To the problem of complementability of a periodic frame to a periodic basis
We obtain sufficient conditions (and necessary conditions in the simplest case) of complementability of a periodic frame to a periodic basis for the Euclidean space in terms of monodromy matrices of some linear system of differential equations built by using this periodic frame. We consider the pro...
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2001
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| Цитувати: | To the problem of complementability of a periodic frame to a periodic basis / O.A. Burylko, A.A. Davydenko // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 458-470. — Бібліогр.: 9 назв. — англ. |
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Burylko, O.A. Davydenko, A.A. 2021-01-27T18:06:02Z 2021-01-27T18:06:02Z 2001 To the problem of complementability of a periodic frame to a periodic basis / O.A. Burylko, A.A. Davydenko // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 458-470. — Бібліогр.: 9 назв. — англ. 1562-3076 AMS Subject Classification: 34C30, 34C40, 37D10 https://nasplib.isofts.kiev.ua/handle/123456789/174762 We obtain sufficient conditions (and necessary conditions in the simplest case) of complementability of a periodic frame to a periodic basis for the Euclidean space in terms of monodromy matrices of some linear system of differential equations built by using this periodic frame. We consider the problem of complementability for introducing local coordinates in a neighbourhood of a smooth m-dimensional invariant torus of a dynamic system in the Euclidean space R n if the dimensions satisfy the inequality m + 1 < n ≤ 2m. This work was partial supported by the State Fundation for Fundamental Research (grant no. 01.07/00047). en Інститут математики НАН України Нелінійні коливання To the problem of complementability of a periodic frame to a periodic basis До проблеми доповнюваності періодичного репера до періодичного базису К проблеме дополняемости периодического репера до периодического базиса Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
To the problem of complementability of a periodic frame to a periodic basis |
| spellingShingle |
To the problem of complementability of a periodic frame to a periodic basis Burylko, O.A. Davydenko, A.A. |
| title_short |
To the problem of complementability of a periodic frame to a periodic basis |
| title_full |
To the problem of complementability of a periodic frame to a periodic basis |
| title_fullStr |
To the problem of complementability of a periodic frame to a periodic basis |
| title_full_unstemmed |
To the problem of complementability of a periodic frame to a periodic basis |
| title_sort |
to the problem of complementability of a periodic frame to a periodic basis |
| author |
Burylko, O.A. Davydenko, A.A. |
| author_facet |
Burylko, O.A. Davydenko, A.A. |
| publishDate |
2001 |
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English |
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Нелінійні коливання |
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Інститут математики НАН України |
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Article |
| title_alt |
До проблеми доповнюваності періодичного репера до періодичного базису К проблеме дополняемости периодического репера до периодического базиса |
| description |
We obtain sufficient conditions (and necessary conditions in the simplest case) of complementability of a periodic frame to a periodic basis for the Euclidean space in terms of monodromy matrices of some linear system of differential equations built by using this periodic
frame. We consider the problem of complementability for introducing local coordinates in a
neighbourhood of a smooth m-dimensional invariant torus of a dynamic system in the Euclidean space R
n
if the dimensions satisfy the inequality m + 1 < n ≤ 2m.
|
| issn |
1562-3076 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/174762 |
| citation_txt |
To the problem of complementability of a periodic frame to a periodic basis / O.A. Burylko, A.A. Davydenko // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 458-470. — Бібліогр.: 9 назв. — англ. |
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Nonlinear Oscillations, Vol. 4, No. 4, 2001
TO THE PROBLEM OF COMPLEMENTABILITY
OF A PERIODIC FRAME TO A PERIODIC BASIS ∗
A. A. Burilko, A. A. Davydenko
Institute of Mathematics, NAS of Ukraine
Tereshchenkivs’ka Str., 3, Kyiv, 01601, Ukraine
e-mail: AndriyDavydenko@mail.ru
burilko@imath.kiev.ua
We obtain sufficient conditions (and necessary conditions in the simplest case) of comple-
mentability of a periodic frame to a periodic basis for the Euclidean space in terms of mon-
odromy matrices of some linear system of differential equations built by using this periodic
frame. We consider the problem of complementability for introducing local coordinates in a
neighbourhood of a smooth m-dimensional invariant torus of a dynamic system in the Eu-
clidean space Rn if the dimensions satisfy the inequality m+ 1 < n ≤ 2m.
AMS Subject Classification: 34C30, 34C40, 37D10
1. Introduction
Let’s consider a system of differential equations,
dx
dt
= X(x), x ∈ Rn, (1)
where X ∈ C l(Rn), l ≥ 1, C l(D) is the space of l times continuously differentiable functions
on a domain D ⊆ Rn. Suppose system (1) has an invariant manifold
M : x = f(ϕ), (2)
where f(ϕ) ∈ C l(Tm), Tm is the m-dimensional torus, rank (∂f(ϕ)/∂ϕ) = m ∀ϕ ∈ Tm.
Many works were devoted to a construction of local coordinates in a neighbourhood of
invariant toroidal manifold (2) (in particular, these problems were considered in [1 – 8]). As
shown in [1, 5], this problem is solved in the case where the system of vectors
∂f(ϕ)
∂ϕi
, i = 1,m, (3)
can be complemented to a 2π-periodic basis for the space Rn and sufficient conditions for
complementability of the system of linearly independent vectors (r-frame)
ui(ϕ), ϕ ∈ Tm, i = 1, r, (4)
∗ This work was partial supported by the State Fundation for Fundamental Research (grant no. 01.07/00047).
