On the Reduction of the Linear System of the Differential Equations with coefficients of oscillating type to the Triangular Kind in the Non-resonant Case

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http://bulmathmc.enu.kz, E-mail: vest_math@enu.kz

МРНТИ:27.29.23

S.A. Shchogolev

Odessa I.I. Mechnikov National University, Odessa, Ukraine (E-mail: sergas1959@gmail.com)

On the Reduction of the Linear System of the Differential Equations with coefficients of oscillating type to the Triangular Kind in the Non-resonant Case

Abstract: For the linear homogeneous differential system, whose coefficients are represented as an absolutely and uniformly convergent Fourier-series with slowly varying coefficients and frequency, the conditions of the existence of the transformation which leads it to triangular kind, are obtained in the non-resonant cases.

Keywords: linear differential systems, Fourier series.

DOI: https://doi.org/10.32523/2616-7182/2020-130-1-82-92 Introduction. In the theory of linear systems of differential equations is well known problem of the consruction for the linear homogeneous system of the differential equations

dx

dt =A(t)x, (1)

where x= colon(x1, ..., xn), A(t) = (a_jk(t))_j,k=1,n, Lyapunov’s transformation x=L(t)y,

which leads the system (1) to the triangular kind dy

dt =T(t)y, where T(t) = (b_jk(t))_j,k=1,n, b_jk(t)≡0 (j < k) [1–4].

In this paper, we assume, that the system (1) already reduced to a kind, close to triangular:

dx

dt = (T(t) +µP(t))x, (2)

where µ – small parameter, and the matrix P(t) has a some special kind. And we study the problem on bringing the system (2) to a purely triangular form

dy

dt =D(t)y, where D(t) = (d_jk(t))_j,k=1,n, d_jk ≡0 (j < k).

Basic notations and definitions.

Let G(ε₀) ={t, ε: 0< ε < ε₀, t∈R}.

Definition 1. We say, that a function p(t, ε) belongs to a class S(m;ε0) (m∈N∪ {0}), if

1) p:G(ε0)→C, 2) p(t, ε)∈C^m(G(ε0)) with respect t; 3) d^kp(t, ε)/dt^k =ε^kp^∗_k(t, ε) (0≤k≤m),

kpk_S(m;ε₀₎^def=

m

X

k=0

sup

G(ε0)

|p^∗_k(t, ε)|<+∞.

Under the slowly varying function we mean the function of the class S(m;ε₀).

82

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Definition 2. We say, that a function f(t, ε, θ(t, ε)) belongs to a class F(m;ε0;θ) (m ∈ N∪ {0}), if this function can be represented as:

f(t, ε, θ(t, ε)) =

∞

X

n=−∞

f_n(t, ε) exp (inθ(t, ε)), and:

1) f_n(t, ε)∈S(m;ε₀); 2)

kfk_F_(m;ε₀_;θ) ^def=

∞

X

n=−∞

kf_nk_S(m;ε₀₎ <+∞, 3) θ(t, ε) =

t

R

0

ϕ(τ, ε)dτ, ϕ(t, ε)∈R⁺, ϕ(t, ε)∈S(m;ε0), inf

G(ε0)ϕ(t, ε) =ϕ0 >0.

State some properties of the functions of the classes S(m;ε0), F0(m;ε0;θ) (the proofs are given in [5]). Let k= const, p, q∈S(m;ε0), u, v∈F(m;ε0;θ). Then kp, p±q, pq belongs to the class S(m;ε₀), ku, u±v, uv belongs to the class F₀(m;ε₀;θ), and

1) kkpk_S(m;ε₀₎=|k| · kpk_S(m;ε₀₎.

2) kp±qk_S(m;ε₀₎ ≤ kpk_S(m;ε₀₎+kqk_S₀_(m,ε₀₎. 3) kpqk_S(m;ε₀₎ ≤2^mkpk_S(m;ε₀₎kqk_S₀_(m;ε₀₎. 4) kkuk_F(m;ε₀_;θ)=|k| · kuk_F_(m;ε₀_;θ).

5) ku±vk_F_(m;ε₀_;θ) ≤ kuk_F_(m;ε₀_;θ)+kvk_F(m;ε₀_;θ). 6) kuvk_F_(m;ε₀_;θ) ≤2^mkuk_F_(m;ε₀_;θ)· kvk_F(m;ε₀_;θ). For f(t, ε, θ)∈F(m;ε₀;θ) we denote:

Γ_n[f] = 1 2π

2π

Z

0

f(t, ε, θ) exp(−inθ)dθ (n∈Z).

In particular

Γ₀[f] = 1 2π

2π

Z

0

f(t, ε, θ)dθ.

Definition 3. For the vector u= colon(u₁, . . . , u_n) with elements from the class F(m;ε₀;θ) we define the norm:

kuk^∗_F_(m;ε

0;θ)=

n

X

k=1

ku_kk_F_(m;ε₀_;θ).

Statement of the Problem. We consider the next system of differential equations:

dx

dt = (B(t, ε) +µP(t, ε, θ))x, (3)

where x= colon(x1, ..., xn), B(t, ε) – lower triangular matrix with the elements from S(m;ε0), and P(t, ε, θ) = (p_jk(t, ε, θ))_j,k=1,n, p_jk(t, ε, θ) ∈ F(m;ε₀;θ) (j, k = 1, n), µ∈ (0, µ₀) – the real parameter.

We study the problem of the existence of a transformation of kind

x= (E_n+µΨ(t, ε, θ, µ))y, (4)

y= colon(y1, ..., yn), En – unit matrix of order n, Ψ – matrix with elements from F(l;ε1;θ) (0< l₁ ≤m, 0< ε₁ < ε₀), which leads at sufficiently small µ the system (3) to the kind:

dy

dt =K(t, ε, θ, µ))y, (5)

where K = (k_jk(t, ε, θ, µ))_j,k=1,n, k_jk ≡0 (j < k), k_jk(t, ε, θ, µ)∈F(l;ε1;θ).

