ISSN 09655425, Computational Mathematics and Mathematical Physics, 2011, Vol. 51, No. 6, pp. 975–986. © Pleiades Publishing, Ltd., 2011.
Original Russian Text © Ngoc Thanh Do, V.B. Levenshtam, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 6,
pp. 1043–1055.
Asymptotic Integration of a System of Differential Equations
with a Large Parameter in the Critical Case
Ngoc Thanh Doa and V. B. Levenshtama, b
a
Department of Mathematics, Mechanics, and Computer Science, Southern Federal University,
ul. Mil’chakova 8a, RostovonDon, 344090 Russia
b
Southern Institute of Mathematics, Vladikavkaz Scientific Center, Russian Academy of Sciences, ul. Markusa 22,
Vladikavkaz, 362027 Russia
email: dothanhngocctsp@gmail.com, vleven@math.rsu.ru
Received August 19, 2010
Abstract—For a linear normal system of ordinary differential equations with rapidly oscillating coef
ficients in a critical case, the existence of a unique periodic solution is proved, its complete asymptotic
expansion is constructed and justified, and Lyapunov stability and instability conditions are found.
The asymptotic series constructed is shown to converge absolutely and uniformly to the solution.
DOI: 10.1134/S0965542511060042
Keywords: linear normal system with rapidly oscillating coefficients, degenerate stationary averaged
system, complete asymptotic expansion of a periodic solution, Lyapunov stability and instability of a
solution.
1. INTRODUCTION
In the theory of the Bogolyubov–Krylov–Mitropol’skii averaging method (see, e.g., [1]), an important
place is occupied by the result concerning the construction and substantiation of complete asymptotics
(higher approximations) of a periodic solution of a nonlinear normal system of ordinary differential equa
tions with rapidly oscillating coefficients, which exists and is unique in a neighborhood of the solution of
the stationary averaged system. The latter solution is assumed to be nondegenerate; i.e., the matrix coef
ficient in the stationary averaged system linearized around the indicated solution is invertible.
In this paper, we consider a periodic solution of a linear normal system of ordinary differential equa
tions with rapidly oscillating coefficients for which the coefficient A0 of the stationary formally averaged
system has the simple eigenvalue λ = 0 with an eigenvector a0 . Using the set of rapidly oscillating and
lower stationary coefficients of the perturbed system, we construct a matrix B0. It is assumed that the vec
tor a0 has no associated vectors with respect to the pair of matrices A0 and B0 (see [2]). The existence of a
unique periodic solution of the perturbed system is proved, and its complete asymptotic expansion in pow
ers of the small parameter equal to the inverse frequency of coefficient oscillations is constructed and sub
stantiated. The asymptotic series is shown to converge uniformly and absolutely to the solution. The
Lyapunov stability and instability of this solution are analyzed.
Note that the results can be extended to the class of linear systems that, in addition to (1), contain zero
mean highfrequency terms proportional to the square root of the frequency (large highfrequency terms).
For reasons of space, this extension is not presented below. Some results developing the theory of the aver
aging method for (basically nonlinear) systems with large highfrequency terms can be found, for exam
ple, in [3, 4].
2. THE MAIN RESULT
We use the following notation: M n (n ∈ ⺞) is the set of all square matrices of order n with complex coef
n
ficients, and ⺓ is the space of n dimensional column vectors with complex elements. For any two column
n
vectors x = (x1, …, x n)T and y = (y1, … , y n ) T from ⺓ , the inner product is defined as
n
(x, y) =
∑x y ,
i i
i =1
and the vector norm is defined as | x| = (x, x).
975
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NGOC THANH DO, LEVENSHTAM
Definition 1. Given matrices A0, B0 ∈ M n, let λ = 0 be an eigenvalue of A0 and a0 be the corresponding
eigenvector. The vector a0 is said to (see [2, p. 9]) correspond to a kdimensional (k ∈ ⺞) chain of vectors
ai i = 1, k associated with respect to A0 and B0 if A0ai = −B0ai −1, for i = 1, k and the equation A0 x = −B0ak
has no solutions. If the equation A0 x = −B0a0 has no solutions, a0 is said to have no vectors associated with
respect to A0 and B0.
(
)
For brevity, the vectors associated with a0 with respect to A0 and B0 are referred to as associated vectors.
Recall that the average of a continuous lperiodic function f (t ) is defined as
l
f = 1 f (t )dt.
l
∫
0
In what follows, we deal with numbers m, n ∈ ⺞ ; matrices A0, A1, B k ∈ M n ; and vectors d 0, d k ∈ ⺓ n ,
where 1 ≤ | k| ≤ m . Moreover, λ = 0 is a simple root of the characteristic equation A0 − λ E = 0 , and a0 is
the corresponding eigenvector such that a0 = 1. Additionally, suppose that a0 does not have associated
B −k B k
vectors; i.e., (B0a0, z 0 ) ≠ 0, where B0 := A1 +
and z 0 is the eigenvector corresponding to the
1≤|k |≤ m ik
eigenvalue λ = 0 of the adjoint A0* of A0.
The fact that λ = 0 is a simple root of the characteristic equation A0 − λ E = 0 is not related to a0 having
no associated vectors; i.e., these facts do not follow from one another. Indeed, there are counterexamples.
⎛1 0 ⎞
⎛ 1 1⎞
Example 1. A0 = ⎜ ⎟ , B0 = ⎜ ⎟ , λ = 0 is a simple root of the characteristic equation of A0, and
⎝1 0 ⎠
⎝ 0 1⎠
τ
τ
a1 = ( −1 − 1) is the associated vector for a0 = ( 0 1) .
