSTEM … 487
D0 = ξ ω ξ ω
r
M
R
M+ ≤ ≤ −{ }2 2
,
do hranyçno] funkci] x t∗( , )ξ i pry vsix natural\nyx m spravdΩugt\sq ocinky
zbiΩnosti
x t x tm
∗ −( , ) ( , )ξ ξ ≤
( )
( )
qK
qK
Mr t
m
1 1−
; (19)
2) hranyçna funkciq x t∗( , )ξ [ ω-periodyçnog po t i nabuva[ poçatkovoho
znaçennq x∗( , )0 ξ = ξ ;
3) funkciq x t∗( ) = x t∗ ∗( , )ξ [ ω -periodyçnym rozv’qzkom systemy dyfe-
rencial\nyx rivnqn\ (11) todi i til\ky todi, koly toçka ξ = ξ*
[ rozv’qzkom
rivnqnnq
∆ ( ξ ) ≡
1 1
0ω
ξ
ω
X s g s x s ds− ∗∫ ( ) ( , ( , )) = 0. (20)
Dovedennq. Spoçatku navedemo deqki dopomiΩni vykladky. Vykorystovug-
çy intehral\nu formu nerivnosti Koßi – Bunqkovs\koho i beruçy do uvahy, wo
X t�( ) = X t−1( ), dlq dovil\no] neperervno] vektor-funkci] g ( t ) = ( g1 ( t ),
g2 ( t ) ) , 0 ≤ a ≤ b ≤ ω otrymu[mo
a
b
X t s g s ds∫ ( , ) ( ) ≤ b a X t s g s ds
a
b
− ∫ ( , ) ( ) 2 =
= b a X t s g s X t s g s ds
a
b
− 〈 〉∫ ( , ) ( ), ( , ) ( ) =
=
b a X t s X t s g s g s ds
a
b
− 〈 〉∫ �( , ) ( , ) ( ), ( ) = b a g s ds
a
b
− ∫ ( ) 2 . (21)
OtΩe,
0 0
1
t
X t s g s X s g d ds∫ ∫−
( , ) ( ) ( , ) ( )
ω
τ τ τ
ω
=
=
0 0
t
X t s g s ds t X t g d∫ ∫−( , ) ( ) ( , ) ( )
ω
τ τ τ
ω
≤
≤ 1
0
−
+∫ ∫t X t s g s ds t X t s g s ds
t
t
ω ω
ω
( , ) ( ) ( , ) ( ) ≤
≤ 1
0
2 2−
+ −∫ ∫t
t g s ds
t
t g s ds
t
tω ω
ω
ω
( ) ( ) . (22)
Vraxovugçy (12), (22), z (9), (15) oderΩu[mo ocinku
x t x t1 0( , ) ( , )ξ ξ− =
=
0
0 0
0
1
t
X t s g s x s X s g x d ds∫ ∫−
( , ) ( , ( , )) ( , ) ( , ( , ))ξ
ω
τ τ τ ξ τ
ω
≤
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
488 I. I. KOROL|
≤ 1
0
0
2
0
2−
+ −∫ ∫t
t g s x s ds
t
t g s x s ds
t
tω
ξ
ω
ω ξ
ω
( , ( , )) ( , ( , )) ≤
≤ M r1 ( t ) ≤
Mω
2
. (23)
Vyberemo dovil\nu toçku ξ z oblasti D0. Oskil\ky x t0( , )ξ = ξ , to z
(23) za pravylom trykutnyka oderΩymo nerivnosti
x t1( , )ξ ≥ x t x t x t0 1 0( , ) ( , ) ( , )ξ ξ ξ− − ≥ r,
x t1( , )ξ ≤ x t x t1 0( , ) ( , )ξ ξ− + x t0( , )ξ ≤ R .
Takym çynom, x1 ( t, ξ ) ∈ D . Za indukci[g moΩna perekonatysq, wo pry vsix
m ≥ 1 ma[mo
x t x tm( , ) ( , )ξ ξ− 0 ≤ M r1 ( t ) ≤
Mω
2
, (24)
a otΩe, r ≤ x tm( , )ξ ≤ R , i vsi çleny poslidovnosti (15) naleΩat\ oblasti D.
