Researches in Mathematics and Mechanics.– 2015.–V. 20, Is. 2(26).–P. 81–89
UDC 517.926
S. A. Shchogolev
I. I. Mechnikov Odesa National University
ON THE EXISTENCE OF AN INTEGRAL MANIFOLD OF A
SPECIAL TYPE OF A SOME NONLINEAR DIFFERENTIAL
SYSTEM
A nonlinear system of the differential equations is considered. This system is a linear
extension of the nonlinear differential equation on a circle. The coefficients of the system
belong to the class F, which are the functions that can be represented in the form of the
absolutely and uniformly convergent Fourier series on the entire axis with slowly varying
coefficients in a certain sense and frequencies. The basic properties of this functions’ class
are investigated. This system is expected as a close one to an inhomogeneous linear system
of the differential equations with a diagonal matrix of the slowly varying coefficients and free
terms, belonging to the class F. The transformations with the coefficients of class F, causing
the system to a system in which the frequency is determined from the nonlinear equations
with slowly varying coefficients, are constructed using the contraction mapping principle. As
a result, the conditions, under which the original system has an integral manifold of class F,
are considered.
MSC: 34A34, 34A25.
Key words: integral manifold, differential, slowly-varying, Fourier series.
Introduction. One of the powerful methods of the study of nonlinear systems
of differential equations is the method of integral manifolds [1–3]. Particularly important role it plays in the research of multi-frequency oscillations, in particular, in
systems containing the slowly varying parameters [4]. An important object of study
in the same time and are single-frequency system [5]. This paper is a continuation
of research initiated in paper [6–9] on the existence of the particular solutions and
integral manifolds of the linear and quasilinear differential systems, are represented as
an absolutely and unifirmly convergent Fourier-series with slowly verying in a certain
sense coefficients.
Notation.
Let
𝐺(𝜀0 ) = {𝑡, 𝜀 : 𝑡 ∈ R, 𝜀 ∈ [0, 𝜀0 ], 𝜀0 ∈ R+ }
Definition 1. We say, that a function 𝑓 (𝑡, 𝜀), in general a complex-valued, belong
to the class 𝑆𝑚 (𝜀0 ), 𝑚 ∈ N ∪ {0}, if
1) 𝑓 : 𝐺(𝜀0 ) → C; 2) 𝑓 (𝑡, 𝜀) ∈ 𝐶 𝑚 (𝐺(𝜀0 )) with respect 𝑡;
3) 𝑑𝑘 𝑓 (𝑡, 𝜀)/𝑑𝑡𝑘 = 𝜀𝑘 𝑓𝑘* (𝑡, 𝜀) (0 6 𝑘 6 𝑚),
𝑑𝑒𝑓
‖𝑓 ‖𝑚 =
𝑚
∑︁
sup |𝑓𝑘* (𝑡, 𝜀)| < +∞
𝑘=0 𝐺(𝜀0 )
Received 15.08.2015
© Shchogolev S. A., 2015
82
Shchogolev S. A.
𝜃
Definition 2. We say, that a function 𝑓 (𝑡, 𝜀, 𝜃) belong to the class 𝐹𝑚,∞
(𝜀0 )
(𝑚 ∈ N ∪ {0}), if this function can be represented in form:
∞
∑︁
𝑓 (𝑡, 𝜀, 𝜃) =
𝑓𝑛 (𝑡, 𝜀)𝑒𝑖𝑛𝜃 ,
𝑛=−∞
and
1) 𝑓𝑛 (𝑡, 𝜀) ∈ 𝑆𝑚 (𝜀0 ), 𝜃 ∈ R;
2)
∞
∑︁
‖𝑓0 ‖𝑚 +
|𝑛|𝑙 · ‖𝑓𝑛 ‖𝑚 < +∞ (𝑙 = 0, 1, 2, ...).
𝑛=−∞
(𝑛̸=0)
If the function 𝑓 (𝑡, 𝜀, 𝜃) is real, then 𝑓−𝑛 (𝑡, 𝜀) = 𝑓𝑛 (𝑡, 𝜀).
We denote
∞
∑︁
‖𝑓 ‖𝑚,𝜃 =
‖𝑓𝑛 ‖𝑚 .
𝑛=−∞
For any 𝑓 ∈
𝜃
𝐹𝑚,∞
(𝜀0 )
we introduce the linear operators:
1
Γ𝑛 [𝑓 (𝑡, 𝜀, 𝜃)] =
2𝜋
ˆ2𝜋
𝑓 (𝑡, 𝜀, 𝜃)𝑒−𝑖𝑛𝜃 𝑑𝜃, 𝑛 ∈ Z,
0
in particular
1
Γ0 [𝑓 (𝑡, 𝜀, 𝜃)] =
2𝜋
ˆ2𝜋
𝑓 (𝑡, 𝜀, 𝜃)𝑑𝜃;
0
∞
∑︁
Γ𝑛 [𝑓 (𝑡, 𝜀, 𝜃)] 𝑖𝑛𝜃
𝑒 .
