Limit behavior of autonomous random oscillating system of third order
The asymptotic behavior of the general type third order autonomous oscillating system under the action of small non-linear random perturbations of ”white” and ”Poisson” types is investigated.
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irk-123456789-45112009-11-25T12:00:28Z Limit behavior of autonomous random oscillating system of third order Borysenko, O.D. Borysenko, O.V. The asymptotic behavior of the general type third order autonomous oscillating system under the action of small non-linear random perturbations of ”white” and ”Poisson” types is investigated. 2007 Article Limit behavior of autonomous random oscillating system of third order / O.D. Borysenko, O.V. Borysenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 19–28. — Бібліогр.: 13 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4511 en Інститут математики НАН України |
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The asymptotic behavior of the general type third order autonomous oscillating system under the action of small non-linear random perturbations of ”white” and ”Poisson” types is investigated. |
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Article |
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Borysenko, O.D. Borysenko, O.V. |
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Borysenko, O.D. Borysenko, O.V. Limit behavior of autonomous random oscillating system of third order |
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Borysenko, O.D. Borysenko, O.V. |
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Borysenko, O.D. |
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Limit behavior of autonomous random oscillating system of third order |
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Limit behavior of autonomous random oscillating system of third order |
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Limit behavior of autonomous random oscillating system of third order |
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Limit behavior of autonomous random oscillating system of third order |
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Limit behavior of autonomous random oscillating system of third order |
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limit behavior of autonomous random oscillating system of third order |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4511 |
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Limit behavior of autonomous random oscillating system of third order / O.D. Borysenko, O.V. Borysenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 19–28. — Бібліогр.: 13 назв.— англ. |
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AT borysenkood limitbehaviorofautonomousrandomoscillatingsystemofthirdorder AT borysenkoov limitbehaviorofautonomousrandomoscillatingsystemofthirdorder |
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2025-07-02T07:44:18Z |
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2025-07-02T07:44:18Z |
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1836520318638751744 |
| fulltext |
Theory of Stochastic Processes
Vol.13 (29), no.4, 2007, pp.19–28
OLEKSANDR D. BORYSENKO AND OLGA V. BORYSENKO
LIMIT BEHAVIOR OF AUTONOMOUS RANDOM
OSCILLATING SYSTEM OF THIRD ORDER
The asymptotic behavior of the general type third order autonomous
oscillating system under the action of small non-linear random per-
turbations of ”white” and ”Poisson” types is investigated.
1. Introduction
The averaging method proposed by N.M.Krylov, N.N.Bogolyubov and
Yu.A.Mytropolskij ([1], [2]) is one of the main tool in studying of the deter-
ministic oscillating systems under the action of small non-linear perturba-
tions. The case of small random ”white noise” type disturbances in oscillat-
ing systems of second order is considered in papers of Yu.A.Mytropolskij,
V.G.Kolomiets ([3]). The autonomous and non-autonomous oscillating sys-
tems of second order under the action of ”white noise” and Poisson type
noise perturbations are studied in the papers of O.V.Borysenko ([4], [5]).
Particular case of the third order oscillating systems are investigated in arti-
cles of O.D.Borysenko, O.V.Borysenko ([6]), O.D.Borysenko, O.V.Borysen-
ko and I.G.Malyshev ([7], [8]).
This paper deals with investigation of the behaviour, as ε → 0, of
the general type third order autonomous oscillating system described by
stochastic differential equation
x′′′(t) + ax′′(t) + b2x′(t) + ab2x(t) =
= εk1f1(x(t), x′(t), x′′(t)) + fε(x(t), x′(t), x′′(t))
(1)
with non-random initial conditions x(0) = x0, x
′(0) = x′
0, x
′′(0) = x′′
0, where
ε > 0 is a small parameter, fε(x, x′, x′′) is a random function such that
∫ t
0
fε(x(s), x′(s), x′′(s)) ds = εk2
∫ t
0
f2(x(s), x′(s), x′′(s)) dw(s)+
2000 Mathematics Subject Classifications. 60H10
Key words and phrases. Asymptotic behavior, third order autonomous oscillating
system, small non-linear random perturbations
19
20 O.D.BORYSENKO AND O.V.BORYSENKO
+εk3
∫ t
0
∫
R
f3(x(s), x′(s), x′′(s), z) ν̃(ds, dz),
ki > 0, i = 1, 2, 3; fi, i = 1, 2, 3 are non-random functions; w(t) is a stan-
dard Wiener process; ν̃(dt, dy) = ν(dt, dy)−Π(dy)dt, Eν(dt, dy) = Π(dy)dt,
ν(dt, dy) is the Poisson measure independent on w(t); Π(A) is a finite mea-
sure on Borel sets A ∈ R, a > 0, b > 0.