458 c© A. A. Burilko, A. A. Davydenko, 2001
TO THE PROBLEM OF COMPLEMENTABILITY OF A PERIODIC FRAME TO A PERIODIC BASIS 459
have been found, e. g.,
{n > m+ r} ∪ {n = r + 1}. (5)
As it was shown in [4], in general, the system of vectors (4) can not be complemented to a
2π-periodic basis for the space Rn. In particular, the vector
u1(ϕ1, ϕ2) = (2 + (2 + sin(ϕ1)) sin(ϕ2)) (1 + 4(2 + sin(ϕ1))(1 + sin(ϕ2)))
−1/2,
u2(ϕ1, ϕ2) = (2 + sin(ϕ1)) cos(ϕ2) (1 + 4(2 + sin(ϕ1))(1 + sin(ϕ2)))
−1/2,
u3(ϕ1, ϕ2) = cos(ϕ1)(1 + 4(2 + sin(ϕ1))(1 + sin(ϕ2)))
−1/2
(6)
can not be complemented to a 2π-periodic basis for the space R3.
The possibility of introducting local coordinates in a neighbourhood of an invariant torus in
the case where conditions (5) are not fulfilled, namely,
m+ 1 < n < 2m+ 1, (7)
was investigated in [7, 8]. It was suggested to complement system (1) with p equations
dy
dt
= Y (y), y ∈ Rp, Y ∈ C l(Rn), (8)
such that system (1), (8) has an invariant torus,
M1 : x = f(ϕ), y = f1(ϕ). (9)
It is possible to introduce local coordinates ϕ ∈ Tm, h ∈ Rn−m+p, in a neighbourhood of M1,
where the dimension p satisfies certain conditions and ‖h‖ =
(
m+1∑
i=1
h2i
)1/2
< δ, δ is a small
positive number, by defining
x = f(ϕ) +B(ϕ)h, y = f1(ϕ) + b(ϕ)h, (10)
where the matrices B(ϕ), b(ϕ) are taken from the space Cr(Tm) and such that
det
∂f(ϕ)/∂ϕ B(ϕ)
∂f1(ϕ)/∂ϕ b(ϕ)
6= 0 ∀ϕ ∈ Tm. (11)
If the condition Y (0) = 0 is fulfilled for the right-hand side of (8), then system (1), (8) always
has an invariant manifold,
M0 : x = f(ϕ), y = 0,
and the construction of local coordinates in a neighbourhood of the torus M0 is simpler than
in general. In this case, the minimal number of equtions in system (8) sufficient to introduce
local coordinates was calculated in [8]. The following question arises: can we find a system (8)
460 A.A. BURILKO, A.A. DAVYDENKO
smaller than obtained in [8] and such that the condition Y (0) = 0 is not fulfilled, but system (1),
(8) can be written in local coordinates. In this article we’ll answer this question in the negative.
System (1) with the smallest dimension such that condition (7) is fulfilled is a system of 4
differential equations that has 2-dimensional invariant torus. Thus the problem of introducing
local coordinates depends on the problem of complementability of the frame for the dimensions
n = 4,m = 2, r = m = 2, to a periodic basis. This problem can be simplified [1, 8] and reduced
to the problem of complementability of the frame with the dimensions n = 3, m = 2, r = 1.
But example (6) shows that this can not be done in the general case. One of sufficient conditions
for complementability of one vector to a periodic basis is that the set of values of this vector
is not everywhere dense in the orb S1(0) [5]. But this condition is not necessary, because, for
example, the vector u(ϕ) = (sinϕ1 sinϕ2, cosϕ1 sinϕ2, cosϕ2) is everywhere dense in orb and
it is obvious that this vector can be complemented. That’s why it is important to get another
necessary and sufficient conditions of complementability.
This paper continues [8]. In Section 2 we’ll get sufficient conditions of complementability
of an r-frame U(ϕ), ϕ ∈ Tm, with the help of the monodromy matrix of a system of linear
differential equations constructed from vectors of this frame so that these conditions will be
necessary in the simplest case (n = 3, m = 2, r = 1). In Section 3 we’ll improve the results
obtained in Section 2 in paper [8].
2. Conditions of Complementability
Let’s consider a system of linearly independent vectors in the space Rn,
U(ϕ) = (u1(ϕ), . . . , ur(ϕ)) , rankU(ϕ) = r, ∀ϕ ∈ Tm, 1 ≤ r < n,
where U(ϕ) ∈ C l(Tm). We say that the r-frame U(ϕ) can be complemented to a periodic basis
of the space Rn if there is an (n− r)-frame V (ϕ) in the space C l(Tm) such that
det (U(ϕ), V (ϕ)) 6= 0 ∀ϕ ∈ Tm.