We will study this problem for a third-order system (n = 3) so as not to clutter up the presentation with secondary technical difficulties associated with the dimension of the system.

All fundamental difficulties take place in this case too.

Bulletin of L.N. Gumilyov ENU. Mathematics. Computer science. Mechanics series, 2020, Vol. 130, №1

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So, consider the system of the differential equations:

dx

dt = (B(t, ε) +µP(t, ε, θ))x, (6)

x= colon(x1, x2, x3),

B(t, ε) =





b11(t, ε) 0 0 b₂₁(t, ε) b₂₂(t, ε) 0 b₃₁(t, ε) b₃₂(t, ε) b₃₃(t, ε)



,

b_jk(t, ε) ∈ S(m;ε0) (j, k = 1,2,3;j ≥ k), P(t, ε, θ) = (p_jk(t, ε, θ))_j,k=1,2,3, p_jk(t, ε, θ) ∈ F(m;ε₀;θ).

Auxiliary results.

Lemma 1. Let we have the system dv

dt = A(t, ε) +

q

X

l=1

Q_l(t, ε, θ)µ^l

!

v, (7)

x= colon(x1, x2, x3), q∈N, A(t, ε) =





im12(t, ε) −c₃₂(t, ε) 0 0 im₁₃(t, ε) 0 0 c21(t, ε) im23(t, ε)





m_jk(t, ε)∈S(m;ε₀), m_jk(t, ε)∈R, c_jk(t, ε)∈S(m;ε₀),and inf

G(ε0)

|m₁₃(t, ε)−m12(t, ε)−nϕ(t, ε)| ≥γ >0, inf

G(ε0)

|m₂₃(t, ε)−m₁₂(t, ε)−nϕ(t, ε)| ≥γ >0, (8) inf

G(ε0)

|m₁₃(t, ε)−m23(t, ε)−nϕ(t, ε)| ≥γ >0,

n ∈ Z, ϕ(t, ε) – the function in the definition of class F(m;ε₀;θ), the elements of matrices Ql (l= 1, q) belongs to the class F(m;ε0;θ).

Then there exists µ1 ∈ (0, µ0), such that for all µ ∈ (0, µ1) there exists the Lyapunov’s transformation of kind

v = E+

q

X

l=1

Ψ_l(t, ε, θ)µ^l

!

w, (9)

where elemens of matrices Ψ_l(t, ε, θ) (l = 1, q) belongs to the class F(m;ε₀;θ), which leads the system (7) to kind:

dw

dt = A(t, ε) +

q

X

l=1

Ul(t, ε)µ^l+ε

q

X

l=1

Vl(t, ε, θ)µ^l+µ^q+1W(t, ε, θ, µ)

!

w, (10)

where U_l(t, ε) – the matrices with elements from S(m;ε₀), V_l, W – the matrices with elements from F(m−1;ε0;θ).

Proof. We substitute the expression (9) into system (7), and require that the transformed system has the kind (10). We obtain the next chain of matrix differential equations for detemining matrices Ψ₁, ...,Ψ_q:

dΨ₁

dt =A(t, ε)Ψ₁−Ψ₁A(t, ε) +Q₁(t, ε, θ)−U₁(t, ε)−εV₁(t, ε, θ), (11) dΨl

dt =A(t, ε)Ψl−ΨlA(t, ε) +Ql(t, ε, θ)−

l−1

X

ν=1

QνΨl−ν−

−

l−1

X

ν=1

Ψ_νUl−ν(t, ε)−ε

l−1

X

ν=1

Ψ_νVl−ν(t, ε, θ)−U_l(t, ε)−εV_l(t, ε, θ), l= 2, q. (12) where Ψ_l= (ψ_jk^l )_j,k=1,2,3, Q_l = (q^l_jk)_j,k=1,2,3, U_l= (u^l_jk)_j,k=1,2,3, V_l= (v^l_jk)_j,k=1,2,3 (l= 1, q).

Л.Н. Гумилев атындағы ЕҰУ Хабаршысы. Математика. Компьютерлiк ғылымдар. Механика сериясы, 2020, Том 130, №1

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Then the matrix W at sufficiently small values µ is determined from the equation:

E+

q

X

l=1

Ψlµ^l

!

W =

q−1

X

s=0



 X

σ+δ=s+q+1

(QσΨδ−ΨσUδ



µ^s−

−

q−1

X

s=0



 X

σ+δ=s+q+1

ΨσV_δ



µ^s. (13) We consider the equation (11). In the component it looks like this:

dψ₁₁¹

dt =−c₃₂(t, ε)ψ₂₁¹ +q¹₁₁(t, ε, θ)−u¹₁₁(t, ε)−εv₁₁¹ (t, ε, θ),

dψ¹₁₂

dt =i(m12(t, ε)−m13(t, ε))ψ¹₁₂−c32(t, ε)(ψ₂₂¹ −ψ₁₁¹ )−c21(t, ε)ψ¹₁₃+ +q¹₁₂(t, ε, θ)−u¹₁₂(t, ε)−εv₁₂¹ (t, ε, θ),

dψ¹₁₃

dt =i(m12(t, ε)−m23(t, ε))ψ₁₃¹ −c32(t, ε)ψ₂₃¹ + +q¹₁₃(t, ε, θ)−u¹₁₃(t, ε)−εv₁₃¹ (t, ε, θ),