Za dopomohog (13), (22) ocinymo riznycg susidnix çleniv poslidovnosti (15):
x t x tm m+ −1( , ) ( , )ξ ξ =
=
0
1
t
m mX t s g s x s g s x s∫ −
−( , ) ( , ( , )) ( , ( , ))ξ ξ –
– 1
0
1ω
τ τ τ ξ τ τ ξ τ
ω
∫ −{ }
−X s g x g x d dsm m( , ) ( , ( , )) ( , ( , )) ≤
≤ 1
0
1
2−
−∫ −
t
t g s x s g s x s ds
t
m mω
ξ ξ( , ( , )) ( , ( , )) +
+
t
t g s x s g s x s ds
t
m mω
ω ξ ξ
ω
− −∫ −( , ( , )) ( , ( , ))1
2
.
Z umovy Lipßycq (13) ma[mo
x t x tm m+ −1( , ) ( , )ξ ξ ≤ K
t
t x s x s ds
t
m m
−
−∫ −1
0
1
2
ω
ξ ξ( , ) ( , ) +
+
t
t x s x s ds
t
m mω
ω ξ ξ
ω
− −
∫ −( , ) ( , )1
2
. (25)
Vraxovugçy (9), (10), z (23), (25) oderΩu[mo
x t x t2 1( , ) ( , )ξ ξ− ≤ K
t
t x s x s ds
t
−
−∫1
0
1 0
2
ω
ξ ξ( , ) ( , ) +
+
t
t x s x s ds
tω
ω ξ ξ
ω
− −
∫ 1 0
2( , ) ( , ) ≤ K
t
t r s ds
t
−
∫1
0
1
2
ω
( ) +
+
t
t r s ds
tω
ω
ω
−
∫ 1
2( ) ≤ K Mr t2( ) ≤ qK Mr t1( ) .
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
PRO PERIODYÇNI ROZV’QZKY ODNOHO KLASU SYSTEM … 489
Metodom matematyçno] indukci] otrymu[mo ocinku
x t x tm m+ −1( , ) ( , )ξ ξ ≤ ( ) ( )qK Mr tm
1 ,
a tomu dlq vsix m ∈ N
, j ≥ 1 ma[mo
x t x tm j m+ −( , ) ( , )ξ ξ ≤
k
j
m k m kx t x t
=
−
+ + +∑ −
0
1
1( , ) ( , )ξ ξ ≤
≤
k
j
m kqK Mr t
=
−
+∑
0
1
1( ) ( ) ≤ ( ) ( ) ( )qK qK Mr tm
k
j
k
=
−
∑
0
1
1 . (26)
Z (14) vyplyva[, wo poslidovnist\ (15) rivnomirno zbiha[t\sq pry m → ∞ v
oblasti ( t, ξ ) ∈ R × D0 do hranyçno] funkci] x*
( t, ξ ) . Perexodqçy v (26) do
hranyci pry j → ∞ , oderΩu[mo ocinku (19).
Oskil\ky vsi funkci] xm ( t, ξ ) poslidovnosti (15) periodyçni po t z periodom
ω i pry t = 0 nabuvagt\ znaçennq xm ( 0, ξ ) = ξ, to i hranyçna funkciq x*
( t, ξ )
teΩ [ ω-periodyçnog i x*
( 0, ξ ) = ξ .
Perexodqçy v (15) do hranyci pry m → ∞ , baçymo, wo hranyçna funkciq
x*
( t, ξ ) [ rozv’qzkom intehral\noho rivnqnnq
x ( t ) = X t X t s g s x s X s g x d ds
t
( ) ( , ) ( , ( )) ( , ) ( , ( ))ξ
ω
τ τ τ τ
ω
+ −
∫ ∫
0 0
1 ,
a tomu za lemogK4 [ ω-periodyçnym rozv’qzkom systemy (11) todi i til\ky todi,
koly vykonu[t\sq umova (17).
Teoremu dovedeno.
