Some problems of the linear theory of systems of ordinary differential equations
We consider problems of the linear theory of systems of ordinary differential equations related to the investigation of invariant hyperplanes of these systems, the notion of equivalence for these systems, and the Floquet – Lyapunov theory for periodic systems of linear equations. In pa...
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| author | Samoilenko, A. M. Самойленко, А. М. |
| author_facet | Samoilenko, A. M. Самойленко, А. М. |
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| description | We consider problems of the linear theory of systems of ordinary differential equations related to the investigation of invariant hyperplanes of these systems, the notion of equivalence for these systems, and the Floquet – Lyapunov theory for periodic systems of linear equations. In particular, we introduce the notion of equivalence of systems of linear differential equations of different orders, propose a new formula of the Floquet form for periodic systems, and present the application of this formula to the introduction of amplitude-phase coordinates in a neighborhood of a periodic trajectory of a dynamical system. |
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UDC 517.9
A. M. Samoilenko (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS
OF ORDINARY DIFFERENTIAL EQUATIONS
ДЕЯКI ПРОБЛЕМИ ЛIНIЙНОЇ ТЕОРIЇ СИСТЕМ
ЗВИЧАЙНИХ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
We consider problems of the linear theory of systems of ordinary differential equations related to the investi-
gation of invariant hyperplanes of these systems, the notion of equivalence for these systems, and the Floquet –
Lyapunov theory for periodic systems of linear equations. In particular, we introduce the notion of equivalence
of systems of linear differential equations of different orders, propose a new formula of the Floquet form for
periodic systems, and present the application of this formula to the introduction of amplitude-phase coordinates
in a neighborhood of a periodic trajectory of a dynamical system.
Розглянуто проблеми лiнiйної теорiї систем звичайних диференцiальних рiвнянь, пов’язанi з дослiджен-
ням iнварiантних гiперплощин таких систем, поняттям еквiвалентностi для вказаних систем та теорiєю
Флоке – Ляпунова для перiодичних систем лiнiйних рiвнянь. Зокрема, введено поняття еквiвалентнос-
тi систем лiнiйних диференцiальних рiвнянь рiзних порядкiв, запропоновано нову формулу вигляду
Флоке для перiодичних систем, наведено застосування цiєї формули для введення амплiтудно-фазових
координат в околi перiодичної траєкторiї динамiчної системи.
Introduction. In this work, we consider problems of the linear theory of systems of
ordinary differential equations related to the investigation of invariant hyperplanes of
these systems, the notion of equivalence for these systems, and the Floquet – Lyapunov
theory for periodic systems of linear equations.
The work is based on the preprints [ 1 ] and [ 2 ] and consists of three parts.
In the first part, we clarify conditions of invariance in the sense of Bogolyubov for
two orthogonal hyperplanes with respect to a linear system of differential equations with
variable coefficients. It is proved that the invariance of these hyperplanes is equivalent
to the separation of the system of differential equations into two independent subsystems
whose orders correspond to the dimensions of the hyperplanes.
On the basis of these investigations, we introduce the notion of equivalence of
systems of differential equations of different orders.
In the second part, we consider a real T -periodic system of linear differential
equations. We study the case, inadequately described by the Floquet – Lyapunov theory,
of necessary period doubling under the reduction of this system to a system with constant
coefficients by a real periodic matrix. We prove the real T -periodic equivalence of this
system and a higher-order system of differential equations with constant coefficients.
In the third part, the results of the second part are used, first, to the reduction of
a nonlinear system of differential equations with separated periodic linear part to a
system of equations with constant coefficients with separated linear part of higher order
and, second, for the introduction of local coordinates in the neighborhood of a periodic
trajectory of an autonomous system of nonlinear differential equations.
1. Two lemmas. Suppose that matrices Φ1(t) and Φ2(t) are continuously differentiable
for all t ∈ R, Φ1(t) ∈ Mnm(R), Φ2(t) ∈ Mpm(R), where m > n > p = m − n, and
the following condition is satisfied:
det
(
Φ1(t)
Φ2(t)
)
6= 0.
c© A. M. SAMOILENKO, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2 237
238 A. M. SAMOILENKO
Let Φ+
1 (t) and Φ+
2 (t) denote matrices pseudoinverse to Φ1(t) and Φ2(t) and defined
by the condition (
Φ1(t)
Φ2(t)
)(
Φ+
1 (t),Φ+
2 (t)
)
= E, (1.1)
Φ1(t) ∈Mnm(R), Φ2(t) ∈Mm−nm(R), E ∈Mm(R).