Lemma 1. A system of vectors U(ϕ) ∈ C l(Tm) has a maximum subsystem of p vectors (n−
m−1 ≤ p ≤ r−1) that can be complemented to a periodic basis of the space Rn (any subsystem
of (p + 1) vectors can not be complemented) if and only if the r-frame U(ϕ) can be represented
in the form
U(ϕ) = M1(ϕ)
(
Ep 0
0 U(ϕ)
)
M2(ϕ),
where the matricesM1(ϕ),M2(ϕ),U(ϕ) ∈ C l(Tm) have the dimensions n×n, r×r, (n−p)×(r−p)
and detMk(ϕ) 6= 0 ∀ϕ ∈ Tm, k = 1, 2, and any vector-column of the matrix U(ϕ) can not be
complemented to a periodic basis of the space Rn−p, Ep is the p-dimensional identity matrix.
This Lemma is a corollary of Lemma 1 from [8]. Besides, due to the proof of the last
lemma we can conclude that the complementability of the frame can be reduced to that of
complementability of separate vectors to a periodic basis. The algorithm of this simlifica-
tion is given in [8]. That’s why we’ll consider the given problem only for a single vector
TO THE PROBLEM OF COMPLEMENTABILITY OF A PERIODIC FRAME TO A PERIODIC BASIS 461
u (ϕ1, . . . , ϕm) ∈ C l(Tm). Without loss of generality we can assume that u∗(ϕ)u(ϕ) = 1 (D∗
is the transpose of the matrix D). Also suppose that 3 ≤ n ≤ m + 1, so that the sufficient
condition (5) is not fulfilled.
Let’s denote ϕ = (ϕ′i, ϕ
′′
i ), ϕ′i = (ϕ1, . . . , ϕi), ϕ′′i = (ϕi+1, . . . , ϕm), i = 1,m. Consider
the vector u(ϕ′j , 0) for some fixed j ≤ m. Suppose that this vector can be complemented to a
periodic basis if one of the variables is fixed (we may assume that this variable is ϕj). Then there
exists a square matrix Oj(ϕ′j−1) =
[
Zj(ϕ
′
j−1), u(ϕ′j−1, 0)
]
∈ C l(Tj−1), O∗j (ϕ
′
j−1)Oj(ϕ
′
j−1) =
En. The vector uj(ϕ′j) = O∗j (ϕ
′
j−1)u(ϕj , 0) ∈ C l(Tj) satisfies the condition uj(ϕ
′
j−1, 0) =
colon(0, . . . , 0, 1).
Let’s construct the following matrix by using the vector uj(ϕ′j),
Kj(ϕ
′
j) =
∂uj(ϕ
′
j)
∂ϕj
u ∗j (ϕ′j)− uj(ϕ′j)
∂u ∗j (ϕ′j)
∂ϕj
∈ C l−1(Tj),
and consider the system of differential equations
∂h
∂ϕj
= Kj(ϕ
′
j)h. (12)
The coefficients of this system are periodic functions of the variables ϕj and the parameters
ϕ′j−1. This system has the following properties:
1. Kj(ϕ
′
j) is a skew-symmetric matrix.
2. The vector uj(ϕ′j) is a solution of system (12). Indeed, taking into account that the vector
uj(ϕ
′
j) has lenght one, we obtain
Kjuj =
∂uj
∂ϕj
u ∗j uj − uj
∂u ∗j
∂ϕj
uj =
∂uj
∂ϕj
.
3. The matriciant Ω
ϕj
0 (ϕ′j−1) of system (12) is a periodic function of the parameters ϕ′j .
4. The matriciant of system (12) is orthonormalized
(
Ω
ϕj
0 (ϕ′j−1)
)∗
Ω
ϕj
0 (ϕ′j−1) = En. This
follows from the first property.
Combining the second and the third properties we get that the matriciant of system (12) can
be represented in the form Ω
ϕj
0 (ϕ′j−1) =
[
Hj(ϕ
′
j), uj(ϕ
′
j)
]
, where Hj(ϕ
′
j) ∈ C l−1 (Tj−1 ×R).
Taking into account that the matriciant Ω
ϕj
0 (ϕ′j−1) is orthonormalized and the vector uj(ϕ′j)
is a 2π-periodic function of ϕj , the monodromy matrix of system (12) can be represented in the
form
Ω2π
0 (ϕ′j−1) =
(
Hj(ϕ
′
j−1, 2π) 0
0 1
)
,
where Hj(ϕ
′
j) is the first (n − 1) rows of the matrix Hj(ϕ
′
j). Note that
detHj(ϕ
′
j−1, 2π) 6= 0 ∀ϕ′j−1 ∈ Tj−1.
462 A.A. BURILKO, A.A. DAVYDENKO
Suppose there exists a real-valued logarithm of the monodromy matrix
Aj(ϕ
′
j−1) = ln
(
Hj(ϕ
′
j−1, 2π)
)
in the space C l−1(Tj−1). Let us remark that such a logarithm
does not always exist. Some sufficient conditions for existence of this logarithm are obtained in
the papers [5, 9]. In this case, Floquet’s matrix of system (12) can be represented in the form
Φ(ϕ′j) = Φ(ϕ′j−1, ϕj) = Ω
ϕj
0 (ϕ′j−1) exp
{
−(2π)−1ϕj ln Ω2π
0 (ϕ′j−1)
}
=
[
Hj(ϕ
′
j), uj(ϕ
′
j)
]
diag
{
exp
{
−(2π)−1A(ϕ′j−1)ϕj
}
, 1
}
=
[
H ′j(ϕ
′
j), uj(ϕ
′
j)
]
,
where H ′j(ϕ
′
j) = Hj(ϕ
′
j) exp
{
−(2π)−1A(ϕ′j−1)ϕj
}
.