dψ₂₁¹

dt =i(m₁₃(t, ε)−m₁₂(t, ε))ψ₂₁¹ +q₂₁¹ (t, ε, θ)−u¹₂₁(t, ε)−εv₂₁¹ (t, ε, θ),

dψ¹₂₂

dt =c₃₂(t, ε)ψ¹₂₁−c₂₁(t, ε)ψ¹₃₂+q¹₂₂(t, ε, θ)−u¹₁₁(t, ε)−εv₂₂¹ (t, ε, θ),

dψ₂₃¹

dt =i(m13(t, ε)−m23(t, ε))ψ₂₃¹ +q₂₃¹ (t, ε, θ)−u¹₂₃(t, ε)−εv₂₃¹ (t, ε, θ),

dψ¹₃₁

dt =i(m23(t, ε)−m12(t, ε))ψ₃₁¹ +c21(t, ε)ψ₂₁¹ + +q¹₃₁(t, ε, θ)−u¹₃₁(t, ε)−εv₃₁¹ (t, ε, θ),

dψ¹₃₂

dt =i(m₂₃(t, ε)−m₁₃(t, ε))ψ¹₃₂+c₂₁(t, ε)(ψ₂₁¹ −ψ₃₃¹ ) +c₃₂(t, ε)ψ¹₃₁+ +q¹₃₂(t, ε, θ)−u¹₃₂(t, ε)−εv₃₂¹ (t, ε, θ),

dψ¹₃₃

dt =c₂₁(t, ε)ψ¹₂₃+q¹₃₃(t, ε, θ)−u¹₃₃(t, ε)−εv¹₃₃(t, ε, θ).

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Define ψ_jk¹ , u¹_jk, v_jk¹ by the following expression:

ψ¹₂₁(t, ε, θ) =−

∞

X

n=−∞

Γn[q¹₂₁(t, ε, θ)]

i(m₁₃(t, ε)−m₁₂(t, ε)−nϕ(t, ε)) e^inθ(t,ε), u¹₂₁(t, ε)≡0,

v₂₁¹ (t, ε, θ) = 1 ε

∞

X

n=−∞

d dt

Γn[q₂₁¹ (t, ε, θ)]

i(m₁₃(t, ε)−m₁₂(t, ε)−nϕ(t, ε))

e^inθ(t,ε), ψ₁₁¹ (t, ε, θ) =

∞

X

n=−∞

(n6=0)

Γn[q₁₁¹ (t, ε, θ)−c32(t, ε)ψ₂₁¹ (t, ε, θ)]

inϕ(t, ε) e^inθ(t,ε),

u¹₁₁(t, ε) = Γ₀[q¹₁₁(t, ε, θ)−c₃₂(t, ε)ψ₂₁¹ (t, ε, θ)], v₁₁¹ (t, ε, θ) =−1

ε

∞

X

n=−∞

(n6=0)

d dt

Γ_n[q₁₁¹ (t, ε, θ)−c₃₂(t, ε)ψ₂₁¹ (t, ε, θ)]

inϕ(t, ε)

e^inθ(t,ε),

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ψ₃₁¹ (t, ε, θ) =−

∞

X

n=−∞

Γn[q₃₁¹ (t, ε, θ) +c21(t, ε)ψ₂₁¹ (t, ε, θ)]

i(m₂₃(t, ε)−m₁₂(t, ε)−nϕ(t, ε)) e^inθ(t,ε), u¹₃₁(t, ε)≡0,

v₃₁¹ (t, ε, θ) = 1 ε

∞

X

n=−∞

d dt

Γ_n[q₃₁¹ (t, ε, θ) +c₂₁(t, ε)ψ₂₁¹ (t, ε, θ)]

i(m13(t, ε)−m12(t, ε)−nϕ(t, ε))

e^inθ(t,ε), ψ¹₂₃(t, ε, θ) =−

∞

X

n=−∞

Γn[q¹₂₃(t, ε, θ)]

i(m₁₃(t, ε)−m₂₃(t, ε)−nϕ(t, ε)) e^inθ(t,ε), u¹₂₃(t, ε)≡0,

v₂₃¹ (t, ε, θ) = 1 ε

∞

X

n=−∞

d dt

Γ_n[q₂₃¹ (t, ε, θ)]

i(m13(t, ε)−m23(t, ε)−nϕ(t, ε))

e^inθ(t,ε), ψ₃₃¹ (t, ε, θ) =

∞

X

n=−∞

(n6=0)

Γn[q₃₃¹ (t, ε, θ) +c21(t, ε)ψ₂₃¹ (t, ε, θ)]

u¹₃₃(t, ε) = Γ0[q¹₃₃(t, ε, θ) +c21(t, ε)ψ₂₃¹ (t, ε, θ)], v₃₃¹ (t, ε, θ) =−1

ε

∞

X

n=−∞

(n6=0)

d dt

Γ_n[q₃₃¹ (t, ε, θ) +c₂₁(t, ε)ψ₂₃¹ (t, ε, θ)]

inϕ(t, ε)

e^inθ(t,ε),

ψ¹₂₂(t, ε, θ) =

∞

X

n=−∞

(n6=0)

Γ_n[q¹₂₂(t, ε, θ) +c₃₂(t, ε)ψ¹₂₁(t, ε, θ)−c₂₁(t, ε)ψ¹₂₃(t, ε, θ)]

u¹₂₂(t, ε) = Γ₀[q₂₂¹ (t, ε, θ) +c₃₂(t, ε)ψ₂₁¹ (t, ε, θ)−c₂₁(t, ε)ψ¹₂₃(t, ε, θ)], v₂₂¹ (t, ε, θ) =−1