Let
M1(t) = Φ+
1 (t)Φ1(t),
M2(t) = Φ+
2 (t)Φ2(t).
If follows from (1.1) and the definitions of the matrices M1(t) and M2(t) that these
matrices satisfy the conditions
M2
ν (t) = M1(t), ν = 1, 2, rankM1(t) = n, rankM2(t) = p,
M1(t)M2(t) = M2(t)M1(t) = 0, M1(t) +M2(t) = E. (1.2)
Let us prove equality (1.2). Indeed, since the matrix (Φ+
1 (t),Φ+
2 (t)) is inverse to the
matrix
(
Φ1(t)
Φ2(t)
)
, multiplying the latter from the left by the matrix
(
Φ+
1 (t),Φ+
2 (t)
)
we
obtain the equality
Φ+
1 (t)Φ1(t) + Φ+
2 (t)Φ2(t) = E,
which is equivalent to equality (1.2).
In the space Rm, we define two subspaces by using the matrix M1(t), namely,
Mn(t) = {y ∈ Rm : y = M1(t)y},
Mm−n(t) = {y ∈ Rm : M1(t)y = 0},
(1.3)
and two subspaces by using the matrix M2(t):
Mp
1 (t) = {y ∈ Rm : y = M2(t)y},
Mm−p
1 (t) = {y ∈ Rm : M2(t)y = 0}.
(1.4)
Lemma 1. The subspaces Mk(t) and Mk
1 (t), k ∈ {n, p}, satisfy the conditions
Mn(t) = Mm−p
1 (t) = ker Φ2(t), Mm−n(t) = Mp
1 (t) = ker Φ1(t).
Indeed, according to properties of the matrix M1(t), the general solution of the
equation defined by subspace (1.3) is the function
y = M1(t)c(t), (1.5)
where c(t) is an arbitrary function with values in Rm. According to properties of the
matrix M2(t), the general solution of the equation defined by subspace (1.4) is the
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SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 239
function
y = (E −M2(t))c1(t) = M1(t)c1(t).
Thus, the general solutions of the considered equations coincide for c(t) = c1(t), which
proves the equality Mn(t) = Mm−p
1 (t).
It follows from the definition of Mm−p
1 that
Mm−p
1 (t) = kerM2(t) = ker(Φ+
2 (t)Φ2(t)).
Taking into account that the equality Φ2(t)x = 0 implies that
Φ2(t)Φ+
2 (t)x = x = 0,
we conclude that ker Φ+
2 (t) = 0 and, hence, ker(Φ+
2 (t)Φ2(t)) = ker Φ2(t). This proves
the first equality of Lemma 1. The second equality is proved by analogy.
Lemma 2. The mapping Φ+
1 (t) : y = Φ+
1 (t)x is a diffeomorphism of Rn into
Mn(t), and the mapping Φ+
2 (t) : y = Φ+
2 (t)x is a diffeomorphism of Rn intoMm−n(t).
We prove only the first assertion of Lemma 2 because the second assertion is proved
by analogy.
The matrix Φ+
1 (t), as a block of the matrix inverse to
(
Φ1(t)
Φ2(t)
)
, is continuously
differentiable and has a continuously differentiable pseudoinverse matrix, namely Φ1(t).
Therefore, to prove the first assertion of Lemma 2, it remains to prove that the mapping
Φ+
1 (t) : Rn →Mn(t) is a homeomorphism.
Since ker Φ+
1 (t) = 0, we conclude that Φ+
1 (t) is a homeomorphism of Rn into the
image of Φ+
1 (t;Rn) under the mapping Φ+
1 (t) : Rn → Rm. It remains to prove that
Φ+
1 (t;Rn) = Mn(t).
Assume that this is not true. Then either there exists a point y ∈ Φ+
1 (t;Rn) such that
y /∈Mn(t) or there exists a point y ∈ Rm such that y /∈ Φ+
1 (t;Rn).
In the first case, y is the image of a certain point from Rn, namely the point
x = Φ1(t)y, according to the equation y = Φ+
1 (t)x. In this case, we have y =
= Φ+
1 (t)Φ1(t)y, y = M1(t)y, y ∈Mn(t), which contradicts the assumption.