Rozhlqnemo moΩlyvist\ vyvçennq pytannq isnuvannq ω-periodyçnyx roz-
v’qzkiv systemy (11) bez znaxodΩennq hranyçno] funkci] x*
( t, ξ ) . Dlq c\oho
rozhlqnemo nablyΩeni vyznaçal\ni rivnqnnq
∆m ( ξ ) ≡
1 1
0ω
ξ
ω
X s g s x s dsm
−∫ ( ) ( , ( , )) = 0. (27)
Nastupne tverdΩennq na pidstavi analizu koreniv rivnqnnq (27) da[ moΩly-
vist\ robyty vysnovky pro isnuvannq rozv’qzkiv rivnqnnq (20).
Teorema,2. Nexaj systema (11) zadovol\nq[ umovy (12) – (14) i, krim toho:
1) pry deqkomu fiksovanomu natural\nomu m rivnqnnq (27) ma[ izol\ova-
nyj rozv’qzok ξ = ξ 0 m ;
2) indeks osoblyvo] toçky ξ 0 m vidobraΩennq ∆ m , porodΩenoho (27), ne
dorivng[ nulg;
3) isnu[ opukla zamknena oblast\ D1 ⊂ D0 taka, wo ξ 0 m [ v D1 [dy-
nym rozv’qzkom rivnqnnq (27) i na ]] hranyci ∂ D1 vykonu[t\sq nerivnist\
inf ( )
ξ
ξ
∈∂D m
1
∆ >
( )qK M
qK
m+
−
1
1
. (28)
Todi systema (11) ma[ [dynyj ω-periodyçnyj rozv’qzok x = x*
( t ) = x*
( t,
ξ*
) z poçatkovog umovog x*
( 0 ) = ξ*
, de ξ* ∈ D1
.
Dovedennq. Vykorystovugçy ocinky (19), (21), pry m ≥ 1 oderΩu[mo spiv-
vidnoßennq
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490 I. I. KOROL|
∆ ∆( ) ( )ξ ξ− m =
1 1
0ω
ξ ξ
ω
X s g s x s g s x s dsm
− ∗ −{ }∫ ( ) ( , ( , )) ( , ( , )) ≤
≤ 1
0
2
ω
ξ ξ
ω
∫ ∗ −g s x s g s x s dsm( , ( , )) ( , ( , )) ≤
≤
( )
( )
( )
qK MK
qK
r s ds
m
1
0
1
2
− ∫ω
ω
≤
( )qK M
qK
m+
−
1
1
.
Dali, vraxovugçy ostanng ocinku, za sxemog dovedennq teoremy 3.1 [7] moΩemo
pokazaty homotopnist\ poliv ∆ ( ξ ) i ∆m ( ξ ) , wo zaverßu[ dovedennq teoremy.
Proilgstru[mo praktyçne zastosuvannq rozroblenoho alhorytmu.
Pryklad. Nexaj potribno znajty 2π-periodyçnyj rozv’qzok systemy dyfe-
rencial\nyx rivnqn\
dx
dt
1 = x
t
x x2 1 2
2
16
1
8
1
32
− − +sin( )
,
(29)
dx
dt
2 = − + −x x x
t
1 1 2
1
5
2
40
sin( )
v oblasti ( t, x ) ∈ Ω = R × D, D = x x x x x= + ≤{ }( , )1 2 1
2
2
2 1 . NevaΩko pere-
konatysq, wo v Ω funkciq
g ( t, x ) = col − − + −
sin( )
,
sin( )t
x x x x
t
16
1
8
1
32
1
5
2
401 2
2
1 2
zadovol\nq[ umovy (12) – (14) zi stalymy M = 0,149, K = 253 40/ .