ε

∞

X

n=−∞

(n6=0)

d dt

Γn[q¹₂₂(t, ε, θ) +c32(t, ε)ψ¹₂₁(t, ε, θ)−c21(t, ε)ψ¹₂₃(t, ε, θ)]

inϕ(t, ε)

e^inθ(t,ε),

ψ₃₂¹ (t, ε, θ) =−

∞

X

n=−∞

Γ_n[q₃₂¹ (t, ε, θ) +c₂₁(t, ε)(ψ¹₂₂−ψ¹₃₃) +c₃₂(t, ε)ψ₃₁¹ ]

i(m23(t, ε)−m13(t, ε)−nϕ(t, ε)) e^inθ(t,ε), u¹₃₂(t, ε)≡0,

v₃₂¹ (t, ε, θ) = 1 ε

∞

X

n=−∞

d dt

Γn[q¹₃₂(t, ε, θ) +c21(t, ε)(ψ₂₂¹ −ψ₃₃¹ ) +c32(t, ε)ψ₃₁¹ ] i(m₂₃(t, ε)−m₁₃(t, ε)−nϕ(t, ε))

e^inθ(t,ε), ψ₁₃¹ (t, ε, θ) =−

∞

X

n=−∞

Γ_n[q₁₃¹ (t, ε, θ)−c₃₂(t, ε)ψ₂₃¹ (t, ε, θ)]

i(m12(t, ε)−m23(t, ε)−nϕ(t, ε)) e^inθ(t,ε), u¹₁₃(t, ε)≡0,

v₁₃¹ (t, ε, θ) = 1 ε

∞

X

n=−∞

d dt

Γn[q₁₃¹ (t, ε, θ)−c32(t, ε)ψ₂₃¹ (t, ε, θ)]

i(m₁₂(t, ε)−m₂₃(t, ε)−nϕ(t, ε))

e^inθ(t,ε), ψ₁₂¹ (t, ε, θ) =−

∞

X

n=−∞

Γ_n[q₁₂¹ (t, ε, θ)−c₃₂(t, ε)(ψ¹₂₂−ψ¹₁₁)−c₂₁(t, ε)ψ₁₃¹ ]

i(m12(t, ε)−m13(t, ε)−nϕ(t, ε)) e^inθ(t,ε), u¹₁₂(t, ε)≡0,

v₁₂¹ (t, ε, θ) = 1 ε

∞

X

n=−∞

d dt

Γn[q¹₁₂(t, ε, θ)−c32(t, ε)(ψ₂₂¹ −ψ₁₁¹ )−c21(t, ε)ψ₁₃¹ ] i(m₁₂(t, ε)−m₁₃(t, ε)−nϕ(t, ε))

e^inθ(t,ε). All the elements of matrix U1 belongs to the class S(m;ε0). All the elements of matrix Ψ1

belongs to the class F(m;ε₀;θ). All the elements of matrix V₁ belongs to the class F(m− 1;ε₀;θ).

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All the equations (12) are considered similarly to equations (11), and so the matrices Ψl, U_l, V_l (l= 1, q) are determined. And also all the elements of matrix Ψ_l belongs to the class F(m;ε₀;θ), all the elements of matrix U_l belongs to the class S(m;ε₀), all the elements of matrix V_l belongs to the class F(m−1;ε0;θ) (l= 1, q). Matrix W are determined from the equations (13).

Lemma 1 are proved.

Problem solving method and basic results.

We seek the transformation of the kind:

x= (E3+µΨ(t, ε, θ, µ))y, (15)

y= colon(y₁, y₂, y₃), E₃ – unit matrix of third order, Ψ(t, ε, θ, µ) =





0 ψ12(t, ε, θ, µ) ψ13(t, ε, θ, µ) 0 0 ψ₂₃(t, ε, θ, µ)

0 0 0



,

ψjk ∈F(m1;ε1;θ) (0≤l1 ≤m; 0≤ε1< ε0), which leads the system (6) to the kind:

dy

dt = (B(t, ε) +µD(t, ε, θ, µ))y, (16)

where

D(t, ε, θ, µ) =





d11(t, ε, θ, µ) 0 0

d₂₁(t, ε, θ, µ) d₂₂(t, ε, θ, µ) 0 d31(t, ε, θ, µ) d32(t, ε, θ, µ) d33(t, ε, θ, µ)



.

We substitute the expression (15) into system (6), and require that the transformed system has the kind (16). We obtain the next system of the differential equations for detemining ψ₁₂, ψ₁₃, ψ₂₃:

dψ12

dt =K12(t, ε, θ, ψ12, ψ13, ψ23, µ),

dψ13

dt =K13(t, ε, θ, ψ12, ψ13, ψ23, µ),

dψ23

dt =K23(t, ε, θ, ψ12, ψ13, ψ23, µ),

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where

K₁₂= (b₁₁(t, ε)−b₂₂(t, ε))ψ₁₂−b₃₂(t, ε)ψ₁₃+p₁₂(t, ε, θ)+

+ +µb₂₁(t, ε)ψ₁₂² +µb₃₂(t, ε)ψ₁₂ψ₂₃−

−µ²p21(t, ε, θ)ψ₁₂² +µ²b31(t, ε)ψ₁₂² ψ23+µ²p32(t, ε, θ)ψ12ψ23+

+µ²b31(t, ε)ψ12ψ13+µ²p32(t, ε, θ)ψ13+µ³p31(t, ε, θ)ψ₁₂² ψ23+µ³p31(t, ε, θ)ψ12ψ13, K13= (b11(t, ε)−b33(t, ε))ψ13+p13(t, ε, θ) +µ(p11(t, ε, θ)−p33(t, ε, θ))ψ13+

+µp12(t, ε, θ)ψ23−µb13(t, ε)ψ₁₃² −µb32(t, ε)ψ13ψ23−

−µ²p₃₁(t, ε, θ)ψ₁₃² −µ²p₃₂(t, ε, θ)ψ₁₃ψ₂₃,

K23= (b22(t, ε)−b33(t, ε))ψ23+b21(t, ε)ψ13+p23(t, ε, θ) +µp21(t, ε, θ)ψ12+ +µ(p22(t, ε, θ)−p33(t, ε, θ))ψ23−µb31(t, ε)ψ13ψ23−