∑
⎛0 1⎞
⎛1 0 ⎞
Example 2. A0 = ⎜ ⎟ , B0 = ⎜ ⎟ , λ = 0 is a multiple root of the characteristic equation of A0, but the
⎝0 0⎠
⎝1 1 ⎠
corresponding eigenvector has no associated vectors.
Consider the homogeneous system
dx = ⎛⎜ A + 1 B ⎞⎟ x +
Bk xe ikωt .
0⎟
⎜ 0
dt ⎝
ω ⎠
1≤|k |≤ m
∑
Performing an infinite number of sequential Krylov–Bogolyubov substitutions (see [1, p. 24])
1 B ye ikωt is the first of them), we obtain the formal system
(x = y +
1≤|k |≤ m ik ω k
∑
dz = ⎛ A + 1 A + 1 A + …⎞ z ≡ A z.
⎜ 0
⎟
1
2
ω
dt ⎝
ω
⎠
ω2
Consider the characteristic polynomial | λE − Aω| = λ n + a1ωλ n−1 + … + anω and the corresponding Hurwitz
matrix
Γ(ω) ≡
⎛ a
1ω
⎜
⎜
a
⎜
3ω
⎜
⎜ …
⎜
⎜⎜
a
⎝ 2n−1,ω
1
a2ω
…
0
a1ω
…
0
1
…
a2n−2,ω a2n−3,ω a2n−4,ω
…
…
…
…
0 ⎞⎟
0 ⎟⎟
⎟,
… ⎟⎟
anω ⎟⎟⎠
where asω = 0 for s > n . All the diagonal minors Diω, i = 1, n of the Hurwitz matrix, which are called Hur
−1
witz determinants, are expanded in formal series in powers of ω :
∞
D1ω = a1ω =
∑
ω − d1q ,
q =0
q
∞
…,
Dnω = | Γ ω| =
∑ω
−q
d nq .
q =0
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Consider the inhomogeneous system
ikωt
dx = ⎛⎜ A + 1 A ⎞⎟ x +
+ d 0.
(Bk x + d k )e
1⎟
⎜ 0
dt ⎝
ω ⎠
1≤|k|≤ m
∑
(1)
Below is the main result of this paper.
Theorem 1. There exists a number ω0 > 0 such that the following assertions hold for ω > ω0 :
1. Equation (1) has a unique 2π / ω periodic solution. This solution can be represented as a converging
series:
x(t) = ωc−1a0 + [ x0 + c0a0 + y0(ωt )] + ω−1 [ x1 + c1a0 + y1(ωt )] + …,
(2)
where c−1, ci ∈ ⺓; xi ∈ ⺓ n; yi : ⺢ → ⺓ n are 2π periodic vector functions with a zero mean ( yi (τ) = 0),
i = 0, 1,…), which can be effectively determined. Moreover, there exists a constant M 0 > 0 such that, for all
ω > ω0 and t ∈ R , it holds that
x(t ) − {ω c−1a0 + [ x 0 + c0a0 + y 0(ω t )] + … + ω−r [ x r + cr a0 + y r (ω t )]} <
M 0r +1
,
ωr +1
r = 0, 1, ….
−1
2. In the expansions in powers of ω of the Hurwitz determinants Diω, i = 1, n, if all the first nonzero coef
ficients are positive, then the solution xω(t) is Lyapunov stable. If at least one of the first nonzero coefficients of
these expansions is negative, then the solution xω(t) is unstable.
By the effectiveness of determining the coefficients of series (2), we mean that the determination of
each of them is reduced to a finite number of arithmetic operations over numbers. More specifically, the
computation of each coefficient in series (2) is reduced to a finite number of problems of two types: sys
tems of algebraic equations A0 x = b and systems of differential equations
∑
ik τ
dx =
ck e ,
d τ 0< k ≤ M
x = 0.
The problems of the first type are solved by Gauss elimination, while the solutions of problems of the
second type can be directly written out:
x=
∑
ck ikτ
e .
ik
0< k ≤ M
The existence of a solution x(t ) of Eq. (1) that is representable as converging series (2) and the estimates
given in the theorem are proved in Section 3, while the uniqueness of this solution is proved in Section 4.
The proof of item 2 in the theorem is omitted, since it repeats the corresponding argument in [5] with the
Krylov–Bogolyubov substitutions taken into account.
3. EXISTENCE OF A SOLUTION REPRESENTABLE
AS A CONVERGING ASYMPTOTIC SERIES
By using the method of twoscale expansions and the Vishik–Lusternik method (see [2]), a formal
asymptotic expansion of the solution to Eq. (1) is sought in the form of (2). Substituting (2) into (1) gives
dy0 dy1 1 dy2
dy
+
+
+ … + 1r r +1 + …
dτ dτ ω dτ
ω dτ
= A0(x0 + y0 ) + 1 A0(x1 + y1) + … + 1r A0(x r + y r )
ω
ω
1
1
… + c−1 A1a0 + A1(x0 + c0a0 + y0 )… + r A1(x r −1 + cr −1a0 + yr −1)
ω
ω
⎡
⎤
⎢ω c B a + B (x + c a + y ) + … + 1 B (x + c a + y ) + … + d ⎥ e ik τ + d ,
…+
−1 k 0
0 0
0
0
k 0
r 0
r
k⎥
r k r
⎢
⎣
⎦
ω
1≤|k|≤ m
ω
∑
where τ = ωt.