Poslidovni 2π-periodyçni nablyΩennq do rozv’qzkiv systemy (29), pobudova-
ni za formulog (15), magt\ vyhlqd
x01 ( t, ξ ) = ξ ξ1 2cos( ) sin( )t t+ ,
x02 ( t, ξ ) = – ξ ξ1 2sin( ) cos( )t t+ ,
xm1 ( t, ξ ) = x t
t t
01
2
120
7
480
( , )
sin( ) sin( )ξ + + –
–
0
11
16
t
mt s s x s
∫
−
−cos( ) sin( ) ( , ), ξ
+
cos( ) ( , ),t s x sm− −1 2
2
8
ξ
–
–
sin( ) ( , ) ( , ), ,t s x s x s
ds
m m−
− −11 1 2
5
ξ ξ
–
t t xm
2 160
2
11
π
σ σ σ ξπ
∫
−
−cos( ) sin( ) ( , ),
+
+
cos( ) ( , ),t xm− −σ σ ξ1 2
2
8
+
sin( ) ( , ) ( , ), ,t x x
d
m m−
− −σ σ ξ σ ξ
σ11 1 2
5
,
(30)
xm2 ( t, ξ ) = x t
t t
02
2
60
7
480
1
32
( , )
cos( ) cos( )ξ + + − +
+
0
11
16
t
mt s s x s
∫
−
−sin( ) sin( ) ( , ), ξ
+
sin( )
( , ),
t s
x sm
−
−8 1 2
2 ξ +
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PRO PERIODYÇNI ROZV’QZKY ODNOHO KLASU SYSTEM … 491
+
cos( ) ( , ) ( , ), ,t s x s x s
ds
m m−
− −11 1 2
5
ξ ξ
+
t t xm
2 160
2
11
π
σ σ σ ξπ
∫
−
−sin( ) sin( ) ( , ),
+
+
sin( ) ( , ),t xm− −σ σ ξ1 2
2
8
+
cos( ) ( , ) ( , ), ,t x x
dm m−
− −σ σ ξ σ ξ
σ1 1 1 2
5
.
NablyΩeni vyznaçal\ni funkci] ∆m ( ξ ) , znajdeni za formulog (27), magt\ vy-
hlqd
∆m ( ξ ) = 1
2
2
32 8 5
16 8
0
2
1 2
2
1 2
0
2 2
1 2
2
1 2
π
ξ ξ ξ ξ
ξ ξ ξ
π
π
∫
∫
+ +
+ −
sin( ) ( , ) cos( ) ( , ) sin( ) ( , ) ( , )
sin ( ) ( , ) sin( ) ( , ) cos( ) ( , ) (
s x s s x s s x s x s
ds
s x s s x s s x s x s
m m m m
m m m m ,, )ξ
5
ds
.
Pry m = 2 otrymu[mo dva rozv’qzky rivnqnnq ∆m ( ξ ) = 0:
�
ξ2 = ( ),
� �
ξ ξ21 22 = ( , , , )3 4895466056 10 0 4999999999611⋅ − ,
ξ2 = ( ),ξ ξ21 22 = ( , , , )2 9438729962 10 0 006509944890911⋅ − .
Pidstavlqgçy ]x u (30), oderΩu[mo pry m = 1 vidpovidno dva perßi nably-
Ωennq:
�
x t1( ) = x t1( ),
�
ξ = ( )( ), ( )
� �
x t x t11 12 = ( )( , ), ( , )x t x t11 12
� �
ξ ξ
i
x t1( ) = x t1( ), ξ = ( )( ), ( )x t x t11 12 = ( )( , ), ( , )x t x t11 12ξ ξ ,
a pry m = 2 — dva druhi nablyΩennq:
�
x t2( ) = x t2( ),
�
ξ = ( )( ), ( )
� �
x t x t21 22 = ( )( , ), ( , )x t x t21 22
� �
ξ ξ
i
x t2( ) = x t2( ), ξ = ( )( ), ( )x t x t21 22 = ( )( , ), ( , )x t x t21 22ξ ξ .
Pry m = 3 nablyΩene vyznaçal\ne rivnqnnq ∆m ( ξ ) = 0 takoΩ ma[ dva roz-
v’qzky:
�
ξ3 = ( ),
� �
ξ ξ31 32 = ( , , , )1 25074999659 10 0 49999999999911⋅ − ,
ξ3 = ( ),ξ ξ31 32 = ( , , , )− ⋅ −2 2087970574 10 0 0091864696781611 .
Obçyslggçy spivvidnoßennq (30) pry m = 3 i pidstavlqgçy v nyx znaçennq
�
ξ3
i ξ3, otrymu[mo vidpovidni ]m treti nablyΩennq:
�
x t3( ) = x t3( ),
�
ξ = ( )( ), ( )
� �
x t x t31 32 = ( )( , ), ( , )x t x t31 32
� �
ξ ξ
i
x t3( ) = x t3( ), ξ = ( )( ), ( )x t x t31 32 = ( )( , ), ( , )x t x t31 32ξ ξ
do 2π-periodyçnoho rozv’qzku systemy (29).