We will consider the equation (1) as the system of stochastic differential
equations
dx(t) = x′(t)dt,
dx′(t) = x′′(t)dt,
dx′′(t) = [−ax′′(t) − b2x′(t) − ab2x(t)+
+ εk1f1(x(t), x′(t), x′′(t))]dt+
+ εk2f2(x(t), x′(t), x′′(t))dw(t)+
+ εk3
∫
R
f3(x(t), x′(t), x′′(t), z)ν̃(dt, dz),
x(0) = x0, x′(0) = x′
0, x′′(0) = x′′
0.
(2)
In what follows we will use the constant K > 0 for the notation of different
constants, which are not depend on ε.
2. Auxiliary result
From Borysenko O. and Malyshev I. [9], using the obvious modifications
we obtain following results
Lemma. Let for each x ∈ Rd there exists
lim
T→∞
1
T
∫ T+A
A
f(t, x) dt = f̄(x)
uniformly with respect to A, the function f̄(x) is bounded, continuous, func-
tion f(t, x) is bounded and continuous in x uniformly with respect to (t, x) in
any region t ∈ [0,∞), |x| ≤ K, and stochastic processes ξ(t) ∈ Rd, η(t) ∈ R
are continuous, then
lim
ε→0
∫ t
0
f
(s
ε
+ η(s), ξ(s)
)
ds =
∫ t
0
f̄(ξ(s)) ds
almost surely for all arbitrary t ∈ [0, T ].
LIMIT BEHAVIOR 21
Remark. Let f(t, x, z) is bounded and uniformly continuous in x with
respect to t ∈ [0,∞) and z ∈ R in every compact set |x| ≤ K, x ∈ Rd. Let
Π(·) be a finite measure on the σ-algebra of Borel sets in R and let
lim
T→∞
1
T
∫ T+A
A
f(t, x, z) dt = f̄(x, z),
uniformly with respect to A for each x ∈ Rd, z ∈ R, where f̄(x, z) is
bounded, uniformly continuous in x with respect to z ∈ R in every compact
set |x| ≤ K. Then for any continuous processes ξ(t) ∈ Rd and η(t) ∈ R we
have
lim
ε→0
∫ t
0
∫
R
f
(s
ε
+ η(s), ξ(s), z
)
Π(dz)ds =
∫ t
0
∫
R
f̄(ξ(s), z) Π(dz)ds.
3. Main result
Let us consider the following representation of processes x(t), x′(t), x′′(t):
x(t) = C(t) exp{−at} + A1(t) cos(bt) + A2(t) sin(bt),
x′(t) = −aC(t) exp{−at} − bA1(t) sin(bt) + bA2(t) cos(bt),
x′′(t) = a2C(t) exp{−at} − b2A1(t) cos(bt) − b2A2(t) sin(bt),
N(t) = C(t) exp{−at}.