−µb₃₂(t, ε)ψ₂₃² −µ²p₃₁(t, ε, θ)ψ₁₃ψ₂₃−µ²p₃₂(t, ε, θ)ψ²₂₃. In this case d_jk(t, ε, θ, µ) (j ≥k) has a kind:

d31(t, ε, θ) =p31(t, ε, θ),

d₃₂(t, ε, θ, µ) =p₃₂(t, ε, θ) +b₃₁(t, ε)ψ₁₂+µp₃₁(t, ε, θ)ψ₁₂,

d33(t, ε, θ, µ) =p33(t, ε, θ) +b31(t, ε)ψ13+b32(t, ε)ψ23+µ(p31(t, ε, θ)ψ13+p32(t, ε, θ)ψ23), d21(t, ε, θ, µ) =p21(t, ε, θ)−b31(t, ε)ψ13−µp31(t, ε, θ)ψ23,

d22(t, ε, θ, µ) =p22(t, ε, θ)−b21(t, ε)ψ12−b32(t, ε)ψ23+µp21(t, ε, θ)ψ12−µd32(t, ε, θ, µ)ψ23, d₁₁(t, ε, θ, µ) =p₁₁(t, ε, θ)−b₂₁(t, ε)ψ₁₂−b₃₁(t, ε)ψ₁₃−µ(d₂₁(t, ε, θ, µ)ψ₁₂+p₃₁(t, ε, θ)ψ₁₃). (18)

The case 1. |Re(b_jj(t, ε)−b_kk(t, ε))| ≥γ >0 (j6=k). From the results of the paper [6] follows the theorems.

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Theorem 1. In the case 1 there exists µ1 ∈(0, µ0) such that for all µ∈(0, µ1) there exists unique particular solution ψ_jk(t, ε, θ, µ) (j < k) of the system (17), all the components of which belongs to the class F(m;ε₀;θ).

Theorem 2. In the case 1 there exists µ1 ∈(0, µ0) such that for all µ∈(0, µ1) there exists the transformation of the kind (15), whose coefficients ψ_jk(t, ε, θ, µ) (j < k) belongs to the class F(m;ε0;θ), which leads the system (6) to the triangular kind (16), where djk(t, ε, θ, µ) (j≥k) are determined by the formulas (18).

The case 2. bjj(t, ε)−bkk(t, ε) =imjk(t, ε), mjk ∈R, inf

G(ε0)

|m_jk(t, ε)−nϕ(t, ε)| ≥γ >0 ∀n∈Z.

Together with the system (17) we consider the auxiliary system:

ϕ(t, ε)^dψ_dθ¹² =K12(t, ε, θ, ψ12, ψ13, ψ23, µ), ϕ(t, ε)^dψ_dθ¹³ =K13(t, ε, θ, ψ12, ψ13, ψ23, µ), ϕ(t, ε)^dψ_dθ²³ =K23(t, ε, θ, ψ12, ψ13, ψ23, µ),

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where ϕ(t, ε) – function in the definition of the class F(m;ε0;θ), and t, ε are considered as constant. Using the method of the small parameter of Poincarais [7], we construct the partial sums of the series in degrees of the small parameter representing the 2π-periodic with respect to θ solution of the system (19):

ψ^∗_jk(t, ε, θ, µ) =ψ⁰_jk(t, ε, θ) +µψ¹_jk(t, ε, θ) +. . .+µ^2q−1ψ_jk^2q−1(t, ε, θ), (20) where ψ^s_jk(t, ε, θ) (s= 0,2q−1) – 2π-periodic with respect to θ functions. Regarding these functions, we obtain the chain of the system of the differential equations:

ϕ(t, ε)^dψ_dθ⁰¹² =im₁₂(t, ε)ψ₁₂⁰ −b₃₂(t, ε)ψ₁₃⁰ +p₁₂(t, ε, θ), ϕ(t, ε)^dψ_dθ¹³⁰ =im₁₃(t, ε)ψ₁₃⁰ +p₁₃(t, ε, θ), ϕ(t, ε)^dψ_dθ⁰²³ =im23(t, ε)ψ₂₃⁰ +b21(t, ε)ψ₁₃⁰ +p23(t, ε, θ),

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ϕ(t, ε)^dψ_dθ¹²^s =im₁₂(t, ε)ψ₁₂^s −b₃₂(t, ε)ψ₁₃^s + +P₁₂^s(t, ε, θ, ψ⁰₁₂, ψ₁₃⁰ , ψ⁰₂₃, . . . , ψ₁₂^s−1, ψ^s−1₁₃ , ψ^s−1₂₃ ),

ϕ(t, ε)^dψ_dθ¹³^s =im₁₃(t, ε)ψ₁₃⁰ +

+Q^s₁₃(t, ε, θ, ψ₁₂⁰ , ψ₁₃⁰ , ψ⁰₂₃, . . . , ψ₁₂^s−1, ψ^s−1₁₃ , ψ₂₃^s−1), ϕ(t, ε)^dψ_dθ²³^s =im23(t, ε)ψ₂₃^s +b21(t, ε)ψ₁₃^s +

+R^s₂₃(t, ε, θ, ψ⁰₁₂, ψ₁₃⁰ , ψ₂₃⁰ , . . . , ψ^s−1₁₂ , ψ₁₃^s−1, ψ^s−1₂₃ ), s= 1,2, . . . ,2q−1.