𝐼[𝑓 (𝑡, 𝜀, 𝜃)] =
𝑖𝑛
𝑛=−∞
(𝑛̸=0)
𝜃
(𝜀0 ). Let 𝑢(𝑡, 𝜀, 𝜃),
We note the some properties of the functions of the class 𝐹𝑚,∞
𝜃
𝑣(𝑡, 𝜀, 𝜃) belongs to class 𝐹𝑚,∞ (𝜀0 ), and 𝑘 = 𝑐𝑜𝑛𝑠𝑡.
1) 𝑢(𝑡, 𝜀, 𝜃) is 2𝜋-periodic with respect 𝜃.
2)
𝜕 𝑙 𝑢(𝑡, 𝜀, 𝜃)
𝜃
∈ 𝐹𝑚,∞
(𝜀0 ) (𝑙 = 0, 1, 2, . . .),
𝜕𝜃𝑙
𝜕 𝑘 𝑢(𝑡, 𝜀, 𝜃)
𝜃
∈ 𝐹𝑘−1,∞
(𝜀0 ) (𝑘 = 1, 𝑚).
𝜕𝑡𝑘
3) Γ𝑛 [𝑢(𝑡, 𝜀, 𝜃)] ∈ 𝑆𝑚 (𝜀0 ) (𝑛 ∈ Z).
4)
]︂
[︂
𝜕𝑢(𝑡, 𝜀, 𝜃)
𝜃
𝜃
= 𝑢(𝑡, 𝜀, 𝜃) − Γ0 [𝑢(𝑡, 𝜀, 𝜃)] ∈ 𝐹𝑚,∞
(𝜀0 ).
𝐼[𝑢(𝑡, 𝜀, 𝜃)] ∈ 𝐹𝑚,∞
(𝜀0 ), 𝐼
𝜕𝜃
5) ‖𝑘𝑢‖𝑚,𝜃 = |𝑘| · ‖𝑢‖𝑚,𝜃 .
6) ‖𝑢 + 𝑣‖𝑚,𝜃 6 ‖𝑢‖𝑚,𝜃 + ‖𝑣‖𝑚,𝜃 .
On the existence of a integral manifold
83
7)
‖𝑢‖𝑚,𝜃
⃦
𝑚 ⃦
∑︁
⃦ 1 𝜕𝑘𝑢 ⃦
⃦
⃦
=
⃦ 𝜀𝑘 𝜕𝑡𝑘 ⃦ .
0,𝜃
𝑘=0
8)
‖𝑢𝑣‖𝑚,𝜃 6 2𝑚 ‖𝑢‖𝑚,𝜃 · ‖𝑣‖𝑚,𝜃 .
We prove the property 8). For 𝑚 = 0 we have:
‖𝑢𝑣‖0,𝜃 6 ‖𝑢‖0,𝜃 · ‖𝑣‖0,𝜃 .
By virtue of the property 7):
‖𝑢𝑣‖𝑚,𝜃
⃦ 𝑘−𝑗 ⃦
⃦
⃦ 𝑗 ⃦
𝑚
𝑘
𝑚 ⃦
∑︁
∑︁
⃦𝜕
⃦ 1 𝜕 𝑘 (𝑢𝑣) ⃦
𝑣⃦
1 ∑︁ 𝑗 ⃦
𝜕 𝑢⃦
⃦
⃦
⃦
⃦ 6
⃦
⃦
·
𝐶
=
𝑘
⃦
⃦
⃦
⃦ 𝜀𝑘 𝜕𝑡𝑘 ⃦ 6
𝜀𝑘 𝑗=0
𝜕𝑡𝑗 0,𝜃
𝜕𝑡𝑘−𝑗 ⃦0,𝜃
0,𝜃
𝑘=0
𝑘=0
⎡
⎤ ⎡
⃦ 𝑗 ⃦
𝑚
𝑚
∑︁
∑︁
⃦
⃦
1
𝜕
𝑢
1
⃦
⃦ ⎦·⎣
6 2𝑚 ⎣
⃦
⃦
𝑗
𝑗
𝜀 𝜕𝑡 0,𝜃
𝜀𝑗
𝑗=0
𝑗=0
⎤
⃦ 𝑗 ⃦
⃦𝜕 𝑣⃦
⃦
⃦ ⎦ = 2𝑚 ‖𝑢‖𝑚,𝜃 · ‖𝑣‖𝑚,𝜃 .
⃦ 𝜕𝑡𝑗 ⃦
0,𝜃
𝜃
(𝜀0 ) is the Banach algebra [10].
We can say that the space 𝐹𝑚,∞
𝜃
9) If 𝑢, 𝑣 are real, then 𝑢(𝑡, 𝜀, 𝜃 + 𝑣(𝑡, 𝜀, 𝜃)) ∈ 𝐹𝑚,∞
(𝜀0 ).