Then
N(t) =
b2x(t) + x′′(t)
a2 + b2
,
A1(t) = cosα cos(bt + α)x(t) − sin bt
b
x′(t) − sin α sin(bt + α)
b2
x′′(t),
A2(t) = cos α sin(bt + α)x(t) +
cos bt
b
x′(t) +
sin α cos(bt + α)
b2
x′′(t),
where α = arctg (b/a). We can apply Ito formula [10] to stochastic process
ξ(t) = (N(t), A1(t), A2(t)) and obtain for the process ξ(t) the system of
stochastic differential equations
dN(t) =
[
−aN(t) +
εk1
a2 + b2
f̃1(t, N(t), A1(t), A2(t))
]
dt+
+
εk2
a2 + b2
f̃2(t, N(t), A1(t), A2(t))dw(t)+
+
εk3
a2 + b2
∫
R
f̃3(t, N(t), A1(t), A2(t), z)ν̃(dt, dz),
22 O.D.BORYSENKO AND O.V.BORYSENKO
dA1(t) = −sin α sin(bt + α)
b2
[εk1 f̃1(t, N(t), A1(t), A2(t))dt+ (3)
+εk2 f̃2(t, N(t), A1(t), A2(t))dw(t)+
+εk3
∫
R
f̃3(t, N(t), A1(t), A2(t), z)ν̃(dt, dz)],
dA2(t) =
sin α cos(bt + α)
b2
[εk1 f̃1(t, N(t), A1(t), A2(t))dt+
+εk2 f̃2(t, N(t), A1(t), A2(t))dw(t)+
+εk3
∫
R
f̃3(t, N(t), A1(t), A2(t), z)ν̃(dt, dz)],
N(0) =
b2x0 + x′′
0
a2 + b2
, A1(0) =
a2x0 − x′′
0
a2 + b2
, A2(0) =
ax′′
0 + (a2 + b2)x′
0 + ab2x0
b(a2 + b2)
,
where f̃i(t, N, A1, A2) = fi(N + A1 cos bt + A2 sin bt,−aN − bA1 sin bt +
bA2 cos bt, a2N − b2A1 cos bt − b2A2 sin bt), i = 1, 2, f̃3(t, N, A1, A2, z) =
f3(N +A1 cos bt+A2 sin bt,−aN −bA1 sin bt+bA2 cos bt, a2N −b2A1 cos bt−
b2A2 sin bt, z).
Theorem. Let Π(R) < ∞, t ∈ [0, t0], k = min(k1, 2k2, 2k3). Let us suppose,
that functions fi, i = 1, 3 bounded and satisfy Lipschitz condition on x, x′, x′′.
If given below matrix σ̄2(A1, A2) is non-negative definite, then
1. For k1 = 2k2 = 2k3 the stochastic process ξε(t) = ξ(t/εk) weakly
converges, as ε → 0, to the stochastic process ξ̄(t) = (0, Ā1(t), Ā2(t)), where
Ā(t) = (Ā1(t), Ā2(t)) is the solution to the system of stochastic differential
equations
dĀ(t) = ᾱ(Ā(t))dt + σ̄(Ā(t))dw̄(t), Ā(0) = (A1(0), A2(0)), (4)
where ᾱ(Ā) = (ᾱ(1)(A1, A2), ᾱ
(2)(A1, A2)),
ᾱ(1)(A1, A2) = − 1
2πb(a2 + b2)
2π∫
0
f̂1(ψ, A1, A2)(a sin ψ + b cos ψ) dψ,
ᾱ(2)(A1, A2) =
1
2πb(a2 + b2)
2π∫
0
f̂1(ψ, A1, A2)(a cos ψ − b sin ψ) dψ,
σ̄(A1, A2) =
{
B̄(A1, A2)
} 1
2 =
⎧⎨
⎩
1
2πb2(a2 + b2)2
2π∫
0
f̂(ψ, A1, A2)B(ψ) dψ
⎫⎬
⎭
1
2
,
B(ψ) = (Bij(ψ), i, j = 1, 2), B11(ψ) = (a sin ψ + b cos ψ)2,
LIMIT BEHAVIOR 23
B12(ψ) = B21(ψ) = −(a sin ψ + b cos ψ)(a cos ψ − b sin ψ),
B22(ψ) = (a cos ψ − b sin ψ)2, f̂i(ψ, A1, A2) = f̃i(ψ, 0, A1, A2), i = 1, 2,
f̂3(ψ, A1, A2, z) = f̃3(ψ, 0, A1, A2, z),
f̂(ψ, A1, A2) = f̂ 2
2 (ψ, A1, A2) +
∫
R
f̂ 2
3 (ψ, A1, A2, z) Π(dz),
w̄(t) = (wi(t), i = 1, 2), wi(t), i = 1, 2 – independent one-dimensional
Wiener processes.