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P₁₂^s, Q^s₁₃, R^s₂₃ – polynomials from ψ₁₂⁰ , . . . , ψ^s−1₂₃ with coefficients from the class F(m;ε₀;θ). Consider a generating system (21). In the case 2 this system has unique 2π-periodic with respect to θ solution:

ψ₁₃⁰ (t, ε, θ) =

∞

X

n=−∞

ψ_13,n⁰ (t, ε) exp(inθ), where

ψ_13,n⁰ (t, ε) =− Γ_n[p₁₃(t, ε, θ)]

i(m₁₃(t, ε)−nϕ(t, ε)),

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ψ₁₂⁰ (t, ε, θ) =

∞

X

n=−∞

ψ⁰_12,n(t, ε) =−Γ_n[p₁₂(t, ε, θ)]−b₃₂(t, ε)ψ_13,n⁰ (t, ε) i(m12(t, ε)−nϕ(t, ε)) , ψ₂₃⁰ (t, ε, θ) =

∞

X

n=−∞

ψ⁰_23,n(t, ε) =−Γn[p23(t, ε, θ)] +b21(t, ε)ψ_13,n⁰ (t, ε) i(m23(t, ε)−nϕ(t, ε)) , and ψ₁₃⁰ (t, ε, θ), ψ₁₂⁰ (t, ε, θ), ψ₂₃⁰ (t, ε, θ) belongs to the class F(m;ε0;θ).

Similarly, all systems in the chain (22) also have a unique 2π-periodic with respect to θ solutions, and all components of theese solutions belongs to the class F(m;ε0;θ).

Consequently, the functions ψ^∗_jk(t, ε, θ, µ) belongs to the class F(m;ε0;θ) also.

We make in the system (17) the substitution:

ψ_jk =ψ_jk^∗ (t, ε, θ, µ) +ξ_jk (j < k). (23) We obtain:

dξ12

dt =im12(t, ε)ξ12−b32(t, ε)ξ13+εg12(t, ε, θ, µ)+

+µ^2qc12(t, ε, θ, µ) +

q

X

l=1

b_12l(t, ε, θ)µ^l

! ξ12+

q

X

l=1

c_12l(t, ε, θ)µ^l

! ξ13+ +

q

X

l=1

d_12l(t, ε, θ)µ^l

!

ξ₂₃+µ^q+1(α₁₂(t, ε, θ, µ)ξ₁₂+β₁₂(t, ε, θ, µ)ξ₁₃+ +γ12(t, ε, θ, µ)ξ23) +µΞ12(t, ε, θ, ξ12, ξ13, ξ23, µ),

dξ₁₃

dt =im₁₃(t, ε)ξ₁₃+εg₁₃(t, ε, θ, µ)+

+µ^2qc₁₃(t, ε, θ, µ) +

q

X

l=1

b_13l(t, ε, θ)µ^l

! ξ₁₂+

q

X

l=1

c_13l(t, ε, θ)µ^l

! ξ₁₃+ +

q

X

l=1

!

ξ23+µ^q+1(α13(t, ε, θ, µ)ξ12+β13(t, ε, θ, µ)ξ13+ +γ13(t, ε, θ, µ)ξ23) +µΞ13(t, ε, θ, ξ12, ξ13, ξ23, µ),

dξ23

dt =im23(t, ε)ξ23+b21(t, ε)ξ13+εg23(t, ε, θ, µ)+

+µ^2qc23(t, ε, θ, µ) +

q

X

l=1

b23l(t, ε, θ)µ^l

! ξ12+

q

X

l=1

c23l(t, ε, θ)µ^l

! ξ13+ +

q

X

l=1

!

ξ₂₃+µ^q+1(α₂₃(t, ε, θ, µ)ξ₁₂+β₂₃(t, ε, θ, µ)ξ₁₃+

+γ₂₃(t, ε, θ, µ)ξ₂₃) +µΞ₂₃(t, ε, θ, ξ₁₂, ξ₁₃, ξ₂₃, µ), (24) where g₁₂, g₁₃, g₂₃∈F(m−1;ε₀;θ), c₁₂, c₁₃, c₂₃, b_jkl, c_jkl, d_jkl, α_jk, β_jk,

γjk ∈F(m;ε0;θ) (j < k), Ξ12,Ξ13,Ξ23 – polynomials with respect to ξ12, ξ13, ξ23 with coefficients from the class F(m;ε₀;θ), containing terms not lower than second order with respect ξ12, ξ13, ξ23.

We introduce ξ = colon(ξ₁₂, ξ₁₃, ξ₂₃), A₁(t, ε) =





im₁₂(t, ε) −b₃₂t, ε) 0 0 im13(t, ε) 0 0 b₂₁(t, ε) im₂₃(t, ε)



,

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g(t, ε, θ, µ) = colon(g12(t, ε, θ, µ), g13(t, ε, θ, µ), g23(t, ε, θ, µ)), c(t, ε, θ, µ) = colon(c₁₂(t, ε, θ, µ), c₁₃(t, ε, θ, µ), c₂₃(t, ε, θ, µ)),

K_l(t, ε, θ) =





b_12l(t, ε, θ) c_12lt, ε, θ) d_12lt, ε, θ) b_13l(t, ε, θ) c_13l(t, ε, θ) d_13l(t, ε, θ) b23l(t, ε, θ) c23l(t, ε, θ) d23l(t, ε, θ)



,

L(t, ε, θ, µ) =





α12(t, ε, θ, µ) β12t, ε, θ, µ) γ12t, ε, θ, µ) α₁₃(t, ε, θ, µ) β₁₃t, ε, θ, µ) γ₁₃t, ε, θ, µ) α23(t, ε, θ, µ) β23t, ε, θ, µ) γ23t, ε, θ, µ)



, Ξ(t, ε, θ, ξ, µ) = colon(Ξ12(t, ε, θ, ξ12, ξ13, ξ23, µ),Ξ13(t, ε, θ, ξ12, ξ13, ξ23, µ), Ξ₂₃(t, ε, θ, ξ₁₂, ξ₁₃, ξ₂₃, µ)).

Then the system (24) can be written as:

dξ

dt = A1(t, ε) +

q

X

l=1

K_l(t, ε, θ)µ^l

!