We prove the property 9). We denote: 𝑤(𝑡, 𝜀, 𝜃) = 𝑢(𝑡, 𝜀, 𝜃 + 𝑣(𝑡, 𝜀, 𝜃)). Since
𝜃
(𝜀0 ), then
𝑢, 𝑣 ∈ 𝐹𝑚,∞
sup
𝑡,𝜀∈𝐺(𝜀0 )
⃒ 𝑠+𝑙
⃒
⃒ 𝜕 𝑢(𝑡, 𝜀, 𝜃) ⃒
⃒
⃒ < +∞,
sup ⃒
⃒
𝜕𝑡𝑠 𝜕𝜃𝑙
𝜃∈R
sup
𝑡,𝜀∈𝐺(𝜀0 )
⃒ 𝑠+𝑙
⃒
⃒ 𝜕 𝑣(𝑡, 𝜀, 𝜃) ⃒
⃒
⃒ < +∞
sup ⃒
𝜕𝑡𝑠 𝜕𝜃𝑙 ⃒
𝜃∈R
(1)
(𝑠 = 0, 𝑚; 𝑙 = 0, 1, 2, . . .).
Converting the expression
1
Γ𝑛 [𝑤(𝑡, 𝜀, 𝜃)] =
2𝜋
ˆ2𝜋
𝑤(𝑡, 𝜀, 𝜃)𝑒−𝑖𝑛𝜃 𝑑𝜃
(𝑛 ̸= 0)
0
by the formula (𝑙 + 2)-fold integration by the parts, we obtain:
1
Γ𝑛 [𝑤(𝑡, 𝜀, 𝜃)] =
2𝜋(𝑖𝑛)𝑙+2
ˆ2𝜋
𝑝(𝑡, 𝜀, 𝜃)𝑒−𝑖𝑛𝜃 𝑑𝜃
(𝑛 ̸= 0),
0
where 𝑝(𝑡, 𝜀, 𝜃) is polynom of degree 𝑙 + 3 with coefficients belong to the class 𝑆𝑚 (𝜀0 )
relatively derivatives 𝜕 𝑟 𝑢(𝑡, 𝜀, 𝜃)/𝜕𝜃𝑟 (𝑟 = 0, 𝑙 + 2), which are calculated by values of
argument 𝜃 is equal 𝜃 + 𝑣(𝑡, 𝜀, 𝜃) and 𝜕 𝑟 𝑣(𝑡, 𝜀, 𝜃)/𝜕𝜃𝑟 (𝑟 = 0, 𝑙 + 2). Given (1) we
𝜃
obtain, that 𝑤(𝑡, 𝜀, 𝜃) ∈ 𝐹𝑚,∞
(𝜀0 ).
10) The follows chains hold:
𝜃
𝜃
𝜃
𝐹0,∞
(𝜀0 ) ⊃ 𝐹1,∞
(𝜀0 ) ⊃ . . . ⊃ 𝐹𝑚,∞
(𝜀0 ),
𝑆0 (𝜀0 ) ⊃ 𝑆1 (𝜀0 ) ⊃ . . . ⊃ 𝑆𝑚 (𝜀0 ).
84
Shchogolev S. A.
Definition 3. We say, that vector 𝑓 = colon(𝑓1 , ..., 𝑓𝑁 ) belong to the class
𝜃
𝜃
𝐹𝑚,∞
(𝜀0 ) (or 𝑆𝑚 (𝜀0 )), if 𝑓𝑗 ∈ 𝐹𝑚,∞
(𝜀0 ) (relatively 𝑓𝑗 ∈ 𝑆𝑚 (𝜀0 )) (𝑗 = 1, 𝑁 ) .
𝜃
(𝜀0 ) (or
Definition 4. We say, that matrix (𝑎𝑗𝑘 )𝑗,𝑘=1,𝑁 belong to the class 𝐹𝑚,∞
𝜃
𝑆𝑚 (𝜀0 )), if 𝑎𝑗,𝑘 ∈ 𝐹𝑚,∞ (𝜀0 ) (relatively 𝑎𝑗,𝑘 ∈ 𝑆𝑚 (𝜀0 )) (𝑗, 𝑘 = 1, 𝑁 ) .
Consider the next system of the differential equations:
𝑑𝑥
𝑑𝑡
= (Λ(𝑡, 𝜀) + 𝜇𝑃 (𝑡, 𝜀, 𝜃))𝑥 + 𝑓 (𝑡, 𝜀, 𝜃),
(2)
𝑑𝜃
𝑑𝑡
= 𝜔(𝑡, 𝜀) + 𝜇𝑎(𝑡, 𝜀, 𝜃),
where (𝑡, 𝜀) ∈ 𝐺(𝜀0 ), 𝑥 = col(𝑥1 , ..., 𝑥𝑁 ), Λ = diag(𝜆1 , ..., 𝜆𝑁 ) ∈ 𝑆𝑚 (𝜀0 ), 𝑃 =
𝜃
𝜃
(𝜀0 ), scalar real functions 𝜔 ∈ 𝑆𝑚 (𝜀0 ), inf 𝜔 > 0, 𝑎 ∈ 𝐹𝑚,∞
(𝜀0 ),
(𝑝𝑗𝑘 )𝑗,𝑘=1,𝑁 ∈ 𝐹𝑚,∞
𝐺(𝜀0 )
𝜇 ∈ (0, 𝜇0 ) ⊂ R+ .
We study the problem determine of the conditions of existence of integral manifold
𝜃
*
𝑥(𝑡, 𝜀, 𝜃, 𝜇) of the system (2), belongs to the class 𝐹𝑚
* ,∞ (𝜀0 ), where 𝑚 6 𝑚.