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Next, we equate the coefficients of like powers of ω on the left and righthand sides of (3). For the
1
highest power ω , we have
∑
dy0
ikτ
=
c−1Bka0e ,
d τ 1≤|k|≤m
y0(τ) = 0,
so that
y0(τ) =
∑c
−1
1≤|k|≤m
Bka0 ikτ
e .
ik
(4)
For ω , Eq. (3) yields
0
∑
dy1
ikτ
= A0 x0 + A0 y0 + c−1 A1a0 +
[Bk (x0 + c0a0 + y0) + dk ]e + d0.
dτ
1≤|k|≤m
Substituting y0(τ) given by (4) and averaging both sides of the result, we obtain
0 = A0 x0 + c−1 A1a0 +
∑c
−1
1≤|k|≤m
B−k Bka0
+ d 0,
ik
or
− A0 x0 = c−1B0a0 + d (0),
d (0) := d0.
(5)
Equation (5) is solvable if and only if (c−1B0a0 + d , z 0 ) = 0 ; i.e., c−1 = −(d , z 0 )/(B0a0, z 0 ). Hence,
c−1 = −(d (0), z 0 )/(B0a0, z 0 ) and x0 = −W (c−1B0a0 + d (0)) , where W is the inverse of the restriction of A0 to the
orthogonal complement of a0 . Furthermore,
⎛
⎞
A0 Bk a0
B Ba
dy1
⎜c
c−1 k l 0 e i(k +l )τ, y1(τ) = 0,
=
+ Bk x0 + c0 Bk a0 + d k ⎟⎟ e ikτ +
⎜ −1
d τ 1≤|k|≤m ⎝
ik
il
⎠
1≤|k|,|l |≤ m,
(0)
∑
(0)
∑
k +l ≠ 0
so that
−
y1(τ) =
⎛ A0 Bk a0 Bk x 0
B a d ⎞ ikτ
+
+ c0 k 0 + k ⎟ e
⎜ c −1
2
ik
ik
ik ⎠
(ik)
1≤|k |≤ m ⎝
∑
Bk Bl a0 i(k +l )τ
Ba
e
:=
c0 k 0 e ikτ +
ak(1)e ikτ,
ik
(k + l )l
1≤|k|≤ m
1≤|k|≤2m
c −1
1≤|k|,|l |≤ m,
k +l ≠ 0
∑
∑
∑
where ak(1) are given vectors and | k| = 1, 2m.
In this manner, we can find all the coefficients, vectors, and vector functions involved in formal asymp
totics (2). Indeed, given some r ≥ 1, assume that c j −2, y j −1, x j −1 have been found for any j ≤ r and, addition
ally, y r has the form
yr (τ) =
∑c
1≤|k|≤m
where
ak(r )
r −1
∑
Bka0 ikτ
(r ) ikτ
e +
ak e ,
ik
1≤|k|≤(r +1)m
(6)
−r
are given vectors and | k| = 1, (r + 1)m. Equating the coefficients of ω in (3) gives
∑
dyr +1
ikτ
= A0(xr + yr ) + A1(xr −1 + cr −1a0 + yr −1) +
Bk (xr + cr a0 + yr )e .
dτ
1≤|k|≤m
(7)
Next, substituting the expression for y r from (6) into (7) and averaging both sides of the resulting relation,
we find
0 = A0 xr + A1xr −1 + cr −1 A1a0 +
∑c
1≤|k|≤m
r −1
∑
B−k Bka0
(r )
+
B−kak ,
ik
1≤|k|≤m
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or
− A0 x r = A1x r −1 + cr −1B0a0 +
∑B
(r )
−k a k
:= cr −1B0a0 + d (r ),
1≤|k |≤ m
d
(r )
∑
:= A1x r −1 +
(9)
B−k ak(r ).
1≤|k|≤ m
This equation is solvable if and only if (cr −1B0a0 + d (r ), z 0 ) = 0 ; i.e., cr −1 = −(d (r ), z 0 )/(B0a0, z 0 ). After deter
mining cr −1 , the vector function y r is found from (6), while (9) is used to determine the vector
x r = −W (cr −1B0a0 + d (r )). Moreover, combining (7) with (6) and (8) produces
∑
∑
B Ba
dy r +1
= A0 y r + A1yr −1 +
(Bk x r + cr Bk a0 )e ikτ +
cr −1 k l 0 e i(k +l )τ +
dτ
il
1≤|k|,|l |≤ m,
1≤|k|≤ m
k +l ≠ 0
∑
Bk al(r )e i(k +l )τ,
1≤|l |≤(r +1)m
1≤|k|≤ m, k +l ≠ 0
or
∑
∑
dyr +1
ikτ
(r +1) ikτ
=
cr Bka0e +
(ik)ak e ,
d τ 1≤|k|≤m
1≤|k|≤(r +2)m
where ak(r +1) are given vectors and | k| = 1, (r + 2)m. Therefore, yr +1 has the form of (6):
yr +1(τ) =
∑c
r
1≤|k|≤m
∑
Bka0 ikτ
(r +1) ikτ
e +
ak e .
ik
1≤|k|≤(r + 2)m
Thus, we have shown that the described iterative process of finding the coefficients of asymptotics (2)
is implementable.
Now we prove the absolute convergence of series (2) for sufficiently large ω and show that (2) is the
solution of Eq. (1). Preliminarily, the norm of a matrix A = (aij ) in M n is defined as
∑ |a | .
A =
2
ij
1≤i, j ≤n
Recall that any vector x from ⺓ satisfies the inequality
n
Ax ≤ A x .
Without loss of generality, we assume that, in Eq. (1),
A0 ≤ 1,
A1 ≤ 1,
Bk ≤ 1,
d 0 ≤ 1,
dk ≤ 1
for all k such that 1 ≤ | k| ≤ m . Indeed, this can always be achieved by making the substitution t = lt1, where
l ∈ ⺢.