2. If k < k1 then in the averaging equation (4) we must put f̃1 ≡ 0; if
k < 2k2 then in the averaging equation (4) we must put f̃2 ≡ 0; if k < 2k3
then in the averaging equation (4) we must put f̃3 ≡ 0.
Proof. Let us make a change of variable t → t/εk in equation (3) and obtain
for the process ξε(t) = (Nε(t), A
ε
1(t), A
ε
2(t)) = (N(t/εk), A1(t/ε
k), A2(t/ε
k))
the system of stochastic differential equations
dNε(t) =
[
− a
εk
Nε(t) +
εk1−k
a2 + b2
f̃1(t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t))
]
dt+
+
εk2−k/2
a2 + b2
f̃2(t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t))dwε(t)+
+
εk3
a2 + b2
∫
R
f̃3(t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t), z)ν̃ε(dt, dz),
dAε
1(t) = −sin α sin(bt/εk + α)
b2
[εk1−kf̃1(t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t))dt+ (5)
+εk2−k/2f̃2(t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t))dwε(t)+
+εk3
∫
R
f̃3(t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t), z)ν̃ε(dt, dz)],
dAε
2(t) =
sin α cos(bt/εk + α)
b2
[εk1−kf̃1(t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t))+
+εk2−k/2f̃2(t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t))dwε(t)+
+εk3
∫
R
f̃3(t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t), z)ν̃ε(dt, dz)],
where wε(t) = εk/2w(t/εk), ν̃ε(t, A) = ν(t/εk, A)−Π(A)t/εk, here A is Borel
set in R. For any ε > 0 the process wε(t) is the Wiener process and ν̃ε(t, A)
is the centered Poisson measure independent on wε(t).
Since we have relationship Nε(t) = exp{−at/εk}C(t/εk) and process
Cε(t) = C(t/εk) satisfies the stochastic equation
Cε(t) = C(0) + εk1−k
∫ t
0
exp{as/εk}
a2 + b2
f̃1(s/ε
k, Nε(s), A
ε
1(s), A
ε
2(s)) ds+
24 O.D.BORYSENKO AND O.V.BORYSENKO
+εk2−k/2
∫ t
0
exp{as/εk}
a2 + b2
f̃2(s/ε
k, Nε(s), A
ε
1(s), A
ε
2(s)) dwε(s)+
+εk3
∫ t
0
∫
R
exp{as/εk}
a2 + b2
f̃3(s/ε
k, Nε(s), A
ε
1(s), A
ε
2(s), z) ν̃ε(dt, dz),
where C(0) =
b2x0+x′′
0
a2+b2
, we can obtain estimate
E|Nε(t)|2 ≤ K[e−2at/εk
+ εk(1 − e−2at/εk
)(tε2(k1−k) + ε2k2−k + ε2k3−k)].
Therefore limε→0 E|Nε(t)|2 = 0 and it is sufficient to study the behaviour,
as ε → 0, of solution to the system of stochastic differential equations
dAε
1(t) = −sin α sin(bt/εk + α)
b2
[εk1−kf̂1(t/ε
k, Aε
1(t), A
ε
2(t))dt+
+εk2−k/2f̂2(t/ε
k, Aε
1(t), A
ε
2(t))dwε(t)+
+εk3
∫
R
f̂3(t/ε
k, Aε
1(t), A
ε
2(t), z)ν̃ε(dt, dz)], (6)
dAε
2(t) =
sin α cos(bt/εk + α)
b2
[εk1−kf̂1(t/ε
k, Aε
1(t), A
ε
2(t))+
+εk2−k/2f̂2(t/ε
k, Aε
1(t), A
ε
2(t))dwε(t)+
+εk3
∫
R
f̂3(t/ε
k, Aε
1(t), A
ε
2(t), z)ν̃ε(dt, dz)],
with initial conditions Aε
1(0) = A1(0), Aε
2(0) = A2(0).
Let us denote Aε(t) = (Aε
1(t), A
ε
2(t)). Using conditions on coefficients of
equation (6) and properties of stochastic integrals we obtain estimates
E||Aε(t)||2 ≤ K[1 + t2ε2(k1−k) + t(ε2k2−k + ε2k3−k)],
E||Aε(t) − Aε(s)||2 ≤ K[|t − s|2ε2(k1−k) + |t − s|(ε2k2−k + ε2k3−k)].