Auxiliary arguments.
Lemma 1. There exists 𝜇1 ∈ (0, 𝜇0 ) such that for all 𝜇 ∈ (0, 𝜇1 ) there exists the
real reversible transformation
𝜙
𝜃 = 𝜙 + 𝜇𝑣(𝑡, 𝜀, 𝜙, 𝜇), 𝑣 ∈ 𝐹𝑚,∞
(𝜀0 ),
(3)
such the system (2) reduces to the following form:
𝑑𝑥
𝑑𝑡
= (Λ(𝑡, 𝜀) + 𝜇𝑄(𝑡, 𝜀, 𝜙, 𝜇))𝑥 + 𝑔(𝑡, 𝜀, 𝜙, 𝜇),
(4)
𝑑𝜙
𝑑𝑡
= 𝜔(𝑡, 𝜀) + 𝜇𝑏(𝑡, 𝜀, 𝜇) + 𝜇𝜀𝛽(𝑡, 𝜀, 𝜙, 𝜇),
𝜙
(𝜀0 ), 𝑔 = 𝑓 (𝑡, 𝜀, 𝜙 + 𝜇𝑣(𝑡, 𝜀, 𝜙, 𝜇)) ∈
where 𝑄 = 𝑃 (𝑡, 𝜀, 𝜙 + 𝜇𝑣(𝑡, 𝜀, 𝜙, 𝜇)) ∈ 𝐹𝑚,∞
𝜙
𝜙
𝐹𝑚,∞ (𝜀0 ), 𝑏 ∈ 𝑆𝑚 (𝜀0 ), 𝛽 ∈ 𝐹𝑚−1,∞ (𝜀0 ).
Proof. We define the function 𝑣 from equation:
𝜔(𝑡, 𝜀)
𝜕𝑣
𝜕𝑣
= 𝑎(𝑡, 𝜀, 𝜙 + 𝜇𝑣) − 𝑏(𝑡, 𝜀, 𝜇) − 𝜇𝑏(𝑡, 𝜀, 𝜇) ,
𝜕𝜙
𝜕𝜙
(5)
where the function 𝑏(𝑡, 𝜀, 𝜇) is to be determined.
𝜙
We seek the solution 𝑣 ∈ 𝐹𝑚,∞
(𝜀0 ) of equation (5) and function 𝑏 ∈ 𝑆𝑚 (𝜀0 ) by
the method of succesive approximations, defining the initial approximations 𝑣0 , 𝑏0
from the equation:
𝜕𝑣0
𝜔(𝑡, 𝜀)
= 𝑎(𝑡, 𝜀, 𝜙) − 𝑏0 (𝑡, 𝜀),
(6)
𝜕𝜙
and the subsequent approximations 𝑣𝑠 , 𝑏𝑠 (𝑠 = 1, 2, . . .) defining from the equations:
𝜔(𝑡, 𝜀)
𝜕𝑣𝑠
𝜕𝑣𝑠−1
= 𝑎(𝑡, 𝜀, 𝜙 + 𝜇𝑣𝑠−1 ) − 𝜇𝑏𝑠−1 (𝑡, 𝜀, 𝜇)
− 𝑏𝑠 (𝑡, 𝜀, 𝜇), 𝑠 = 1, 2, . . . . (7)
𝜕𝜙
𝜕𝜙
We denote:
𝑏0 (𝑡, 𝜀) = Γ0 [𝑎(𝑡, 𝜀, 𝜙)],
(8)
On the existence of a integral manifold
𝑣0 (𝑡, 𝜀, 𝜙) =
1
𝐼[𝑎(𝑡, 𝜀, 𝜙)],
𝜔(𝑡, 𝜀)
]︂
[︂
𝜕𝑣𝑠−1
,
𝑏𝑠 (𝑡, 𝜀, 𝜇) = Γ0 𝑎(𝑡, 𝜀, 𝜙 + 𝜇𝑣𝑠−1 ) − 𝜇𝑏𝑠−1
𝜕𝜙
[︂
]︂
1
𝜕𝑣𝑠−1
𝑣𝑠 (𝑡, 𝜀, 𝜙, 𝜇) =
𝐼 𝑎(𝑡, 𝜀, 𝜙 + 𝜇𝑣𝑠−1 ) − 𝜇𝑏𝑠−1
=
𝜔(𝑡, 𝜀)
𝜕𝜙
=
1
𝑏𝑠−1 (𝑡, 𝜀, 𝜇)
𝐼 [𝑎(𝑡, 𝜀, 𝜙 + 𝜇𝑣𝑠−1 )] − 𝜇
𝑣𝑠−1 (𝑡, 𝜀, 𝜙, 𝜇), 𝑠 = 1, 2, . . . .
𝜔(𝑡, 𝜀)
𝜔(𝑡, 𝜀)
85
(9)
(10)
(11)
We show, that all the approximations 𝑏𝑠 (𝑡, 𝜀, 𝜇), defined by the formulas (8), (10),
belongs to the class 𝑆𝑚 (𝜀0 ), and all the approximations 𝑣𝑠 (𝑡, 𝜀, 𝜙, 𝜇), defined by the
𝜙
formulas (9), (11), belongs to the class 𝐹𝑚,∞
(𝜀0 ).