Pidstavyvßy oderΩani nablyΩennq u systemu (29), oderΩymo, wo vidxylen-
nq dlq
�
x t1( ) ,
�
x t2( ),
�
x t3( ) ne perevywu[ 7 ⋅ 10–10, tobto poxybku obçyslen\, a
dlq x t1( ) , x t2( ), x t3( ) ne perevywu[ vidpovidno 1 307 10 3, ⋅ − , 8 005 10 5, ⋅ − ,
3 291 10 6, ⋅ − .
ZauvaΩymo, wo x*
( t ) = ( sin ( t ) / 2 , cos ( t ) / 2 ) [ toçnym rozv’qzkom systemy
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492 I. I. KOROL|
(29). Jomu vidpovidagt\ poslidovni nablyΩennq
�
x t1( ) ,
�
x t2( ),
�
x t3( ), i pry c\omu
]x pokoordynatni vidxylennq ne perevywugt\ 3 10 11⋅ − .
3. Systemy z malym parametrom. Rozhlqnemo systemu
dx
dt
= P ( t ) x + f ( t ) + ε g ( t, x ) , (31)
de ε — malyj dodatnyj parametr, ( t, x ) ∈ Ω , P ( t ) — matrycq vyhlqdu (2), (4),
funkci] f ( t ) , g ( t, x ) neperervni za svo]my zminnymy, ω-periodyçni po t, f ( t )
zadovol\nq[ umovu ortohonal\nosti (5), a dlq g ( t, x ) vykonugt\sq umovy (12),
(13).
Pobudu[mo poslidovnist\ ω-periodyçnyx funkcij
ˆ ( , )x tm ξ = ˆ ( , ) ( , ) , ˆ ( , )( )x t X t s g s x s
t
m0
0
1ξ ε ξ+
∫ − –
– 1
0
1ω
τ τ τ ξ τ
ω
∫ −
X s g x d dsm( , ) , ˆ ( , )( ) , (32)
ˆ ( , )x t0 ξ = X t X t s f s
t
( ) ( , ) ( )ξ + ∫
0
, m = 1, 2, … .
Pry dostatn\o malyx ε dlq systemy (31) vykonugt\sq ocinky
ε M ≤
R r−
ω
, ε q K < 1,
a tomu teoremy 1 i 2 moΩut\ buty pereneseni na systemu (31), i pry c\omu neriv-
nosti (19) i (28) matymut\ vyhlqd
ˆ ( , ) ˆ ( , )x t x tm
∗ −ξ ξ ≤
( )
( )
ε
ε
εqK
qK
Mr t
m
1 1−
,
inf ˆ ( )
ξ
ξ
∈∂D m
1
∆ >
( )ε
ε
qK M
qK
m+
−
1
1
, (33)
de
ˆ ( , )x t∗ ξ = lim ˆ ( , )
m mx t
→∞
ξ , ˆ ( )∆m ξ =
1 1
0ω
ξ
ω
X s g s x s dsm
−∫ ( ) ( , ˆ ( , )) .
Krim toho, dlq ci[] systemy ma[ misce nastupne tverdΩennq.
Teorema,3. Nexaj systema (31) zadovol\nq[ navedeni vywe umovy i vidobra-
Ωennq
ˆ ( )∆0 ξ =
1 1
0
0ω
ξ
ω
X s g s x s ds−∫ ( ) ( , ˆ ( , ))
ma[ v oblasti D0 izol\ovanu osoblyvu toçku ξ = ξ0 nenul\ovoho indeksu.
Todi isnu[ take ε0, wo pry vsix 0 < ε < ε0 systema (31) ma[ ω-perio-
dyçnyj rozv’qzok.
Dovedennq. Dlq poslidovnosti (32) za indukci[g moΩemo oderΩaty ocinky,
analohiçni do (24):
ˆ ( , ) ˆ ( , )x t x tm ξ ξ− 0 ≤ ε M r1 ( t ) ,
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PRO PERIODYÇNI ROZV’QZKY ODNOHO KLASU SYSTEM … 493
a tomu, perexodqçy do hranyci pry m → ∞ , ma[mo
ˆ ( , ) ˆ ( , )x t x t∗ −ξ ξ0 ≤ ε M r1 ( t ) .