The matrix Φ(ϕ′j) is continuously differentiable of order l−1 and is a periodic function of the
parameters ϕ′j−1 (being a superposition of a periodic matrix of parameters) and the variable ϕj
(as follows from the Floquet theory). In other words, Φ(ϕ′j) is an element of the class C l−1(Tj)
and, thus, the matrix H ′j(ϕ
′
j) is an element of this class, too. The last matrix is a complement of
the vector uj(ϕ′j) to a periodic basis of the space Rn, because Φ(ϕ′j) is a nondegenerate matrix.
In the same way, the vector uj+1(ϕ
′
j+1), where j + 1 ≤ m, under the given conditions, can
be complemented if there exists a matrix Aj+1(ϕj) from the space C l−1(Tj) (this is a sufficient
condition). In this case, we can take H ′j(ϕ
′
j) as the matrix Zj+1(ϕ
′
j). And so on. Finally we get
the following theorem.
Theorem 1. Suppose a vector u(ϕ) ∈ C l(Tm) can be complemented to a periodic basis if
some m− s variables are fixed (without loss of generality, we can take them to be ϕs+1, . . . , ϕm).
Besides, there exist matrices Aj(ϕ′j−1) ∈ C l−1(Tj−1), j = s+ 1,m. Then the vector u(ϕ) can be
complemented by vectors from the class C l−1(Tm) to a periodic basis of the space Rn.
Note 1. If s ≤ n − 2, then the first condition of the previous theorem is fulfilled. Indeed,
conditions (5) are fulfilled in this case.
Note 2. Suppose a vector u(ϕ) can be complemented by vectors from the class C l−1(Tm)
using the previous theorem; then taking into account Theorem 1 from [7], there also exist
complementing vectors from the space C l(Tm).
Now we look at the case of 3-dimension vectors of two variables in more details
u(ϕ) = colon (u1(ϕ), u2(ϕ), u3(ϕ)) ∈ C l(T2).
Let u(ϕ) be orthonormalized. It can easily be checked that for this vector the first condition
of Theorem 1 is satisfied for any variables ϕ1, ϕ2. Then without loss of generality we assume
that u(ϕ1, 0) = colon[0; 0; 1]. Let us construct the matrix K2(ϕ1, ϕ2) from this vector. So, we
get the matriciant Ωϕ2
0 (ϕ1) = [h′(ϕ), h′′(ϕ), u(ϕ)] of the corresponding system (12). Taking into
account that this matriciant is orthonormalized, we can represent the monodromy matrix in the
form
Ω2π
0 (ϕ1) =
a(ϕ1) −b(ϕ1) 0
b(ϕ1) a(ϕ1) 0
0 0 1
, where a2(ϕ1) + b2(ϕ1) ≡ 1. (13)
TO THE PROBLEM OF COMPLEMENTABILITY OF A PERIODIC FRAME TO A PERIODIC BASIS 463
Let’s denote by z(ϕ1) the complex-valued function
z(ϕ1) = a(ϕ1) + ib(ϕ1).
Thus, the logarithm of this monodromy matrix is always real-valued for any ϕ1, namely,
Ln Ω2π
0 (ϕ1) =
0 −Arg(z(ϕ1)) 0
Arg(z(ϕ1)) 0 0
0 0 0
.
Let’s prove the following lemma.
Lemma 2. Let z(ϕ1) ∈ C l−1(T1) (l ≥ 2), z(ϕ) 6= 0. There exists a logarithm ln z(ϕ1) from
the space C l−1(T1) if and only if the point z(ϕ1) moving along a closed curve C about the origin
makes the same number of clockwise and counterclockwise revolutions as the variable ϕ1 ranges
over [0; 2π].
Proof. It is clear that ln |z(ϕ1)| ∈ C l−1(T1). Because z(ϕ1) ∈ C l−1(T1), we get that the
point z(ϕ1) moves continuesly along some closed curve C while the variable ϕ1 ranges over R.
Let’s fix some point ϕ(0)
1 ∈ [0; 2π] and denote by ω(ϕ1) : R → R the continuous function
that is equal to the angle between the vector z(ϕ1) and the positive real half axis such that
ω(ϕ
(0)
1 ) = ω0 ∈ [0; 2π]. The function ω(ϕ1) belongs to the same space as z(ϕ1), because
dω(ϕ1)
dϕ1
=
db(ϕ1)
dϕ1
a(ϕ1)−
da(ϕ1)
dϕ1
b(ϕ1).
Thus, Argz(ϕ1) consists of all smoth curves ω(ϕ1) + 2πk, k ∈ Z. If the vector z(ϕ1) makes a
complete turn around zero moving clockwise (counterclockwise) from z(ϕ
(0)
1 ) then the value of
the function ω(ϕ1) changes from ω0 to ω0+2π (ω0−2π). Hence, ω(ϕ1) is a 2π-periodic function
(that is, ω(ϕ
(0)
1 + 2π) = ω0) if and only if the point z(ϕ1) makes the same numbers of complete
turns around zero in each of the two directions while ϕ ∈ [ϕ
(0)
1 ;ϕ
(0)
1 + 2π]. Replacing variable
ϕ1 = ϕ1 − ϕ(0)
1 we complete the proof.