𝜙
Obviously, that 𝑏0 ∈ 𝑆𝑚 (𝜀0 ), 𝑣0 ∈ 𝐹𝑚,∞ (𝜀0 ), 𝑣0 ∈ R, and Γ0 [𝑣0 ] ≡ 0. Then, using
𝜙
the property 9) of the functions of the class 𝐹𝑚,∞
(𝜀0 ), we obtain, that 𝑎(𝑡, 𝜀, 𝜙 +
𝜙
𝜙
𝜇𝑣0 (𝑡, 𝜀, 𝜙)) ∈ 𝐹𝑚,∞ (𝜀0 ), and 𝑏1 (𝑡, 𝜀, 𝜇) ∈ 𝑆𝑚 (𝜀0 ), 𝑣1 (𝑡, 𝜀, 𝜙, 𝜇) ∈ 𝐹𝑚,∞
(𝜀0 ).
𝜙
Suppose by the induction, that 𝑏𝑠 (𝑡, 𝜀, 𝜇) ∈ 𝑆𝑚 (𝜀0 ), 𝑣𝑠 (𝑡, 𝜀, 𝜙, 𝜇) ∈ 𝐹𝑚,∞
(𝜀0 )
(𝑠 = 2, 𝑘) we, by virtue of the property 9), show, that then 𝑏𝑘+1 (𝑡, 𝜀, 𝜇) ∈ 𝑆𝑚 (𝜀0 ),
𝜙
(𝜀0 ).
𝑣𝑘+1 (𝑡, 𝜀, 𝜙, 𝜇) ∈ 𝐹𝑚,∞
We introduce the sets:
{︀
}︀
𝜙
Ω1 = { 𝑏 ∈ 𝑆𝑚 (𝜀0 ) : ‖𝑏‖𝑚 6 𝑑 } , Ω2 = 𝑣 ∈ 𝐹𝑚,∞
(𝜀0 ) : ‖𝑣‖𝑚,𝜙 6 𝑑 , 𝑑 > 0.
Using techniques contraction mapping principle [10] it is easy to show that for
sufficiently small values 𝜇 fulfilled 𝑏𝑠 ∈ Ω1 , 𝑣𝑠 ∈ Ω2 (𝑠 = 0, 1, 2, . . .), and process
𝜙
(𝜀0 ) of the equation (5), in which
(8)–(11) converges to the solution of class 𝐹𝑚,∞
function 𝑏 ∈ 𝑆𝑚 (𝜀0 ).
Then for sufficiently small values 𝜇 the function 𝛽 is determined from the equation:
)︂
(︂
𝜕𝑣(𝑡, 𝜀, 𝜃, 𝜇)
1 𝜕𝑣(𝑡, 𝜀, 𝜃, 𝜇)
1+𝜇
𝛽=−
.
𝜕𝜃
𝜀
𝜕𝑡
Now we show, that the transformation (3) are reversible for sufficiently small
values 𝜇. Means for sufficiently small values 𝜇 there exists the reverse transformation
𝜙 = 𝜃 + 𝑞(𝑡, 𝜀, 𝜃, 𝜇),
(12)
𝜃
(𝜀0 ).
where 𝑞 ∈ 𝐹𝑚,∞
We substitute (12) in (3) and obtain the equation respectively 𝑞:
𝑞 = −𝜇𝑣(𝑡, 𝜀, 𝜃 + 𝑞, 𝜇).
(13)
𝜃
We seek the solution 𝑞 ∈ 𝐹𝑚,∞
(𝜀0 ) of equation (13) by the method of succesive
approximations, defining:
𝑞0 ≡ 0,
𝑞𝑠+1 = −𝜇𝑣(𝑡, 𝜀, 𝜃 + 𝑞𝑠 , 𝜇), 𝑠 = 0, 1, 2, . . . .
(14)
Using techniques contraction mapping principle [10] it is easy to show that for sufficiently small values 𝜇 the process (14) converges by the norm ‖ · ‖𝑚,𝜃 to the solution
𝜃
of class 𝐹𝑚,∞
(𝜀0 ) of the equation (13).
86
Shchogolev S. A.