Takym çynom, pry m = 0 zaming[mo (33) na nerivnist\
inf ˆ ( )
ξ
ξ
∈∂D1
0∆ > ε q K M . (34)
Viz\memo v qkosti oblasti D1 kolo radiusa ρ z centrom u toçci ξ0 .
Oskil\ky ξ0 [ izol\ovanog osoblyvog toçkog, to pry dostatn\o malomu ρ v
D1 nema[ inßyx osoblyvyx toçok vodobraΩennq
ˆ ( )∆0 ξ i
inf ˆ ( )
ξ ξ ρ
ξ
− =0
0∆ = η > 0.
Todi nerivnist\ (34) zapyßet\sq tak:
inf ˆ ( )
ξ ξ ρ
ξ
− =0
0∆ = η > ε q K M .
OtΩe, pry
ε < ε0 =
η
qK M
isnu[ ω-periodyçnyj rozv’qzok ˆ( , )x t ξ systemy (31).
4. n - Vymirni systemy. OderΩani vywe rezul\taty moΩna uzahal\nyty na
systemy
dy
dt
= A ( t ) y + F ( t ) , (35)
de F ( t ) — neperervna ω-periodyçna vektor-funkciq, F ∈ Rn, A ( t ) [ nepererv-
nog ω-periodyçnog kososymetryçnog ( n × n ) -matryceg i zadovol\nq[ umovu
Lappo – Danylevs\koho:
A t A s ds
t
( ) ( )⋅ ∫
0
=
0
t
A s ds A t∫ ⋅( ) ( ).
Vidomo [11, c. 117], wo pry c\omu matrycant Y ( t ) vidpovidno] odnoridno] sys-
temy
dy
dt
= A ( t ) y (36)
ma[ vyhlqd
Y ( t ) = e
A s ds
t
( )
0∫ . (37)
Oçevydno, wo matrycq
A0 =
0
ω
∫ A s ds( )
teΩ [ kososymetryçnog, a tomu [12, c. 118] isnu[ taka dijsna ortohonal\na mat-
rycq S ( )( )S S S= =−� 1 , wo
A0 = SZ S−1, Z = diag
0
0
0
0
0 0
1
1
p
p
p
p
d
d−
…
−
…
, , , , , .
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494 I. I. KOROL|
OtΩe, dijsna çastyna vsix vlasnyx znaçen\ λj matryci A0 dorivng[ nulg:
λj = i γj
, i = −1, γj =
± =
+ ≤ ≤
p j d
d j n
j , , ,
, .
1
0 2 1
Pry c\omu matrycq monodromi] Y ( ω ) ma[ vyhlqd
Y ( ω ) = Se SZ −1,
eZ = diag
cos( ) sin( )
sin( ) cos( )
, ,
cos( ) sin( )
sin( ) cos( )
, , ,
p p
p p
p p
p p
d d
d d
1 1
1 1
1 1
−
…
−
…
,
a mul\typlikatoramy [ ρj = cos γj + i sin γj , j = 1, n .
Oçevydno, wo koΩnomu λj = i 2 π lj , de lj — deqki cili çysla, vidpovida[ ω-
periodyçnyj rozv’qzok ψj ( t ) systemy (35). Takym çynom, moΩemo sformulgva-
ty nastupni tverdΩennq.
Lema,5. Nexaj A ( t ) — neperervna ω-periodyçna kososymetryçna ( n × n ) -
matrycq, wo zadovol\nq[ umovu Lappo – Danylevs\koho.
Todi vsi vlasni znaçennq λ j matryci A0 magt\ nul\ovu dijsnu çastynu.