In the second case, we have y = M1(t)y and, according to (1.5), y = M1(t)c(t)
for a certain c(t) ∈ Rm. Thus, y = M1(t)c(t) = Φ+
1 (t)Φ1(t)c(t) = Φ+
1 (t)x, where
x = Φ1(t)c(t), i.e., y is the image of the point x under the mapping Φ+
1 (t) : Rn → Rm,
whence y ∈ Φ+
1 (t;Rn). This contradicts the assumption.
2. Main theorem. According to the results presented above, every continuously
differentiable nonsingular matrix
(
Φ1(t)
Φ2(t)
)
defines, in the (t, y)-space R × Rm, m =
= dim
(
Φ1(t)
Φ2(t)
)
, the two subspaces
ker Φ1(t) = Mn
1 (t), n = dim Φ1(t),
ker Φ2(t) = Mm−n
1 (t), p = dim Φ2(t) = m− n,
and two diffeomorphisms
Φ+
1 (t) : Rn →Mn
1 (t),
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
240 A. M. SAMOILENKO
Φ+
2 (t) : Rp →Mm−n
1 (t).
Using a matrixQ(t) ∈Mm(R) continuous for all t ∈ R, we introduce, in the (t, y)-space
R × Rm, a linear vector field (t, y′ = Q(t)y) the integral curves of which are defined
by the solutions y = y(t) of the differential equation
dy
dt
= Q(t)y. (2.1)
If the union of the subspaces Mn
1 (t) and Mm−n
1 (t) (or one of these subspaces) is
the union of integral curves of the vector field (t, y′), then these subspaces are called
invariant manifolds of the differential equation (2.1) or the vector field (t, y′). If the
subspace Mk
1 (t), k ∈ {n, p}, is an invariant manifold of Eq. (2.1), then “the motion of
its points y in the (t, y)-space is independent of the motion of the points y outside the
subspace Mn(t) for both t > 0 and t < 0”.
We pose the problem as follows: Find conditions under which the subspace Mk
1 (t)
is an invariant manifold of Eq. (2.1). An equivalent statement of this problem is the
following: Find conditions under which the solutions y = y(t) of Eq. (2.1) satisfy one
of the additional conditions
y = M1(t)y
and
M1(t)y = 0
for any t ∈ R.
Finally, according to the terminology of Krylov – Bogolyubov nonlinear mechanics
[3, 4], the invariant manifold Mk(t) of Eq. (2.1) is an integral manifold of Eq. (2.1) if,
for any solution y = y(t) of Eq. (2.1), the fact that the inclusion
y(t) ∈Mk(t),
holds for a certain t = t0 implies that this inclusion is true for any t ∈ R.
Therefore, the posed problem is equivalent to the problem of finding conditions under
which the subspace Mk(t) is an integral manifold of Eq. (2.1).
Theorem 1. Suppose that Q(t) ∈ Mm(R), Φ(t) ∈ Mmn(R), m > n, Φ+(t) ∈
∈ Mmn(R), and rank Φ(t) = n. Let Q(t) be a continuous function and let Φ(t) and
Φ+(t) be continuously differentiable functions for all t ∈ R. Also assume that Φ+(t) is
a matrix pseudoinverse to the matrix Φ(t) and
Mn(t) = {y ∈ Rm : y = M(t)y},
Mm−n(t) = {y ∈ Rm : M(t)y = 0},
M(t) = Φ+(t)Φ(t),
L(M,Q) =
dM
dt
+MQ−QM.
Then the following assertions are true:
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SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 241
1. The subspaces Mn(t) and Mm−n(t), taken together, are invariant manifolds of
the differential equation
dy
dt
= Q(t)y (I)
if and only if
L(M(t), Q(t)) = 0.
2. The subspace Mn(t) is an invariant manifold of the differential equation (I) if
and only if
L(M(t), Q(t))M(t) = 0
for any t ∈ R. Moreover, if Mn(t) is an invariant manifold of Eq. (I), then, on Mn(t)
defined by the diffeomorphism Φ+(t),
y = Φ+(t)x, x ∈ Rn,
Eq. (I) is equivalent to the equation
dx
dt
= P (t)x (II)
with the coefficient matrix
P (t) =
(
dΦ(t)
dt
+ Φ(t)Q(t)
)
Φ+(t),
i.e., the fundamental matrices of solutions of Eqs. (I) and (II) Y (t) and X(t), Y (0) = E,
X(0) = E ∈Mn(R), satisfy the relations
Y (t)Φ+(0) = Φ+(t)X(t),
X(t) = Φ(t)Y (t)Φ+(0)
for any t ∈ R.