Lemma 2. There exists 𝜇2 ∈ (0, 𝜇1 ) such that for all 𝜇 ∈ (0, 𝜇2 ) there exists the
chain of reversible transformations of kind:
𝜙 = 𝜓1 + 𝜇𝜀𝑤1 (𝑡, 𝜀, 𝜓1 , 𝜇),
(15)
𝜓1 = 𝜓2 + 𝜇𝜀2 𝑤2 (𝑡, 𝜀, 𝜓2 , 𝜇),
(16)
···
𝜓𝑚1 −2 = 𝜓𝑚1 −1 + 𝜇𝜀
where 𝑚1 < 𝑚, 𝑤𝑘 ∈
to the following form:
𝑑𝑥
𝑑𝑡
𝜓𝑘
𝐹𝑚−𝑘,∞
(𝜀0 )
𝑚1 −1
𝑤𝑚1 −1 (𝑡, 𝜀, 𝜓𝑚1 −1 , 𝜇),
(17)
(𝑘 = 1, 𝑚1 − 1), such that the system (4) reduces
= (Λ(𝑡, 𝜀) + 𝜇𝑅𝑚1 −1 (𝑡, 𝜀, 𝜓𝑚1 −1 , 𝜇))𝑥 + ℎ𝑚1 −1 (𝑡, 𝜀, 𝜓𝑚1 −1 , 𝜇),
𝑑𝜓𝑚1 −1
𝑑𝑡
= 𝜔(𝑡, 𝜀) + 𝜇𝑏(𝑡, 𝜀, 𝜇) + 𝜇
∑︀𝑚1 −1
𝑙=1
𝜀𝑘 𝛽𝑘 (𝑡, 𝜀, 𝜇)+
(18)
+𝜇𝜀𝑚1 𝛽̃︀𝑚1 −1 (𝑡, 𝜀, 𝜓𝑚1 −1 , 𝜇),
𝜓
𝜓
𝑚1 −1
𝑚1 −1
where 𝑅𝑚1 −1 ∈ 𝐹𝑚−𝑚
(𝜀0 ), ℎ𝑚1 −1 ∈ 𝐹𝑚−𝑚
(𝜀0 ), 𝛽𝑘 ∈ 𝑆𝑚−𝑘 (𝜀0 ),
1 +1
1 +1
𝜓𝑚1 −1
(𝜀
)
(𝑘
=
𝛽̃︀𝑚1 −1 ∈ 𝐹𝑚−𝑚
1,
𝑚
−
1).
0
1
1 ,∞
Proof. Consider the transformation (15). The application of this is reduce the
system (4) to the kind:
𝑑𝑥
𝑑𝑡
= (Λ(𝑡, 𝜀) + 𝜇𝑅1 (𝑡, 𝜀, 𝜓1 , 𝜇))𝑥 + ℎ1 (𝑡, 𝜀, 𝜓1 , 𝜇),
(19)
𝑑𝜓1
𝑑𝑡
= 𝜔(𝑡, 𝜀) + 𝜇𝑏(𝑡, 𝜀, 𝜇) + 𝜇𝜀𝛽1 (𝑡, 𝜀, 𝜇) + 𝜇𝜀2 𝛽̃︀1 (𝑡, 𝜀, 𝜓1 , 𝜇),
where 𝛽1 , 𝛽̃︀1 must be defined.
We define the functions 𝑤1 , 𝛽1 from the equation:
(𝜔(𝑡, 𝜀) + 𝜇𝑏(𝑡, 𝜀, 𝜇))
𝜕𝑤1
𝜕𝑤1
+ 𝛽1 (𝑡, 𝜀, 𝜇) = 𝛽(𝑡, 𝜀, 𝜓1 + 𝜇𝜀𝑤1 , 𝜇) − 𝜇𝜀𝛽1 (𝑡, 𝜀, 𝜇)
. (20)
𝜕𝜓1
𝜕𝜓1
Then the function 𝛽̃︀1 are defined from the equation:
(︂
)︂
𝜕𝑤1 ̃︀
1 𝜕𝑤1
.
1 + 𝜇𝜀
𝛽1 = −
𝜕𝜓1
𝜀 𝜕𝑡
(21)
Similarly to the proof of Lemma 1 we show that for sufficiently small values 𝜇
𝜓1
the equation (20) has a solution 𝑤1 (𝑡, 𝜀, 𝜓1 , 𝜇) ∈ 𝐹𝑚−1,∞
(𝜀0 ). In this 𝛽1 ∈ 𝑆𝑚−1 (𝜀0 ).
𝜓1
And for sufficiently small values 𝜇 the equation (21) has a solution 𝛽̃︀1 ∈ 𝐹𝑚−2,∞
(𝜀0 ).
̃︀
̃︀
Similarly are defined the functions 𝑤2 , . . . , 𝑤𝑚1 −1 , 𝛽2 , . . . , 𝛽𝑚1 −1 , 𝛽2 , . . . , 𝛽𝑚1 −1 .
Main Results
Theorem. Let the elements 𝜆𝑗 (𝑡, 𝜀) (𝑗 = 1, 𝑁 ) of matrix Λ(𝑡, 𝜀) in system (2) are
such that inf |Re𝜆𝑗 (𝑡, 𝜀)| > 𝛾 > 0 (𝑗 = 1, 𝑁 ). Then there exists 𝜇* ∈ (0, 𝜇0 ) such that
𝐺(𝜀0 )
𝜃
for all 𝜇 ∈ (0, 𝜇* ) the system (2) has the integral manifold 𝑥(𝑡,
̃︀ 𝜀, 𝜃, 𝜇) ∈ 𝐹𝑚
(𝜀0 ),
1 ,∞
where 2𝑚1 6 𝑚 (𝑚1 ∈ N ∪ {0}).