ξ+εg(t, ε, θ, µ) +µ^2qc(t, ε, θ, µ)+

+µ^q+1L(t, ε, θ, µ)ξ+ Ξ(t, ε, θ, ξ, µ). (25)

Based on Lemma 1, using the conditions (8) and the transformation of kind:

ξ= E+

q

X

l=1

Ψ_l(t, ε, θ)µ^l

!

η, (26)

where η= colon(η1, η2, η3), we leads the system (25) to the kind:

dη

dt = A₁(t, ε) +

q

X

l=1

U_l(t, ε)µ^l

!

η+εg¹(t, ε, θ, µ) +µ^2qc¹(t, ε, θ, µ)+

+ε

q

X

l=1

V_l(t, ε, θ)µ^l

!

η+µ^q+1L¹(t, ε, θ, µ) +µH(t, ε, θ, η, µ), (27) where Ul(t, ε) = diag(u1l(t, ε), u2l(t, ε), u3l(t, ε)), and ujl(t, ε)∈S(m;ε0) (j= 1,2,3; l= 1, q).

Lemma 2. Let the system (27) satisfy the next conditions: 1) the eigenvalues λj(t, ε, µ) (j = 1,2,3) of the matrix

U(t, ε, µ) =A₁(t, ε) +

q

X

l=1

U_l(t, ε)µ^l such that

inf

G(ε0)

|Reλ_j(t, ε, θ)| ≥γ0µ^q⁰ (γ0 ≥0, 0< q0 ≤q);

2) for the matrix U(t, ε, µ) there exists the matrix Y(t, ε, µ) such that a) inf

G(ε0)

|detY(t, ε, µ)|>0,

b) Y⁻¹U Y = Λ(t, ε, µ) – diagonal matrix.

Then there exists µ₂ ∈ (0, µ₀), ε₁(µ) ∈ (0, ε₀) such that for all µ ∈ (0, µ₂) and for all ε∈(0, ε1(µ)) there exists the particular solution of the system (27), all the components of which belongs to the class F(m−1;ε1(µ);θ).

Proof. Based on condition 2) of Lemma, we make in the system (27) the substitution:

η= ε+µ^2q

µ^q⁰ Y(t, ε, µ)χ. (28)

We obtain:

dχ

dt = Λ(t, ε, µ)χ+ εµ^q⁰

ε+µ^2q g²(t, ε, θ, µ) + µ^2q+q⁰

ε+µ^2q c²(t, ε, θ, µ)+

+εA₂(t, ε, θ, µ)χ+µ^q+1C(t, ε, θ, µ)χ+ε+µ^2q

µ^q⁰⁻¹ X(t, ε, θ, χ, µ), (29)

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where elements of vector g² and matrix A2 belongs to the class F(m−1;ε0;θ), elements of vector c² and matrix C belongs to the class F(m;ε₀;θ), elements of vector-function X belongs to the class F(m;ε₀;θ) in respect to t, ε, θ and polynomials in respect to elements of vector χ.

Together with the system (29) we consider the linear nonhomogeneous system:

dχ⁰

dt = Λ(t, ε, µ)χ⁰+ εµ^q⁰

ε+µ^2q g²(t, ε, θ, µ) + µ^2q+q⁰

ε+µ^2q c²(t, ε, θ, µ). (30) From the results of the paper [6], based on conditions 1) of Lemma, we obtain, that there exists particular solution χ⁰(t, ε, θ, µ) of the system (30), all elements of which belongs to the class F(m−1;ε₀;θ), and there exists K∈(0,+∞) such that:

kχ⁰k^∗_F_(m−1;ε

0;θ) ≤ K γµ^q⁰

εµ^q⁰

ε+µ^2q kg²k^∗_F(m−1;ε

0;θ)+ µ^2q+q⁰

ε+µ^2q kc²k^∗_F_(m−1;ε

0;θ)

<

< K γ

kg²k^∗_F_(m−1;ε

0;θ)+kc²k^∗_F_(m−1;ε

0;θ)

.

We construct the process of succesive approximation, defininng as initial approximation χ⁰, and subsequent approximations defining as solutions from the class F(m−1;ε₀;θ) of the systems:

dχ^j+1

dt = Λ(t, ε, µ)χ^j+1+ εµ^q⁰

ε+µ^2q g²(t, ε, θ, µ) + µ^2q+q⁰

ε+µ^2q c²(t, ε, θ, µ)+

+εA₂(t, ε, θ, µ)χ^j+µ^q+1C(t, ε, θ, µ)χ^j+ ε+µ^2q

µ^q⁰⁻¹ X(t, ε, θ, χ^j, µ), j= 0,1,2, .... (31) Using an usual techniques contraction mapping principle [8] it is easy to show that there exists µ3∈(0, µ0) and ε1(µ) =K2µ, where K2 – sufficiently small constant, such that for all µ∈(0, µ₃) and for all ε∈(0, ε₁(µ)) the process (31) converges to the solution χ(t, ε, θ, µ) of the system (29), and all components of this solution belongs to the class F(m−1;ε1(µ);θ).

Lemma 2 are proved.

The following statements are an immediate consequences of Lemma 2.

Lemma 3. Let the system (17) be such that:

1) conditions (8) are satisfies;

2) for the system (27), obtained from the system (17) using the transformation (23), (26), all conditions of Lemma 2 are satisfies.

Then there exists µ₄ ∈ (0, µ₀), ε₂(µ) ∈ (0, ε₀) such that for all µ ∈ (0, µ₄) and for all ε∈(0, ε2(µ)) there exists the particular solution of the system (17), all the components of which belongs to the class F(m−1;ε₂(µ);θ).