Since
(
)
cr −1 = − d (r ), z 0 / ( B0a0, z 0 )
and
(
)
x r = −W cr −1B0a0 + d (r ) ,
for any integer r ≥ 0 , there are positive numbers C1, C2, and C3 independent of r such that cr −1 ≤ C1 d (r )
and x r ≤ C2 cr −1B0a0 + d (r ) ≤ C3 d (r ) , r ≥ 0 . Let C = max {1, C1, C3}. Then cr −1 ≤ C d (r ) and x r ≤ C d (r ) .
Define
L0 = 0,
S0 = 2m| c−1|,
Lr =
∑
ak(r ) ,
S r = Lr + 2m| cr −1|,
r = 1, 2,….
1≤|k|≤(r +1)m
The inequalities d (r ) ≤ (2m + 2)2r C r and | S r | ≤ (2m + 2)2r +1C r +1 for any r ≥ 0 are proved by induction. Obvi
ously, they hold for r = 0 . Let us prove that they also hold for r = 1. Preliminarily, we note that
d
(r )
= A1xr −1 +
∑B
(r )
−k ak
1≤|k|≤m
≤ A1xr −1 +
∑
ak ≤ xr −1 + Lr
(r )
1≤|k|≤m(r +1)
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NGOC THANH DO, LEVENSHTAM
for any r from ⺞. We have
∑c
y1(τ) =
0
1≤|k|≤ m
⎛ A0 Bk a0 Bk x 0 d k ⎞ ikτ
Bk a0 ikτ
B Ba
e +
c −1
c−1 k l 0 e i(k +l )τ,
+
+ ⎟e +
⎜
2
ik
ik
ik ⎠
i(k + l )(il )
(ik)
1≤|k|≤ m ⎝
1≤|k |,|l |≤ m,
∑
∑
k +l ≠ 0
so that
L1 =
⎛
A0 Bk a0 Bk x 0 d k ⎞
B Ba
c −1 k l 0
+
+
+
⎜ c −1
⎟
2
ik
ik ⎠ 1≤|k|,|l |≤m,
i(k + l )(il )
(ik)
1≤|k |≤ m ⎝
∑
ak ≤
∑
∑
(| c−1|+| x 0 |+| d k |) +
(1)
1≤|k |≤2m
≤
∑
1≤|k |≤ m
∑
k +l ≠ 0
| c−1| ≤ 2m3C + 4m C = (4m + 6m)C.
2
2
1≤|k |,|l |≤ m,
k +l ≠0
It follows that
d (1) ≤ | x0| + L1 ≤ C + (4m2 + 6m)C ≤ (2m + 2)2 C
and
S1 = L1 + 2m | c0 |≤ (4m2 + 6m)C + 2m | c0 |≤ (4m2 + 6m)C + 2mC | d (1) |
≤ (4m2 + 6m)C 2 + 2m(2m + 2)2 C 2 ≤ (2m + 2)3 C 2.
Thus, we have proved the inequalities for r = 1. Now, assuming that they hold for all j ≤ r − 1 and r ≥ 2 ,
we prove them for j = r . Since
∑
∑
B Ba
dy r
= A0 y r −1 + A1y r −2 +
Bk (x r −1 + cr −1a0 )e ikτ +
cr −2 k l 0 e i(k +l )τ +
dτ
il
1≤|k|,|l |≤ m
1≤|k|≤ m
⎛
⎜
= A0 ⎜⎜
∑
c r −2
⎜1≤|k|≤ m
⎝
∑ B (x
+
k
r −1
⎞
Bk a0 ikτ
(r −1) ik τ ⎟
e +
ak e ⎟⎟ +
ik
⎟
1≤|k|≤ rm
∑
+ cr −1a0 )e
ik τ
1≤|k|≤ m
⎠
k +l ≠ 0
⎛
⎜
A1 ⎜⎜
cr −3
⎜ 1≤|k|≤ m
⎝
∑
∑
B Ba
+
cr −2 k l 0 e i(k +l )τ +
il
1≤|k|,|l |≤m
k +l ≠ 0
∑
Bk al(r −1)e i(k +l )τ
1≤|l |≤ rm,
1≤|k|≤ m, k +l ≠ 0
⎞
⎟
Bk a0 ikτ
e +
ak(r −2)e ikτ ⎟⎟
ik
⎟
1≤|k|≤(r −1)m
∑
∑
⎠
Bk al(r −1)e i(k +l )τ,
y r (τ) = 0,
1≤|l |≤rm,
1≤|k|≤ m,k +l ≠ 0
we conclude that
∑c
y r (τ) =
1≤|k|≤m
+
∑
1≤|k|≤(r −1)m
(
r −2
A0 Bk a0 ikτ
A0ak(r −1) ikτ
A B a ikτ
e +
e +
cr −3 1 k 2 0 e
2
ik
(ik)
(ik)
1≤|k|≤rm
1≤|k|≤m
∑
∑
)
A1ak(r −2) ikτ
Bk x r −1
B a ikτ
B B a i(k +l )τ
e +
c r −2 k l 0 e
+ cr −1 k 0 e +
+
ik
ik
ik
i(k + l )il
1≤|k|,|l |≤m
1≤|k|≤m
∑
∑
k +l ≠ 0
∑
1≤|l |≤rm,
1≤|k|≤m,k +l ≠0
Bk al(r −1) i(k +l )τ
e
.