On the existence of a integral manifold
87
Proof. Based on Lemmas 1,2, we reduce the system (2) to the kind (18). We
denote:
𝜓 = 𝜓𝑚1 −1 ,
𝑅(𝑡, 𝜀, 𝜓, 𝜇) = 𝑅𝑚1 −1 (𝑡, 𝜀, 𝜓𝑚1 −1 , 𝜇), ℎ(𝑡, 𝜀, 𝜓, 𝜇) = ℎ𝑚1 −1 (𝑡, 𝜀, 𝜓𝑚1 −1 , 𝜇),
𝜔1 (𝑡, 𝜀, 𝜇) = 𝜔(𝑡, 𝜀0 ) + 𝜇𝑏(𝑡, 𝜀, 𝜇) +
𝑚
1 −1
∑︁
𝜀𝑘 𝛽𝑘 (𝑡, 𝜀, 𝜇) + 𝜀𝑚1 𝛽̃︀𝑚1 −1 (𝑡, 𝜀, 𝜓𝑚1 −1 , 𝜇).
𝑙=1
𝜃
Using condition of Theorem and property 10) of the functions of class 𝐹𝑚,∞
(𝜀0 ),
𝜓
we can state, that 𝑅(𝑡, 𝜀, 𝜓, 𝜇), ℎ(𝑡, 𝜀, 𝜓, 𝜇) ∈ 𝐹𝑚1 ,∞ (𝜀0 ), 𝜔1 (𝑡, 𝜀, 𝜇) ∈ 𝑆𝑚1 (𝜀0 ). Then
we write the system (18) in form:
𝑑𝑥
𝑑𝑡
= (Λ(𝑡, 𝜀) + 𝜇𝑅(𝑡, 𝜀, 𝜓, 𝜇))𝑥 + ℎ(𝑡, 𝜀, 𝜓, 𝜇),
(22)
𝑑𝜓
𝑑𝑡
= 𝜔1 (𝑡, 𝜀, 𝜇).
With the system (22) we consider the system:
𝑑𝑥0
𝑑𝑡
= Λ(𝑡, 𝜀)𝑥0 + ℎ(𝑡, 𝜀, 𝜓, 𝜇),
(23)
𝑑𝜓
𝑑𝑡
= 𝜔1 (𝑡, 𝜀, 𝜇).
By results [6] and condition of Theorem, we can state, that the system (23) has
𝜓
(𝜀0 ). And there exists 𝐾 ∈ (0, +∞) such
the integral manifold 𝑥0 (𝑡, 𝜀, 𝜓, 𝜇) ∈ 𝐹𝑚
1 ,∞
that
‖𝑥0 ‖𝑚1 ,𝜓 6 𝐾‖ℎ‖𝑚1 ,𝜓 .
(24)
We seek the integral manifold of system (22) by the method of succesive approximations, defining as an initial approximation 𝑥0 , and the subsequents approximations
defining from the systems:
𝑑𝑥𝑠+1
𝑑𝑡
= Λ(𝑡, 𝜀)𝑥𝑠+1 + ℎ(𝑡, 𝜀, 𝜓, 𝜇) + 𝜇𝑅(𝑡, 𝜀, 𝜓, 𝜇)𝑥𝑠 , ,
(25)
𝑑𝜓
𝑑𝑡
= 𝜔1 (𝑡, 𝜀, 𝜇), 𝑠 = 0, 1, 2, . . . .
Taking in to account the unequality (24) and using the ordinary technicue of
the contraction mapping principle [10], it is easy to show, that there exists 𝜇3 ∈
𝜓
(0, 𝜇0 ) such that for all 𝜇 ∈ (0, 𝜇3 ) all approximations 𝑥𝑠 belong to class 𝐹𝑚
(𝜀0 ),
1 ,∞
and process (25) converges by the norm ‖ · ‖𝑚,𝜓 to integral manifold 𝑥(𝑡, 𝜀, 𝜓, 𝜇) ∈
𝜓
𝐹𝑚
(𝜀0 ) of the system (22).
1 ,∞
By virtue of the revesibility of the transformations (3), (15) – (17), we can state the
𝜃
existence for sufficiently small values 𝜇 of the integral manifold 𝑥(𝑡,
̃︀ 𝜀, 𝜃, 𝜇) ∈ 𝐹𝑚,∞
(𝜀0 )
of the system (2).
Conclusion. Thus, for the system (2) the existence of the integral manifold of
𝜃
the class 𝐹𝑚
(𝜀0 ) (2𝑚1 6 𝑚) we prove for sufficiently small values of 𝜇.
1 ,∞
88
Shchogolev S. A.
1.
Bogolubov N. N., Mitropol’skii Yu. A., Samoilenko A. M. The method of
accelerated convergence in nonlinear mechanics [in Russian], Naukova dumka, Kiev,
1969. – 247 p.
2.
Mitropol’skii Yu. A., Lykova O. B. The method of Integral Manifolds in nonlinear
mechanics [in Russian], Nauka, Moscow, 1973. – 512 p.
3.
Samoilenko A. M. Elements of the mathematical theory of multifrequency oscillations.
Invariant Tori [in Russian], Nauka, Moscow, 1987. – 304 p.
4.
Samoilenko A. M., Petryshin R. I. Mathematical Aspects of theory of nonlinear
oscillations [in Ukrainian], Naukova dumka, Kiev, 2004. – 474 p.