Krim toho,
1) qkwo pry vsix j = 1, n
Im ( λj ) ≠ 2 π l ,
de l — cile, to systema (35) ma[ [dynyj ω-periodyçnyj rozv’qzok;
2) qkwo dlq j = 1, ν, ν ≤ n, ma[mo
Im ( λj ) = 2 π lj , (38)
de lj — deqki cili çysla, to linijna odnoridna systema (36) ma[ ν linijno ne-
zaleΩnyx ω-periodyçnyx rozv’qzkiv ψ ψν1( ), , ( )t t… , a vidpovidna neodnoridna
systema (35) ma[ ω-periodyçni rozv’qzky todi i til\ky todi, koly
0
ω
ψ∫ 〈 〉j s F s ds( ), ( ) = 0, j = 1, ν,
i pry c\omu vony utvorggt\ ν-parametryçnu sim’g.
Lema,6. Nexaj vykonu[t\sq umova (38). Todi dlq systemy (35) zavΩdy
isnu[ ω-periodyçna funkciq Θ ( t ) taka, wo systema
dy
dt
= A ( t ) y + F ( t ) – Θ ( t )
ma[ ν-parametryçnu sim’g ω-periodyçnyx rozv’qzkiv.
Dovodqt\sq ostanni dva tverdΩennq analohiçno do lemK1 i 2. Zokrema, za
funkcig Θ ( t ) moΩemo vzqty
Θ ( t ) =
1
0ω ν
ω
νY t Y s F s ds( ) ( ) ( )∫ � ,
de Y tν( ) — ( n × ν ) -matrycq, stovpcqmy qko] [ linijno nezaleΩni ω-periodyçni
rozv’qzky ψ ψν1( ), , ( )t t… systemy (36).
Krim toho, rozhlqnemo nelinijni n-vymirni ω-periodyçni systemy
dy
dt
= A ( t ) y + G ( t, y ) . (39)
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PRO PERIODYÇNI ROZV’QZKY ODNOHO KLASU SYSTEM … 495
Qkwo vsi mul\typlikatory dorivnggt\ odynyci, to navedeni vywe teoremy
zalyßagt\sq spravedlyvymy i dlq systemy (39). Pry c\omu ]] ω-periodyçnyj
rozv’qzok ßuka[t\sq qk hranycq y*
( t ) = y*
( t, ξ*
) poslidovnosti
ym ( t, ξ ) = y t Y t s G s y s Y s G y d ds
t
m m0
0
1
0
1
1( , ) ( , ) ( , ( , )) ( , ) , ( , )( )ξ ξ
ω
τ τ τ ξ τ
ω
+ −
∫ ∫− − ,
y0 ( t, ξ ) = Y ( t ) ξ , m = 1, 2, … ,
de Y ( t ) znaxodyt\sq zhidno z (37), Y ( t, s ) = Y ( t ) Y
–
1
( s) , a poçatkove znaçennq
y*
( 0 ) = ξ*
[ rozv’qzkom systemy rivnqn\
Y s G s y s ds− ∗∫ 1
0
( ) ( , ( , ))ξ
ω
= 0.
Vysnovky. U roboti doslidΩeno pytannq isnuvannq, [dynosti periodyçnyx
rozv’qzkiv odnoho klasu qk linijnyx neodnoridnyx, tak i nelinijnyx system dy-
ferencial\nyx rivnqn\. Rozrobleno alhorytmy nablyΩeno] pobudovy periodyç-
nyx rozv’qzkiv vidpovidnyx nelinijnyx system, vstanovleno neobxidni i dostatni
umovy isnuvannq rozv’qzku, znajdeno ocinky zbiΩnosti poslidovnyx nablyΩen\.
1. Boholgbov N. N., Mytropol\skyj G. A. Asymptotyçeskye metod¥ v teoryy nelynejn¥x
kolebanyj. – 4-e yzd. – M.: Nauka, 1974. – 503 s.
2. Hrebenykov E. A., Mytropol\skyj G. A., Rqbov G. A. Vvedenye v rezonansnug analytyçes-
kug dynamyku. – M.: Qnus-K, 1999. – 320 s.
3. Krasnosel\skyj M. A. Operator sdvyha po traektoryqm dyfferencyal\n¥x uravnenyj. –
M.: Nauka, 1966. – 332 s.
4. Samojlenko A. M., Laptynskyj V. N., KenΩebaev K. K. Konstruktyvn¥e metod¥ yssle-
dovanyq peryodyçeskyx y mnohotoçeçn¥x zadaç // Trud¥ Yn-ta matematyky NAN Ukrayn¥. –
1999. – 29. – 220 s.