i(k + l )
Therefore,
Lr ≤
∑
c r −2
1≤|k|≤m
+
≤ 2m cr −2 +
A0Bka0
A0ak(r −1)
A1Bka0
A1ak(r −2)
+
+
+
c
r −3
ik
ik
(ik)2
(ik)2
1≤|k|≤rm
1≤|k|≤m
1≤|k|≤(r −1)m
∑
∑
∑
∑
∑
Bk xr −1
B Ba
+
c r −2 k l 0 +
ik
i(k + l)il
1≤|k|,|l |≤m
1≤|k|≤m
∑
(r −1)
ak
1≤|k|≤rm
≤ S r −1 + S r −2 + 2mC d
k +l ≠ 0
+ 2m| cr −3| +
∑
(r −2)
ak
∑
1≤|l |≤rm,
1≤|k|≤m,k +l ≠0
(r −1)
Bkal
i(k + l)
+ 2m xr −1 + 4m cr −2 + 2m
2
1≤|k|≤(r −1)m
(r −1)
+ 2mS r −1 ≤ (2m +
2r −3
2) ⎡⎢⎣(2m
∑
(r −1)
al
1≤|l |≤rm
+ 2) + 1 + 2m(2m + 2) + 2m(2m + 2) ⎤⎥⎦ C .
2
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Thus,
d (r ) ≤ | x r −1| + Lr < C d (r −1) + Lr ≤ (2m + 2)2r −3 C r[(2m + 2)
+ (2m + 2) + 1 + 2m(2m + 2) + 2m(2m + 2) ] ≤ (2m + 2) r C r ,
2
2
2
and
S r = 2m| cr −1| + Lr ≤ 2m(2m + 2)2r C r +1 + (2m + 2)2r C r ≤ (2m + 2)2r +1C r +1.
The required inequalities have been proved. Therefore, for any t ∈ ⺢ and r = 0, 1,… ,
| x r + cr a0 + y r | ≤ | x r |+| cr |+| yr | ≤ C d (r ) +| cr |+| yr |
≤ (2m + 2)2r C r +1 + (2m + 2)2r +2 C r +2 + (2m + 2)2r +1C r +1 ≤ M r
for some constant M. Thus, for ω > M , series (2) converges absolutely and uniformly. The estimates in the
theorem follow straightforwardly. Obviously, the vector function (2) is 2π / ω periodic.
It remains to verify that the sum of series (2) solves Eq. (1) for sufficiently large ω . For this purpose, we
set
r
S (t) = ωc−1a0 +
(r )
∑ ω1 [x
k
k
+ ck a0 + y k (t )],
r = 0, 1,….
k =0
Then, for sufficiently large ω , S
(r )
converges uniformly with respect to t ∈ ⺢ to a vector function x(t ):
(r)
⺢
S ( t ) → x ( t ).
→
Noting that
⎛
⎞
ikωt ⎟ (r +1)
(r )
dS + = ⎜⎜ A +
⎟S
B
e
dke ikωt ,
(t) + 1 A1S (t) +
0
k
⎜
⎟
dt
ω
⎜
⎟
1≤|k|≤m
0≤|k|≤m
(r 2)
∑
⎝
∑
⎠
r = 0,1,…
we find
(r )
dS
dt
⎞
⺢ ⎛
→ ⎜⎜ A0 + 1 A1 ⎟⎟ x +
Bk xe ikωt +
d k e ikωt ,
→⎝
ω ⎠
1≤|k|≤ m
0≤|k|≤ m
∑
∑
It follows that
⺢
S (t) − S (0) →
→
(r )
(r )
t ⎡
⎢⎛
⎢⎜ A0
⎢⎜⎝
0 ⎢⎣
∫
⎤
⎞
ikωη
ikωη ⎥
+ 1 A1 ⎟⎟ x(η) +
Bk x(η)e
+
d k e ⎥⎥d η,
ω ⎠
⎥
1≤|k |≤ m
0 ≤|k |≤ m
∑
∑
⎦
so that
(r )
(r )
x(t ) − x(0) = lim ⎡⎣S (t ) − S (0)⎤⎦ =
r →+∞
t ⎡
⎢⎛
⎢⎜ A0
⎢⎜⎝
0 ⎢⎣
∫
⎤
⎞
ikωη
ikωη ⎥
+ 1 A1 ⎟⎟ x(η) +
Bk x(η)e
+
d k e ⎥⎥ dη.
ω ⎠
⎥
1≤|k |≤ m
0≤|k |≤ m
∑
∑
⎦
Differentiating both sides of this equality yields
dx = ⎛⎜ A + 1 A ⎞⎟ x +
(Bk x + d k )e ikωt + d 0;
1⎟
⎜ 0
⎝
⎠
ω
dt
1≤|k|≤ m
∑
i.e., x(t ) is the solution of Eq. (1).
4. UNIQUENESS
We prove that Eq. (1) has at most one 2π / ω periodic solution for sufficiently large ω .