5.
Mitropol’skii Yu. A. Nonlinear mechanics. Single-frecuency oscillations [in Russian],
Inst Math., Kiev, 1997. – 388 p.
6.
Kostin A. V., Shchogolev S. A. On the Stability of Oscillations Representable by
Fourier series with Slowly Varying parameters [in Russian] // Differential equations. –
2008. V. 44, No 1. – P. 45 – 51.
7.
Shchogolev S. A. On existence of a special kind’s integral manifold of the nonlinear
differential system with slowly varying parameters // Vysnyk Odesk. Nats. Univers. Mat
i Mekh. – 2013. – V. 18, Is. 2(18). – P. 80 – 96.
8.
Shchogolev S. A. On a reduction of nonlinear first-order differential equation with
oscillating coefficients to a some special kind // Vysnyk Odesk. Nats. Univers. Mat i
Mekh. – 2013. – V. 18, Is. 4(20). – P. 60 – 67.
9.
Shchogolev S. A. On a reduction of a linear homogeneous differential system with
oscillating coefficients to a system with slowly varying coefficients in resonance case //
Vysnyk Odesk. Nats. Univers. Mat i Mekh. – 2014. – V. 19, Is. 3(23). – P. 48 – 56.
10. Kolmogorov A. M., Fomin S. V. Elements of the theory of functions and functional
analysis [in Russian], Nauka, Moscow (1972). – 496 p.
Щоголев С. А.
Про iснування iнтегрального многовиду спецiального вигляду однiєї нелiнiйної диференцiальної системи
Резюме
Для нелiнiйної диференцiальної системи, коефiцiєнти якої представленi у виглядi абсолютно та рiвномiрно збiжних рядiв Фур’є з повiльно змiнними коефiцiєнтами та частотою, отримано умови iснування iнтегрального многовиду аналогiчної структури.
Ключовi слова: диференцiальний, повiльно змiнний, ряди Фур’є.
Щёголев С. А.
О существовании интегрального многообразия специального вида одной нелинейной дифференциальной системы
Резюме
Для нелинейной дифференциальной системы, коэффициенты которой представимы в
виде абсолютно и равномерно сходящихся рядов Фурье с медленно меняющимися коэффициентами и частотой, получены условия существования интегрального многообразия
аналогичной структуры.
Ключевые слова: дифференциальный, медленно меняющийся, ряды Фурье.
On the existence of a integral manifold
89
References
1.
Bogolubov, N. N., Mitropol’skii, Yu. A., & Samoilenko, A. M. (1969). Metod uskoreniy
shodimosti v nelineynoy mehanike [The method of accelerated convergence in nonlinear
mechanics] . Kyiv: Naukova Dumka [in Russian].
2.
Mitropol’skii, Yu. A., Lykova, O. B. (1973). Matematicheskie aspekti teorii nelineynih
kolebaniy [The method of Integral Manifolds in nonlinear mechanics]. Moscow: Nauka
[in Russian].
3.
Samoilenko, A. M. (1987). Elementi matematicheskoy teorii mnogochastnih kolebaniy.
Invariantnie tori [Elements of the mathematical theory of multifrequency oscillations.
Invariant Tori]. Moscow: Nauka [in Russian].
4.
Samoilenko, A. M., & Petryshin, R. I. (2004). Matematichni aspekti teorii neliniynih kolivan [Mathematical Aspects of theory of nonlinear oscillations]. Kiev: Naukova Dumka
[in Ukrainian].
5.
Mitropol’skii, Yu., A. (1997). Neliniyna mehanika. Odnochastotni kolivannya [Nonlinear
mechanics. Single-frequency oscillations. Kiev: Inst. Math. [in Ukrainian].
6.
Kostin, A. V., & Shchogolev, S. A. (2008). Ob ustoychivosti kolebaniy, predstavimih
ryadami Fur’e s medlenno menyayushchimisya parametrami [On the Stability of Oscillations Representable by Fourier series with Slowly Varying parameters]. Differencial’nie
uravneniya - Differential equations, Vol. 44 (1), 45-51 [in Russian].
7.
Shchogolev, S. A. (2013). On existence of a special kind’s integral manifold of the nonlinear differential system with slowly varying parameters. Vysnyk Odesk. Nats. Univers.
Mat. i Mekh., Vol. 18, 2(18), 80–96.
8.
Shchogolev, S. A. (2013). On a reduction of nonlinear first-order differential equation
with oscillating coefficients to a some special kind. Vysnyk Odesk. Nats. Univers. Mat.
i Mekh., Vol. 18, 4(20), 60–67.
9.
Shchogolev, S. A. (2014). On a reduction of a linear homogeneous differential system
with oscillating coefficients to a system with slowly varying coefficients in resonance
case. Vysnyk Odesk. Nats. Univers. Mat. i Mekh., Vol. 19, 3(23), 48–56.
10. Kolmogorov, A. M., & Fomin, S. V. (1972). Elementi teorii funkciy i funkcional’nogo
analiza [Elements of the theory of functions and functional analysis]. Moscow: Nauka
[in Russian].