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ob¥knovenn¥x dyfferencyal\n¥x uravnenyj. 1 // Ukr. mat. Ωurn. – 1965. – 17, # 4. –
S.K16 – 23.
6. Samojlenko A. M. Çyslenno-analytyçeskyj metod yssledovanyq peryodyçeskyx system
ob¥knovenn¥x dyfferencyal\n¥x uravnenyj. 2 // Tam Ωe. – 1966. – 18, # 2. – S. 9 – 18.
7. Samojlenko A. M., Ronto N. Y. Çyslenno-analytyçeskye metod¥ v teoryy kraev¥x zadaç
ob¥knovenn¥x dyfferencyal\n¥x uravnenyj. – Kyev: Nauk. dumka, 1992. – 280 s.
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stvyem. – Kyev: V¥wa ßk., 1987. – 288 s.
9. Mytropol\skyj G. A., Samojlenko A. M., Mart¥ngk D. Y. System¥ πvolgcyonn¥x uravne-
nyj s peryodyçeskymy y uslovno-peryodyçeskymy koπffycyentamy. – Kyev: Nauk. dumka,
1985. – 226 s.
10. Perestgk N. A. O peryodyçeskyx reßenyqx nekotor¥x system dyfferencyal\n¥x uravne-
nyj // Asymptotyçeskye y kaçestvenn¥e metod¥ v teoryy nelynejn¥x kolebanyj. – Kyev:
Yn-t matematyky AN URSR, 1971. – S. 136 – 146.
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OderΩano 05.04.2004
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
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| id | umjimathkievua-article-3615 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:45:50Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/40/e9074e7612d7e1e08b392eb4f81d6c40.pdf |
| spelling | umjimathkievua-article-36152020-03-18T20:00:05Z On Periodic Solutions of One Class of Systems of Differential Equations Про періодичні розв'язки одного класу систем диференціальних рівнянь Korol', I. I. Король, І. І. We study the problem of the existence of periodic solutions of two-dimensional linear inhomogeneous periodic systems of differential equations for which the corresponding homogeneous system is Hamiltonian. We propose a new numerical-analytic algorithm for the investigation of the problem of the existence of periodic solutions of two-dimensional nonlinear differential systems with Hamiltonian linear part and their construction. The results obtained are generalized to systems of higher orders. Вивчаються питання існування періодичних розв'язків двовимірних лінійних неоднорідних періодичних систем диференціальних рівнянь, у яких відповідна однорідна система є гамільтоно-вою. Запропоновано новий чисельно-аналітичний алгоритм дослідження існування і побудови періодичних розв'язків двовимірних нелінійних диференціальних систем із гамільтоновою лінійною частиною. Одержані результати узагальнено на системи вищих порядків. Institute of Mathematics, NAS of Ukraine 2005-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3615 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 4 (2005); 483–495 Український математичний журнал; Том 57 № 4 (2005); 483–495 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3615/3961 https://umj.imath.kiev.ua/index.php/umj/article/view/3615/3962 Copyright (c) 2005 Korol' I. I. |
| spellingShingle | Korol', I. I. Король, І. І. On Periodic Solutions of One Class of Systems of Differential Equations |
| title | On Periodic Solutions of One Class of Systems of Differential Equations |
| title_alt | Про періодичні розв'язки одного класу систем диференціальних рівнянь |
| title_full | On Periodic Solutions of One Class of Systems of Differential Equations |
| title_fullStr | On Periodic Solutions of One Class of Systems of Differential Equations |
| title_full_unstemmed | On Periodic Solutions of One Class of Systems of Differential Equations |
| title_short | On Periodic Solutions of One Class of Systems of Differential Equations |
| title_sort | on periodic solutions of one class of systems of differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3615 |
| work_keys_str_mv | AT korol039ii onperiodicsolutionsofoneclassofsystemsofdifferentialequations AT korolʹíí onperiodicsolutionsofoneclassofsystemsofdifferentialequations AT korol039ii properíodičnírozv039âzkiodnogoklasusistemdiferencíalʹnihrívnânʹ AT korolʹíí properíodičnírozv039âzkiodnogoklasusistemdiferencíalʹnihrívnânʹ |