Let’s choose
Λ(ϕ1) =
1
2π
0 −ω(ϕ1) 0
ω(ϕ1) 0 0
0 0 0
,
where ω(ϕ1) ∈ C l−1(T1) is the function from the previous lemma. In this case the Floquet’s
matrix of system (12) can be represented in the form
Φ(ϕ) = Ωϕ2
0 (ϕ1) exp{−Λ(ϕ1)ϕ2}
464 A.A. BURILKO, A.A. DAVYDENKO
=
[
h′(ϕ), h′′(ϕ), u(ϕ)
]
cos (ω(ϕ1)ϕ2/2π) sin(ω (ϕ1)ϕ2/2π) 0
− sin (ω(ϕ1)ϕ2/2π) cos (ω(ϕ1)ϕ2/2π) 0
0 0 1
=
[
w′(ϕ), w′′(ϕ), u(ϕ)
]
,
where
w′(ϕ) = cos (ω(ϕ1)ϕ2/2π)h′(ϕ)− sin (ω(ϕ1)ϕ2/2π)h′′(ϕ),
w′′(ϕ) = sin (ω(ϕ1)ϕ2/2π)h′(ϕ) + cos (ω(ϕ1)ϕ2/2π)h′′(ϕ).
Therefore, from the above-stated properties, the vectors w′(ϕ), w′′(ϕ) are orthonormalized,
belong to the class C l−1(T2), and complete the vector u(ϕ) to a periodic basis of the space R3.
The main sufficient condition for constructing these vectors is existence of a continuous 2π-
periodic function ω(ϕ1), which is an equivalent to the condition of Lemma 2. Now we show
that this condition is necessary.
Let’s consider any 3 × 3-matrix Ψ(ϕ) ∈ C l(T2) such that the columns make an orthonor-
mal periodic basis of the space R3, that is Ψ
∗
(ϕ)Ψ(ϕ) ≡ E3. Then the columns of the matrix
Ψ(ϕ) = Ψ
∗
(ϕ1, 0)Ψ(ϕ) also make an orthonormal periodic basis of the space R3 and the con-
dition Ψ(ϕ1, 0) = E3 is satisfied. Denoting Ψ(ϕ) = [v′(ϕ), v′′(ϕ), u(ϕ)] and using the given
vector u(ϕ), we construct system (12). Then the matrix Ω
ϕ2
0 (ϕ1) = Ψ(ϕ) is a matriciant of the
system
∂h
∂ϕ2
= K2(ϕ)h+ hΓ(ϕ), h ∈ R3, (14)
where K2(ϕ) is the matrix of coefficients in system (12) and
Γ(ϕ) =
0 −γ(ϕ) 0
γ(ϕ) 0 0
0 0 0
,
where γ(ϕ) = (v′′(ϕ))∗∂v′(ϕ)/∂ϕ2 = −(v′(ϕ))∗∂v′′(ϕ)/∂ϕ2 ∈ C l−1(T2).
It is easy to prove by a direct calculation that the matrix Γ(ϕ) and the integral of this ma-
trix with respect to ϕ2 commute. Then, the Lappo – Danilevsky condition for the system of
differential equations
∂h
∂ϕ2
= hΓ(ϕ), h ∈ R3, (15)
is filfilled and the matriciant of this system is the following:
Ω̃ϕ2
0 (ϕ1) = exp
ϕ2∫
0
Γ(ϕ1, τ)dτ
. (16)
Hence, the matriciant of system (14) can be written as a product of the matriciants of systems
(12), (14),
Ω
ϕ2
0 (ϕ1) = Ωϕ2
0 (ϕ1)Ω̃
ϕ2
0 (ϕ1). (17)
TO THE PROBLEM OF COMPLEMENTABILITY OF A PERIODIC FRAME TO A PERIODIC BASIS 465
Let’s show that the matrix Ln Ω2π
0 (ϕ1), which consists of zero elements and continuously dif-
ferential functions ω(ϕ1) (as was shown in Lemma 2) is 2π-periodic. Consider the matrix
L(ϕ1) = Φ∗(ϕ)Ψ(ϕ). Using (16), (17), the form of Floquet’s matrix Φ(ϕ) of system (12), and
commutativity of the matrices Λ(ϕ1) i Γ(ϕ), we obtain
L(ϕ1) = exp
ϕ2∫
0
Γ(ϕ1, τ)dτ + Λ(ϕ1)ϕ2
. (18)
This matrix belongs to the class C l−1(R2) and is a periodic function of the variable ϕ2, because
Φ(ϕ) and Ψ(ϕ) have such properties. Taking into account the form of the matrix in the exponent
of the exponential function in the right-hand side of (18), we can write
L(ϕ) =
cos l(ϕ) − sin l(ϕ) 0
sin l(ϕ) cos l(ϕ) 0
0 0 1
,
where
l(ϕ) =
ϕ2∫
0
γ(ϕ1, τ)dτ +
1
2π
ω(ϕ1)ϕ2.