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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Preliminarily, we prove certain auxiliary results. Let ε > 0 and Aε = A0 + ε B0 . Without lost of generality,
the matrix A0 is assumed to be given in Jordan normal form:
A0 =
⎛0
⎜
⎜
⎜0
⎜
⎜
⎜
⎜
⎜
⎜⎜
⎝0
0 0 λ1 β1 0 0 0 ⎞⎟
0 ⎟⎟
⎟
⎟⎟ ,
β n−1 ⎟⎟
⎟
λ n−1 ⎟⎠
B0 = (bij ),
where, by the assumption, λ1 … λ n−1 ≠ 0 . Then a0 = (1 0 … 0) is the eigenvector of A0 corresponding to
the eigenvalue λ 0 = 0. The equation A0 x = −B0a0 has no solution; i.e., the equation
T
⎛0
⎜
⎜
⎜0
⎜
⎜
⎜
⎜0
⎜
⎜⎜
⎝0
0 0 λ1 β1 0 0 0 0 ⎛b ⎞
0 ⎞⎟ ⎛⎜ x1 ⎞⎟
⎜ 11 ⎟
⎟⎜
⎟
⎜
⎟
0 ⎟ ⎜ x2 ⎟
⎜ b21 ⎟
⎟⎜
⎟
⎜
⎟
⎟⎟ ⎜⎜ x3 ⎟⎟ = − ⎜⎜ b31 ⎟⎟ ,
⎜⎟
β n−1 ⎟⎟ ⎜⎜ … ⎟⎟
⎜
⎟
⎟⎟ ⎜⎜
⎟⎟
⎜⎜
⎟⎟
λ n−1 ⎠ ⎝ x n ⎠
⎝ bn1 ⎠
where B0 = (bij ), is unsolvable. Therefore, b11 ≠ 0. According to the regular perturbation theory, there exists
an indexing of the eigenvalues λ j and λ 'j
( j = 0, n − 1)of the matrices A0 and Aε, respectively, such that
λ 'j = λ j + o(1) as ε → 0, j = 0, n − 1 ; moreover, λ 0 = 0 and λ j ≠ 0, j = 1, n − 1. The following result holds.
Lemma 1. For sufficiently small ε , the eigenvalue λ '0 of the matrix Aε can be represented as a converging
∞
series λ '0 = ∑ ϕ i ε i , where ϕ1 ≠ 0.
i =1
Proof. Let F (ε, λ) be the characteristic polynomial of Aε ; i.e.,
λ − ε b11
−εb12
−ε b1n
−εb21 λ−λ1−εb22 −ε b2n
F (ε, λ) = | λ E − Aε | = .
−β n−1 − ε bn−1,n
λ − λ n−1 − ε bnn
−εbn1
−εbn2
Since F (ε, λ) is a polynomial in λ and ε such that
n −1
F (0, 0) = | A0 | = 0
and
∂F (0, 0) = (−1)n−1 λ ≠ 0,
i
∂λ
i =1
∏
the classical implicit function theorem (see [6, p. 27; 7, p. 452]) implies that there exists a neighborhood
U 0 × V0 of the point (0, 0) in which the functional equation F (ε, λ) = 0 defines an implicit function λ :
U 0 → V0, λ = λ(ε), where λ is continuously differentiable on U 0 and satisfies the conditions λ(0) = 0 and
(
)(
)
−1
. Moreover, the neighborhood U 0 can always be chosen so that λ in it can
λ '(ε) = − ∂F (ε, λ) ∂F (ε, λ)
∂ε
∂λ
λ=λ(ε)
be expanded in a converging series in powers of ε ; i.e.,
∞
λ(ε) =
∑ϕ ε .
i
i
i =1
Then λ(0) = 0 and
(
)(
)
ϕ1 = λ '(0) = − ∂F (0, 0) ∂F (0, 0)
∂ε
∂λ
−1
n −1
= b11 ≠ 0 ,
since
∂F (0, 0) = (−1)n b
λi .
11
∂ε
i =1
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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Thus, for sufficiently small ε , the eigenvalue λ '0 of Aε can be represented as
∞
λ '0 = λ(ε) =
∑ϕ ε ,
i
i
i =1
where ϕ1 = b11 ≠ 0, as required.
Let R = max | λ j | + 1. Then there exists a number T > 0 such that TR ≠ 2k π , T λ j ≠ 2k π i ∀k ∈ ⺪ ,
1≤ j ≤ n−1
+
j = 1, n − 1 (since the set ⺢ is uncountable, while the set {2k π i / λ j , j = 1, n − 1, k ∈ ⺪ } ∪ {2k π/ R , k ∈ ⺪ }
is countable). Let Tε = ⎡T 1 ⎤ 2πε , where [a] denotes the integer part of a. Then Tε = T + O(ε) as ε → 0.
⎣⎢ 2πε ⎦⎥
Obviously, for sufficiently small ε , we have Tελ 'j ≠ 2k π i and Tε R ≠ 2k π ∀ k ∈ ⺪ , j = 0, n − 1. Therefore,
Tελ 'j = [T + O(ε)] [λ j + o(1)] = T λ j + o(1)
as
ε → 0,
j = 1, n − 1,
Tελ '0 = [T + O(ε)][εb11 + o(ε)] = εb11T + o(ε) as ε → 0,
Tε R = [T + O(ε)] R = TR + O(ε) as ε → 0.
Lemma 2. (eTε Aε − E )−1 < L/ε for some L = const > 0 for sufficiently small ε .
{
}
Let Γ1 = λ ∈ ⺓: | λ| = 1 | b11| ε and Γ 2 = { λ ∈ ⺓: | λ| = R}. Then, since
2
∞
λ '0 = b11ε +
∑ϕ ε
i
i
λ 'j = λ j + O(1)
and
as
ε → 0,
j = 1, n − 1
i =2
(see Lemma 1), we conclude that, for sufficiently small ε , all the eigenvalues of Aε lie in the domain Ω
T A
−1
bounded by the contours Γ1 and Γ2 . Define f ( A) = (e ε − E ) , and let Γ1 and Γ2 be positively oriented;
i.e., when Γ1 and Γ2 are traversed, Ω is on the left. Then (see [9])
f ( Aε ) = 1 (eTελ − 1) −1(λ E − Aε )−1d λ + 1 (eTελ − 1)−1(λ E − Aε )−1d λ.