Theorem 3. Let the system (17) satisfies all conditions of Lemma 3. Then in the case 2 there exists µ₄ ∈(0, µ₀), ε₂(µ)∈(0, µ₀) such that for all µ∈(0, µ₄) and for all ε∈(0, ε₂(µ)) there exists the transformation of the kind (15), whose coefficients ψjk(t, ε, θ, µ) (j < k) belongs to the class F(m−1;ε₂(µ);θ), which leads the system (6) to a triangular kind (16), where d_jk(t, ε, θ, µ) (j≥k) are determines by the formulas (18).

Conclusions. Thus, for the system (2) the conditions of the existence of the transformation with coefficients are represented as an absolutely and uniformly convergent Fourier-series with slowly varying coefficients and frequency, which leads it to triangular kind, are obtained in the non-resonant cases.

References

1 Perron O. Uber eine Matrixtransformation// Math. Zeitschr. – 1930. – V.32. – pp. 465–473.

2 Персидский К.П. О характеристичных числах дифференциальных уравнений // Изв. АН КазССР, сер.

матем. и механ. – 1947, вып. 1. – P. 5–47.

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3 Изобов Н. А. О канонической форме линейной двумерной дифференциальной системы // Дифференц.

уравн. – 1971. – Т. 7, № 12. – С. 2136–2142.

4 Костин А. В. Устойчивость и асимптотика квазилинейных неавтономных дифференциальных систем.

Одесса, ОГУ, 1984. – 95 с.

5 Щоголев С.А. Деякi задачi теорiї коливань для диференцiальних систем, якi мiстять повiльно змiннi параметри. Дисертацiя на здобуття наукового ступеня доктора фiзико-математичних наук. Київ, 2012.

– 290 с.

6 Костин А.В., Щёголев С.А. Об устойчивости колебаний, представимых рядами Фурье с медленно меняющимися параметрами // Дифференц. уравн. – 2008. – Т. 44, № 1. – С. 45 – 51.

7 Малкин И. Г. Некоторые задачи теории нелинейных колебаний. – М.: Гостехиздат, 1956. – 491 с.

8 Треногин В. А. Функциональный анализ. М.: Наука, 1980. – 496 с.

С.А. Щёголев

И.И. Мечников атындағы Одесса ұлттық университетi, Одесса, Украина

Резонансты емес жағдайда осцилляциялы типтi коэффициенттi сызықты дифференциалдық теңдеулер жүйесiн үшбұрышты түрге келтiру туралы

Аннотация: Коэффициенттерi мен жиiлiктерi баяу өзгереiп, абсолюттi және бiрқалыпты жинақталатын Фурье қатарлары түрiнде өрнектелетiн сызықты бiртектi дифференциалдық теңдеулер жүйесi үшiн осы жүйенi резонансты емес жағдайда үшбұрышты түрге келтiретiн түрлендiрудiң бар болу шарттары алынған.

Түйiн сөздерсызықты дифференциалдық жүйелер, Фурье қатарлары.

С. А. Щёголев

Одесский национальный университет имени И. И. Мечникова, Украина

О приведении линейной системы дифференциальных уравнений с коэффициентами осциллирующего типа к треугольному виду в нерезонансном случае

Аннотация: Для линейной однородной дифференциальной системы, коэффициенты которой представимы в виде абсолютно и равномерно сходящихся рядов Фурье с медленно меняющимися коэффициентами и частотой, получены условия существования преобразования, приводящего эту систему к треугольному виду в нерезонансном случае.

Ключевые слова: линейные дифференциальные системы, ряды Фурье.

References

1 Perron O. Uber eine Matrixtransformation, Math. Zeitschr. 1930. V.32. P. 465–473.

2 Persidsky K.P. O kharakteristicnyh chislah differentsialnyh uravnenyi [On the characteristic numbers of the differential equations], Izv. AN KazSSR, ser. math. and mechan. 1947. № 1. P. 5–47.

3 Izobov N.A. O kanonicheskoi forme lineynoi dvumernoi differentsialnoi sistemy [On the canonical form of the linear two-dimensional differential system], Differencial’nye uravneniya [Differential equations]. 1971. V. 7. № 12. P. 2136–2142.

4 Кostin A.V. Ustoychivostj i asymptotica kvazilineynyh neavtonomnyh differentsialnyh sistem [The stability and asyptotics of the nonautonomous differential systems] (Odessa, OGU, 1984, 95 p.).

5 Shchogolev S.A. Dejaki zadachi teorii kolyvanj dlja differentsialnyh sistem, yaki mistyatj povilno zminni parametry [The some problem of the theory of oscillations for the differential systems, containing slowly varying parameters], The thesis for obtaining the scientific degree of Doctor of physical and mathematical sciencies. Kyiv, 2012. 290 p.

6 Kostin A.V., Shchogolev S.A. Ob ustoychivosti kolebaniy, predstavimyh ryadami Furye s medlenno menya- juchimisya parametravi [On the Stability of Oscillations Representable by Fourier Series with Slowly Varying Parameters], Differencial’nye uravneniya [Differential equations]. 2008. V. 44. № 1. P. 45-51.

7 Мalkin I.G. Nekotorye zadachi teorii nelineynyih kolebaniy [Some problems of the theory of nonlinear oscillations] (Gostehizdat, Moscow, 1956, 491 p.).

8 Trenogin V.A. Funktsionalnyi analiz [Functional analysis] (Nauka, Moscow, 1980, 496 p.).

Сведения об авторах:

Щёголев С.А. – физика-математика ғылымдарының докторы, профессор, И.И. Мечников атындағы Одесса ұлттық университетi, Дворянская көш., 2, Одесса, 65026, Украина.

Shchogolev S.A.– Prof., Doctor of Phys. -Math. Sciences, Odessa I.I. Machnikov National University, Dvoryanskaya str., 2, Odessa, 65082, Ukraine.

Поступила в редакцию 17.02.2020