Since L(ϕ) is a 2π-periodic function of the second variable, we get sin l(ϕ1, 2π) = sin l(ϕ1, 0) =
0 and
l(ϕ1, 2π) =
2π∫
0
γ(ϕ1, τ)dτ + ω(ϕ1) = k(ϕ1)π, k(ϕ1) ∈ Z.
The functions γ(ϕ), ω(ϕ1) are continuous in the parameter ϕ1 (γ(ϕ) is a continuous function of
the variable ϕ2). That’s why, for all ϕ1 ∈ R, the function l(ϕ1, 2π) is continuous. This implies
that k(ϕ1) = k = const ∀ϕ1 ∈ R. Thus, we get
ω(ϕ1 + 2π)− ω(ϕ1) = l(ϕ1 + 2π, 2π)−
2π∫
0
γ(ϕ1 + 2π, τ)dτ − l(ϕ1, 2π)
+
2π∫
0
γ(ϕ1, τ)dτ = l(ϕ1 + 2π, 2π)− l(ϕ1, 2π) = kπ − kπ = 0,
which proves that the matrix Ln Ω2π
0 (ϕ1) belongs to the class C l−1(T1).
So, we can formulate the next theorem.
466 A.A. BURILKO, A.A. DAVYDENKO
Theorem 2. A necessary and sufficient condition for a unit vector u(ϕ) ∈ C l(T2), l ≥ 1, to be
complemented to a periodic basis of the space R3 is that the point z(ϕ1) = a(ϕ1)+ib(ϕ1), moving
along some curve C encircling the origin, makes the same number of turns in the clockwise and
counterclockwise directions while the variable ϕ1 ranges over [0; 2π]. Here functions a(ϕ1), b(ϕ1)
are coefficients of the monodromy matrix (13) of system (12) constructed from the vector u(ϕ).
The vectors that complement this vector belong to the space C l−1(T2).
Note 3. According to Lemma 1, the problem of complementability of a vector u(ϕ) ∈
C l(T2) to a periodic basis of the space R3 is the main condition for introducing local coordinates
in a neighbourhood of the invariant 2-dimensional torus in the space R4 (as was shown above
that this is the simplest case when this is impossible). That’s why, the previous theorem is very
important.
3. Complementability of a Frame to a Periodic Basis in the Extended Space
Let us consider a system of differential equation (1) that has invariant manifold (2) under
condition (7). Suppose there exists a system (8) such that system (1), (8) has invariant torus
(9). We’ll investigate the problem of constructing local coordinates in system (1), (8) in the
case when manifold (9) does not coincide with M0. The following lemmas about constructing a
periodic basis for the extended space are needed in the sequel.
Lemma 3. Suppose U(ϕ) = [u1(ϕ), . . . , ur(ϕ)], where each of the vectors uj(ϕ), j = 1, r,
can not be complemented to a periodic basis of the space Rn, U∗(ϕ)U(ϕ) = Er, belongs to the
class C l(Tm), r < n. Then for any r-dimensional square matrix N(ϕ) ∈ C l(Tm) there exists a
periodic basis in the space Rn+r,
F (ϕ) =
(
U(ϕ) En − U(ϕ)U∗(ϕ)− U(ϕ)N∗(ϕ)T (ϕ)U∗(ϕ)
N(ϕ) T (ϕ)U∗(ϕ)
)
∈ C l(Tm), (19)
where T (ϕ) ∈ C l(Tm) is a triangular r-dimensional square matrix used in the Gramm – Schmidt
orthonormalization process for the system of vectors [U∗(ϕ), N(ϕ)]∗.
Proof. By a direct calculation we get the following product:
F ∗(ϕ)F (ϕ) = diag{Er +N∗(ϕ)N(ϕ), En − U(ϕ)U∗(ϕ)
+ U(ϕ)T ∗(ϕ)(N(ϕ)N∗(ϕ) + Er)T (ϕ)U∗(ϕ)}.
According to the conditions of the lemma, the system of vectors P (ϕ) = [U∗(ϕ), N(ϕ)]∗ T (ϕ) is
orthonormalized. So, P ∗(ϕ)P (ϕ) = T ∗(ϕ)(Er + N(ϕ)N∗(ϕ))T (ϕ) = Er. Besides, taking into
account that the quadratic form η∗(Er + N∗(ϕ)N(ϕ))η = ‖η‖2 + ‖N(ϕ)η‖2 ≥ ‖η‖2, η ∈ Rr,
is positive definite, we obtain det(Er + N∗(ϕ)N(ϕ)) > 0 ∀ϕ ∈ Tm. Thus, detF ∗(ϕ)F (ϕ) =
det(Er + N∗(ϕ)N(ϕ)) > 0 ∀ϕ ∈ Tm. This means that the matrix F (ϕ) is a periodic basis in
the space Rn+r, which proves the lemma.
The previous lemma has useful corollaries.
TO THE PROBLEM OF COMPLEMENTABILITY OF A PERIODIC FRAME TO A PERIODIC BASIS 467
Corollary 1. Suppose an n-dimensional vector u(ϕ) ∈ C l(Tm) can not be complemented to
a periodic basis. Then for any function ũ(ϕ) ∈ C l(Tm) there exists the following periodic basis
in the space Rn+1:
F̃ (ϕ) =
u(ϕ) En − (1 + ũ(ϕ)
(
1 + ũ2(ϕ))−1/2
)
u(ϕ)u∗(ϕ)
ũ(ϕ) (1 + ũ2(ϕ))−1/2u∗(ϕ)
.