Similarly for the process ζε(t) = (ζε
1(t), ζ
ε
2(t)), where
ζε
1(t) = −εk2−k/2
∫ t
0
sin α sin(bs/εk + α)
b2
f̂2(s/ε
k, Aε
1(s), A
ε
2(s))dwε(s)−
−εk3
∫ t
0
∫
R
sin α sin(bs/εk + α)
b2
f̂3(s/ε
k, Aε
1(s), A
ε
2(s), z)ν̃ε(ds, dz)],
ζε
2(t) = εk2−k/2
∫ t
0
sin α cos(bs/εk + α)
b2
f̂2(s/ε
k, Aε
1(s), A
ε
2(s))dwε(s)+
+εk3
∫ t
0
∫
R
sin α cos(bs/εk + α)
b2
f̂3(s/ε
k, Aε
1(s), A
ε
2(s), z)ν̃ε(ds, dz)]
LIMIT BEHAVIOR 25
we derive estimates
E||ζε(t)||2 ≤ Kt(ε2k2−k+ε2k3−k), E||ζε(t)−ζε(s)||2 ≤ K|t−s|(ε2k2−k+ε2k3−k).
Therefore for stochastic process ηε(t) = (Aε(t), ζε(t)) conditions of weak
compactness [11] are fulfilled
lim
h↓0
lim
ε→0
sup
|t−s|<h
P{|ηε(t) − ηε(s)| > δ} = 0 for any δ > 0, t, s ∈ [0, T ],
lim
N→∞
lim
ε→0
sup
t∈[0,T ]
P{|ηε(t)| > N} = 0,
and for any sequence εn → 0, n = 1, 2, . . . there exists a subsequence
εm = εn(m) → 0, m = 1, 2, . . ., probability space, stochastic processes
Āεm(t) = (Āεm
1 (t), Āεm
2 (t)), ζ̄εm(t), Ā(t) = (Ā1(t), Ā2(t)), ζ̄(t) defined on this
space, such that Āεm(t) → Ā(t), ζ̄εm(t) → ζ̄(t) in probability, as εm → 0,
and finite-dimensional distributions of Āεm(t), ζ̄εm(t) are coincide with finite-
dimensional distributions of Aεm(t), ζεm(t). Since we interesting in limit
behaviour of distributions, we can consider processes Aεm(t), and ζεm(t)
instead of Āεm(t), ζ̄εm(t). From (6) we obtain equation
Aεm(t) = A(0) +
t∫
0
αεm(s, Aεm(s)) ds + ζεm(t), A0 = (A1(0), A2(0)), (7)
where αε(t, A) = (α
(1)
ε (t, A1, A2), α
(2)
ε (t, A1, A2)),
α(1)
ε (t, A1, A2) = −εk1−k sin α sin(bt/εk + α)
b2
f̂1(t/ε
k, A1, A2),
α(2)
ε (t, A1, A2) = εk1−k sin α cos(bt/εk + α)
b2
f̂1(t/ε
k, A1, A2).
It should be noted that process ζε(t) is the vector-valued square integrable
martingale with matrix characteristic
〈ζ (i)
ε , ζ (j)
ε 〉(t) =
t∫
0
σ(i)
ε (s, Aε
1(s), A
ε
2(s))σ
(j)
ε (s, Aε
1(s), A
ε
2(s)) ds+
+
1
εk
t∫
0
∫
R
γ(i)
ε (s, Aε
1(s), A
ε
2(s), z)γ(j)
ε (s, Aε
1(s), A
ε
2(s), z) Π(dz)ds, i, j = 1, 2,
where
σ(1)
ε (s, A1, A2) = −εk2−k/2 sin α sin(bs/εk + α)
b2
f̂2(s/ε
k, A1, A2),
26 O.D.BORYSENKO AND O.V.BORYSENKO
σ(2)
ε (s, A1, A2) = εk2−k/2 sin α cos(bs/εk + α)
b2
f̂2(s/ε
k, A1, A2),
γ(1)
ε (s, A1, A2, z) = −εk3
sin α sin(bs/εk + α)
b2
f̂3(s/ε
k, A1, A2, z),
γ(2)
ε (s, A1, A2, z) = εk3
sin α cos(bs/εk + α)
b2
f̂3(s/ε
k, A1, A2, z).