2πi
2πi
∫
∫
Γ2
Γ1
We have Tε R − 2k π = [T + O(ε)] R − 2k π = TR − 2k π + O(ε) as ε → 0. Therefore, for sufficiently small
ε , we
have | Tε R − 2k π| > δ/2 > 0 ∀k ∈ ⺪, where δ = min | TR − 2k π| . It follows that
k∈⺪
Tλ
| Tελ − 2k π| > δ/2 ∀λ ∈ Γ 2, ∀k ∈ ⺪. Then there exists σ = σ(δ) > 0 such that | e ε − 1| > σ for sufficiently
small ε , which means that the function |(eTελ − 1)−1| of λ is uniformly bounded on Γ2 with respect to ε . Fur
thermore, we have
(λ E − Aε ) −1 =
1
( Aij ),
det(λ E − Aε )
where Aij are the cofactors of the elements of the matrix λ E − Aε . For λ ∈ Γ 2, we have the obvious equal
ities
n −1
∏(λ − λ ) + O(ε)
det(λ E − Aε ) = λ
j
as
ε → 0,
ε → 0,
i = 1, n,
j =1
n −1
Aii =
∏ (λ − λ ) + O(ε)
j
as
j =0, j ≠ i
Aij = O(ε)
as
ε → 0,
1 ≤ i, j ≤ n,
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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NGOC THANH DO, LEVENSHTAM
Therefore, the norm
(λ E − Aε )
−1
⎛ n
1/2
⎞
⎝
⎠
⎜
⎟
1
⎜
=
| Aij | 2 ⎟⎟ ,
⎜
| det(λ E − Aε ) | ⎜ i, j =1
⎟
∑
is also uniformly bounded on Γ2 with respect to small ε . Thus, there exists L1 = const > 0 such that
∫ (e
Tελ
(
) ∫ dλ < L
− 1) −1(λ E − Aε ) −1d λ ≤ sup (eTε − 1) −1 (λ E − Aε ) −1
λ∈Γ 2
Γ2
1
Γ2
for sufficiently small ε .
(
)
For λ ∈ Γ1, according to L’Hopital’s rule, e |Tελ| − 1 Tελ → 1 as ε → 0. Thus, for sufficiently small ε ,
(
)
we have the estimate eTελ − 1
−1
≤ L2 / ε , where L2 = const > 0 is independent of ε . Since
⎛ λ − ε b11 + O(ε 2 ) −ε b12 ⎞
−ε b1n
⎜
⎟
−εb21
−λ1+O(ε) −ε b2n ⎟
⎜
λ E − Aε =
⎜
⎟
…
⎜
⎟
−ε bn1
−εbn2 −λ n−1 + O(ε) ⎠
⎝
as
ε → 0,
we have
det(λ E − Aε ) = (λ − ε b11)(− 1)
n −1
∏λ
n −1
j
+ O(ε 2 ) = O(ε)
ε → 0,
as
j =1
n −1
∏λ
j
as
ε→0
A11 = (− 1) n−1
+ O(ε)
as
ε → 0,
j =1
Aij = O(ε)
∀(i, j) ≠ (1, 1),
which implies the inequality
1/2
(λ E − Aε )
−1
⎛ n
⎞
1
⎜ | Aij | 2 ⎟
=
⎟
| det(λ E − Aε )| ⎜⎝ i, j =1
⎠
∑
≤
L3
ε
for some L3 = const > 0 for sufficiently small ε . Thus,
∫ (e
Tελ
−1
−1
Tλ
−1
−1
− 1) (λ E − Aε ) d λ ≤ sup (e ε − 1) (λ E − Aε )
λ∈Γ1
Γ1
where
∫ dλ ≤
Γ1
L2 L3 ε b11 L4
2π
= ,
ε ε
2
ε
L4 = π| b11| L2 L3 > 0.
Therefore,
f ( A) ≤
∫ (e
Tελ
− 1)−1(λ E − Aε )−1d λ +
Γ1
∫ (e
Tελ
− 1) −1(λ E − Aε )−1d λ < L1 +
Γ2
L4 L
<
ε ε
for some L = const > 0 for sufficiently small ε . The lemma is proved.
Now we prove that, for sufficiently large ω , Eq. (1) has at most one Tω−1 periodic solution. Let
x (1)(t ) and x (2)(t ) be two Tω−1 periodic solutions of Eq. (1). Then
( j)
( j)
ikωt
dx = ⎛⎜ A + 1 A ⎞⎟ x ( j) +
(Bk x + dk )e + d0,
1
⎜ 0
⎝
dt
ω ⎟⎠
1≤|k|≤m
∑
j = 1, 2,
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ASYMPTOTIC INTEGRATION OF A SYSTEM OF DIFFERENTIAL EQUATIONS
985
(1)
(2)
and the vector function y = x − x satisfies the equation
⎞
dy ⎛⎜
ikωt
= ⎜ A0 + 1 A1 ⎟⎟ y +
Bk ye .
ω ⎠
dt ⎝
1≤|k|≤m
∑
(10)
Note that, for sufficiently large ω , the matrix
∑ ik1ω B e
E+
ikωt
k
1≤|k|≤m
is nonsingular and its inverse has the form
⎛
⎜
⎜E
⎜
⎜
⎝
⎞
−1
1 B e ikωt ⎟⎟ = E +
+
k
⎟
ikω
⎟
1≤|k|≤m
∑
⎠
+∞
⎛
⎞
j⎜
1 B e ikωt ⎟⎟
(−1) ⎜⎜
k
⎟
⎜1≤|k|≤m ikω
⎟
j =1
⎝
⎠
∑
∑
j
.