Corollary 2. Suppose the conditions for the matrix U(ϕ) in Lemma 3 are fulfilled. Then
there exists an orthonormalized periodic basis in space Rn+r,
F (ϕ) =
(
U(ϕ) En − U(ϕ)U∗(ϕ)
Or U∗(ϕ)
)
∈ C l(Tm), (20)
there Or is an r-dimensional square zero matrix.
The last corollary gives the simplest constructive way to complement the frame U(ϕ) under
the condition of Lemma 3 to a periodic basis. Besides, using Proposition 1 from paper [8], we
get that in this case r is the smallest, which means that if we add k coordinates to every vector
of this frame, where k < r, then the given frame can not be complemented to a periodic basis
in the extended space. The following question arises: can we add, to every vector of the frame
U(ϕ), q nonzero coordinates, where q < r, such that the extended frame can be complemented
to a periodic basis in the space Rn+q. This question is important in the theory of invariant
toroidal manifolds, because the answer in the affirmative would allow to decrease the number
of additional equations in system (8) to construct local coordinates [7, 8]. Now we’ll show that
we can answer in the negative.
Lemma 4. Suppose the matrix
F (ϕ) =
U(ϕ) B(ϕ)
N(ϕ) b(ϕ)
∈ C l(Tm)
consists of vectors that make a periodic basis in the space Rn,
detF (ϕ) 6= 0 ∀ϕ ∈ Tm,
where N(ϕ), B(ϕ) are square matrices of dimensions r and n, r < n. Besides, let the systems
of vectors [U∗(ϕ), N∗(ϕ)]∗ and [B∗(ϕ), b∗(ϕ)]∗ belong to two orthogonal subspaces of the space
Rn. Then any vector from the frame U(ϕ) can not be complemented to a periodic basis in the
space Rn−k if and only if any row of the matrix b(ϕ) can not be complemented to a periodic basis
in the space Rn−k.
Proof. First we shall prove this lemma for the matrix F (ϕ) from Lemma 3 with b(ϕ) =
T (ϕ)U∗(ϕ). Using the proof Lemma 3, it can be easily checked that all conditions of Lemma 4
468 A.A. BURILKO, A.A. DAVYDENKO
are satisfied for the matrix F (ϕ). Suppose any vector of the frame U(ϕ) can not be com-
plemented to a periodic basis in the space Rn. Taking into account properties of the ma-
trix T (ϕ) = {Tkj(ϕ)}rk,j=1 from Lemma 3 we obtain that for any coefficient on the diagonal
Tjj(ϕ) 6= 0 ∀ϕ ∈ Tm, and the rows of the matrix b(ϕ) can be written as
ũ∗k(ϕ) =
r∑
j=k
Tkj(ϕ)u∗j (ϕ), k = 1, r.
Suppose there exists some vector ũk0(ϕ) that can be complemented to a periodic basis in
the space Rn. Then there exists an orthonormalized n × (n − 1)-matrix B̃k0(ϕ) such that
det
[
ũk0(ϕ), B̃k0(ϕ)
]
6= 0, ũ∗k0(ϕ)B̃k0(ϕ) = 0 ∀ϕ ∈ Tm. Thus, we get
(
u∗k0(ϕ)
B̃∗k0(ϕ)
)[
ũk0(ϕ), B̃k0(ϕ)
]
=
(
Tkk(ϕ) u∗k0(ϕ)B̃k0(ϕ)
0 En−1
)
.
It follows from the last identity that
[
uk0(ϕ), B̃k0(ϕ)
]
is a nondegenerate matrix for all ϕ ∈ Tm,
because the other matrices in this identity are nondegenerate. Thus, there exists a vector uk0(ϕ)
that can be complemented to a periodic basis in the space Rn, which contradicts the assumed
properties of the matrix U(ϕ).
In the same way, we can prove that if any row of the matrix b(ϕ) can not be complemented
then any vector from U(ϕ) can not be complemented (it is sufficient to transpose the matrix
F (ϕ) with respect to the second diagonal and use U∗(ϕ) = T−1(ϕ)T (ϕ)U∗(ϕ), where T−1(ϕ)
has the same properties as T−1(ϕ)).
Now note that the matrix F (ϕ) can be represented in the form F (ϕ) = F (ϕ)S(ϕ), where
S(ϕ) ∈ C l(Tm) is an n+ r-dimensional nondegenerate square matrix. Taking into account that
the matrices F (ϕ) and F (ϕ) are orthonormalized, we get
S(ϕ) = (F ∗(ϕ)F (ϕ))−1 F ∗(ϕ)F (ϕ) = diag{Er; S(ϕ)},
where S = (En − UU∗ − UT ∗NU∗)B + UT ∗b. Hence, the matrix b(ϕ) can be represented in
the form b(ϕ) = T (ϕ)U∗(ϕ)S(ϕ), where S(ϕ) is a nondegenerate matrix . Suppose there exists
a row bj(ϕ) in the matrix b(ϕ) that can be complemented to a periodic basis in the space Rn.