For processes Aε(t) and ζε(t) following estimates hold
E||Aε(t) − Aε(s)||4 ≤ K
[
ε4(k1−k)|t − s|4 + E||ζε(t) − ζε(s)||4
]
, (8)
E||ζε(t) − ζε(s)||4 ≤ K
[
(ε4k2−2k + ε4k3−2k)|t − s|2+
+ε4k3−3k/2|t − s|3/2 + ε4k3−k|t − s|] , (9)
E||Aε(t) − Aε(s)||8 ≤ K, E||ζε(t) − ζε(s)||8 ≤ K. (10)
Since Aεm(t) → Ā(t), ζεm(t) → ζ̄(t) in probability, as εm → 0, then, using
(10), from (8) and (9) we obtain estimates
E||Ā(t) − Ā(s)||4 ≤ K(|t − s|4 + |t − s|2), E||ζ̄(t) − ζ̄(s)||4 ≤ C|t − s|2.
Therefore processes Ā(t) and ζ̄(t) satisfy the Kolmogorov’s continuity con-
dition [12].
Let us consider the case k1 = 2k2 = 2k3. Under these conditions we
have for i, j = 1, 2
lim
ε→0
1
t
t∫
0
α(i)
ε (s, A1, A2)ds = ᾱ(i)(A1, A2),
lim
ε→0
1
t
t∫
0
[
σ(i)
ε (s, A1, A2)σ
(j)
ε (s, A1, A2)+ (11)
+
1
εk
∫
R
γ(i)
ε (s, A1, A2, z)γ(j)
ε (s, A1, A2, z)Π(dz)
⎤
⎦ ds = B̄ij(A1, A2),
where functions ᾱ(i)(A1, A2) and B̄(A1, A2) = {B̄ij(A1, A2), i, j = 1, 2} are
defined in the condition of theorem. Since processes Ā(t), ζ̄(t) are continu-
ous, then from Lemma and relationships (7), (11) it follows
Ā(t) = A(0) +
t∫
0
ᾱ(Ā1(s), Ā2(s))ds + ζ̄(t), A(0) = (A1(0), A2(0)), (12)
LIMIT BEHAVIOR 27
where ζ̄(t) is continuous vector-valued martingale with matrix characteristic
〈ζ̄ (i), ζ̄ (j)〉(t) =
t∫
0
B̄ij(Ā1(s), Ā2(s))ds, i, j = 1, 2.
Hence [13] there exists Wiener process w̄(t) = (wi(t), i = 1, 2), such that
ζ̄(t) =
t∫
0
σ̄(Ā1(s), Ā2(s)) dw̄(s), σ̄(A1, A2) =
{
B̄(A1, A2)
}1/2
. (13)
Relationships (12), (13) mean that process Ā(t) satisfies equation (4). Un-
der conditions of theorem the equation (4) has unique solution. There-
fore process Ā(t) does not depend on choosing of sub-sequence εm → 0,
and finite-dimensional distributions of process Aεm(t) converge to finite-
dimensional distributions of process Ā(t). Since processes Aεm(t) and Ā(t)
are Markov processes then using the conditions for weak convergence of
Markov processes [12] we finish the proof of statement 1 of theorem.
Let us consider the case k < k1. Then coefficients α
(i)
ε (t, A1, A2), i = 1, 2
of equation (7) tend to zero, as ε → 0. Repeating with obvious modifications
the proof of statement 1) of theorem we obtain proof of statement 2).
In the case k < 2k2 in (11) we have σ
(i)
ε (t, A1, A2)σ
(j)
ε (t, A1, A2) =
O(ε2k2−k), i, j = 1, 2. Then we finish the proof in this case as above. In the
same way we consider the case k < 2k3.
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Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail: odb@univ.kiev.ua
Department of Mathematical Physics, National Technical Univer-
sity ”KPI“, Kyiv, Ukraine
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