Making the substitution
y(t) = z(t) +
∑ ik1ω B z(t)e
ikωt
k
1≤|k|≤m
in Eq. (10) yields
⎛
⎞
⎛
⎞
⎛
⎞
⎜
⎛
⎞⎜
ikωt
ikωt ⎟
ikωt ⎟ il ωt
1
1
1
1
⎜
⎟
⎜
⎟
⎜
d ⎜ z + ikω
Bk ze ⎟ dt = ⎜ A0 + A1 ⎟ ⎜ z +
Bk ze ⎟ +
Bl ⎜ z +
Bk ze ⎟⎟ e ,
⎜
⎟
⎝
⎠⎜
ω
ω
ω
ik
ik
⎟
⎜
⎟
1≤|k|≤m
1≤|k|≤m
1≤|l |≤m ⎝
1≤|k|≤m
⎝
⎠
⎝
⎠
⎠
∑
∑
∑
∑
or
⎛
⎜
⎜E
⎜
⎜
⎝
+
⎞
⎛
⎠
⎝
⎞
1 B e ikωt ⎟⎟ dz = ⎜⎛ A + 1 A ⎟⎞ ⎜⎜ E +
1 B e ikωt ⎟⎟ z +
1 B B ze i(k +l )ωt .
k
k
l k
1⎟ ⎜
⎜ 0
⎟
⎟
⎝
⎠
ik
dt
ik
ik
ω
ω
ω
ω
⎟
⎜
⎟
1≤|k|≤m
1≤|k|≤m
1≤|k|,|l |≤m
∑
∑
∑
⎠
Therefore,
⎛
⎞
dz = ⎜⎜ E +
1 B e ikωt ⎟⎟
k
⎟
dt ⎜⎜
ikω
⎟
1≤|k|≤m
⎝
∑
⎠
−1
⎡
⎢⎛
⎢⎜ A0
⎢⎜⎝
⎢⎣
⎛
⎞
⎤
⎞⎜
1 B e ikωt ⎟⎟ +
1 B B e i(k +l )ωt ⎥⎥ z,
+ 1 A1 ⎟⎟ ⎜⎜ E +
k
l k
⎟
⎥
ikω
ikω
ω ⎠⎜
⎟
⎥
1≤|k|≤m
1≤|k|,|l |≤m
∑
⎝
∑
⎠
⎦
or
B−k Bk ⎞⎤
dz = ⎡⎢ A + 1 ⎜⎛ A +
⎟⎥ z + 1
Dk ze ikωt + 12 F (t, ω)
0
1
dt ⎢⎣
ik ⎟⎠⎦⎥
ω ⎜⎝
ω 1≤|k|≤2m
ω
1≤|k|≤ m
ikω t
= Aω−1 z + 1
+ 12 F (t, ω),
Dk ze
ω 1≤|k|≤2m
ω
∑
∑
(11)
∑
where F (t, ω) is a 2π / ω periodic matrix function of t with a uniformly bounded norm with respect to
ω 1 , t ∈ ⺢. Applying the transformation
z(t) = w(t) +
∑
1 D w(t)e ikωt
k
ikω2
1≤|k|≤2m
to Eq. (11) gives the equation
dw = A w + 1 G(t, ω)w,
(12)
ω−1
dt
ω2
where G (t, ω) is a 2π / ω periodic matrix function with a uniformly bounded norm: G(t, ω) < C0, where
C0 > 0 is independent ofω 1 , t ∈ ⺢. Equation (12) is equivalent to the integral equation
t
w(t ) = e
tA
ω
−1
(t − s ) A −1
ω
w(0) + 12 e
G(s, ω)w(s)ds.
ω
∫
0
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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NGOC THANH DO, LEVENSHTAM
(1)
(2)
Since x and x are periodic solutions of Eq. (1), we conclude that y(t ), z(t ) , and w(t ) are also Tω−1 peri
odic solutions of the corresponding equations. Therefore,
T
e
T
A
ω − 1 ω− 1
w(0) + 12
ω
ω− 1
∫e
(T
ω− 1
− s) A
ω− 1
G(s, ω)w(s)ds = w(Tω−1 ) = w(0),
0
or
(
w(0) = E − e
T
A
ω − 1 ω− 1
)
−1
T
1
ω2
ω− 1
∫e
(T
ω− 1
− s) A
ω−1
G(s, ω)w(s)ds.
0
Substituting the found expression into (11), we obtain (see [8])
T −1
t
ω
⎡ tA −1
⎤
(T −1 − s) A −1
(t − s) A −1
T −1 A −1 − 1
1
ω
ω
ω
ω
ω
ω
w(t) = 2 ⎢e (E − e
)
e
G(s, ω)w(s)ds + e
G(s, ω)w(s)ds ⎥ .
ω ⎢
⎥
0
0
⎣
⎦
∫
∫
By Lemma 2, since Tω−1 < T and Tω−1 Aω−1 ≤ C1 for ω ≥ 1, where C1 > 0 is a constant independent of ω ,
we have, for ω 1 ,
(
tA −1
T −1 A −1
| w| = sup w(t ) ≤ sup 12 e ω e ω ω − E
t∈[0,T −1 ]
t∈[0,T −1 ] ω
ω
ω
)
−1
T
ω−1
∫e
(T
ω−1
− s) A
ω−1
G(s, ω)w(s)ds
0
t
(t − s) A −1
ω
e
G(s, ω)w(s)ds ≤ C | w|,
+ 12 sup
ω
ω t∈[0,Tω−1]
∫
0
where C > 0 is a constant independent of ω . Thus, | w| ≤ C | w| , whence w ≡ 0 for large ω and, hence,
ω
x (1)(t ) ≡ x (2)(t ). Therefore, Eq. (1) has at most one Tω−1 periodic solution, which was to be proved.
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2011