Then there exists an (n− 1)×n–matrix Bj(ϕ) ∈ C l(Tm) such that det
[
b∗j (ϕ), B∗j (ϕ)
]∗
6= 0 for
all ϕ ∈ Tm. For the j-th column of the matrix T (ϕ)U∗(ϕ), we get
det
(
(T (ϕ)U∗(ϕ))j
Bj(ϕ)S
−1
(ϕ)
)
= det
(
(T (ϕ)U∗(ϕ))j S(ϕ)
Bj(ϕ)
)
S
−1
(ϕ) = det
(
bj(ϕ)
Bj(ϕ)
)
S
−1
(ϕ) 6= 0
for all ϕ ∈ Tm. This shows that one row of the matrix b(ϕ) can be complemented to a periodic
basis in the space Rn and we get the contradiction.
Similarly, if any row of the matrix b(ϕ) can not be complemented to a periodic basis, then
any row of the matrix T (ϕ)U∗(ϕ) can not be complemented to a periodic basis, and as above, it
follows that any vector from U(ϕ) can not be complemented to a periodic basis. This completes
the proof.
TO THE PROBLEM OF COMPLEMENTABILITY OF A PERIODIC FRAME TO A PERIODIC BASIS 469
Lemma 5. Suppose the matrix U(ϕ) satisfies the conditions of Lemma 3. Then for any q× r-
dimensional matrix N̂(ϕ) ∈ C l(Tm), q < r, the r-frame
[
U∗(ϕ), N̂∗(ϕ)
]∗
can not be comple-
mented to a periodic basis in the space Rn+q.
Proof. Assume the converse. Then there exist matrices B̂(ϕ), b̂(ϕ) ∈ C l(Tm) of dimen-
sions n× (n+ q − r) and q × (n+ q − r) such that
det F̂ (ϕ) 6= 0 ∀ϕ ∈ Tm, where F̂ (ϕ) =
(
U(ϕ) B̂(ϕ)
N̂(ϕ) b̂(ϕ)
)
.
Let us consider the matrix F̂1(ϕ) = diag
{
F̂1(ϕ), Er−q
}
. We orthonormalize this matrix using
the Gramm – Schmidt procedure. So, we obtain
F̂1(ϕ) =
(
Û1(ϕ) B̂1(ϕ)
N̂1(ϕ) b̂1(ϕ)
)
=
(
U(ϕ) B̂(ϕ)
N̂0(ϕ) b̂0(ϕ)
)(
R11(ϕ) R12(ϕ)
0 R22(ϕ)
)
,
where R11(ϕ) and R22(ϕ) are the nondegenerate square matrices of dimensions r and n,
N̂0(ϕ) =
(
N̂(ϕ)
0
)
, b̂0(ϕ) =
(
b̂(ϕ) 0
0 Er−q
)
.
From the last identity we obtain
b̂0 = −N̂1R
−1
11 R12R
−1
22 + b̂1R
−1
22 =
(
−N̂0R12 + b̂1
)
R−122 .
Denote by b̃0, b̃1, Ñ0 the last r − q rows of the matrices b̂0, b̂1, N̂0. Since N̂0 = 0, we get
b̃0(ϕ) = b̃1(ϕ)R−122 (ϕ). (21)
The matrix F̂1(ϕ) satisfies all conditions of the previous lemma. That’s why any row of b̂1(ϕ)
and also any row of b̃1(ϕ) can not be complemented to a periodic basis in the space Rn. As
in the proof of Lemma 4, we obtain that any row of the matrix b̃1(ϕ)R−122 (ϕ) can not be com-
plemented to a periodic basis. But according to (21), this matrix is represented in the form
b̃0(ϕ) = [0, Er−q], where any row can be complemented to a periodic basis in a trivial way.
This contradiction concludes the proof.
Using Lemmas 1, 4, 5 and the proof of Theorem 1 from [8] we obtain the following conclu-
sion.
Theorem 3. Suppose system (1) has invariant torus (2) of dimension satisfying condition (7).
Besides, let system of vectors (3) can not be complemented to a periodic basis in the space Rn, but
have a maximal subsystem ofm−p vectors (1 ≤ p ≤ 2m+1−n) that can be complemented to a
periodic basis in the space Rn (any subsystem of (m− p+ 1) vectors can not be complemented).
Then in the extended system (1), (8), where y ∈ Rp, with invariant manifold (9), it is possible to
introduce local coordinates ϕ ∈ Tm, h ∈ Rl, ‖h‖ < δ, l = n −m + p, according (10) such that
(11) is fulfilled, where the matrices B(ϕ) and b(ϕ) of dimension n× l and p× l belong to the class
Cr(Tm), any row of the matrix b(ϕ) can not be complemented to a periodic basis in the space Rl.
Besides, p is the smallest number of differential equations in system (8).
470 A.A. BURILKO, A.A. DAVYDENKO
Note 4. Theorem 3 shows that if system (8) is such that the extended system (1), (8) has
invariant manifold (9) that doesn’t coincide with M0, then the dimension p of the variable y is
not less then in [8]. That’s why the construction of local coordinates as given in Theorem 1 in
[8] is optimal.
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Received 10.09.2001
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