3. If Mm−n(t) is an invariant manifold of Eq. (I), then
kerL(M(t), Q(t)) ⊃Mm−n(t)
for any t ∈ R.
We now pass to the proof of the theorem. Let
L(M(t), Q(t)) = 0 ∀t ∈ R.
Consider the function
r = (E −M(t))y(t), (2.2)
where y = y(t) is a solution of Eq. (2.1) corresponding to the initial conditions
y(t0) = M(t0)c (2.3)
and c is an arbitrary point of the space Rm. According to definition (2.2), the function
r is equal to zero for t = t0 :
r = r0 = (E −M(t0))y(t0) = (E −M(t0))M(t0)c = 0. (2.4)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
242 A. M. SAMOILENKO
Differentiating function (2.2), we obtain
dr
dt
= −dM(t)
dt
+ (E −M(t))Q(t)y(t) =
= Q(t)y(t)−
(
dM(t)
dt
+M(t)Q(t)−Q(t)M(t)
)
y(t)−Q(t)M(t)y(t) =
= −L(M(t), Q(t)y(t) +Q(t)(y(t)−M(t)y(t)) = Q(t)r. (2.5)
According to (2.4), it follows from (2.5) that r(t) = 0. Therefore,
y(t) = M(t)y(t) ∀t ∈ R. (2.6)
On the one hand, we have
rankM(t) = rank(Φ+(t)Φ(t)) ≤ min(rank Φ+(t), rank Φ(t)) = n,
while, on the other hand,
n = rank(Φ(t)Φ+(t)Φ(t)Φ+(t)) = rank(Φ(t)M(t)Φ+(t)) ≤ rankM(t).
Therefore, rankM(t) = n for any t ∈ R. Then the subspace of Rm defined by points
(2.2) is n-dimensional. Since points (2.3) belong to the subspace Mn(t0), the subspace
Mn(t0) coincides with the subspace defined by Eq. (2.3). In this case, equality (2.6)
means that
y(t) ∈Mn(t) ∀t ∈ R
for any solution of Eq. (2.1) with initial value y(t0) = M(t0)c for an arbitrary c ∈ Rm.
Thus, the subspace Mn(t) is an invariant manifold of Eq. (2.1).
We now find the solution y = y(t) of Eq. (2.1) with the initial conditions
y(t0) = (E −M(t0))c, (2.7)
where c is an arbitrary point of Rm, and consider the function
r1 = M(t)y(t).
Differentiating this function, we get
dr1
dt
=
dM(t)
dt
y(t) +M(t)Q(t)y(t) =
=
(
dM(t)
dt
+M(t)Q(t)−Q(t)M(t)
)
y(t) +Q(t)M(t)y(t) =
= L(M(t), Q(t))y(t) +Q(t)r = Q(t)r. (2.8)
By definition, the function r1 is equal to zero at the point t = t0 :
r1 = r01 = M(t0)y(t0) = M(t0)(E −M(t0))c = 0. (2.9)
Therefore, it follows from (2.8) and (2.9) that r1(t) = 0 for any t ∈ R. Thus,
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SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 243
M(t)y(t) = 0 ∀t ∈ R,
which completes the proof of the inclusion
y(t) ∈Mm−n(t) (2.10)
for any t ∈ R.
Consider rank(E−M(t0)) = m−rankM(t0) = m−n. Thus, the subspace formed
by points (2.7) is (m− n)-dimensional and coincides with the subspace Mm−n(t0). In
this case, inclusion (2.10) means that the subspace Mm−n(t) is an invariant manifold of
Eq. (2.1).
We have proved that the condition L(M(t), Q(t)) = 0 is sufficient for the subspaces
Mn(t) and Mm−n(t), taken together, to be invariant manifolds of Eq. (2.1).
Let the subspace Mn(t) be an invariant manifold of Eq. (2.1). Consider the solutions
of Eq. (2.1)
y = Y (t)Φ+(0)c,
where c is an arbitrary point of Rm. The relation
y(0) = Y (0)Φ+(0)c = Φ+(0)c ∈Mn(0)
yields the inclusion
Y (t)Φ+(0)c ∈Mn ∀t ∈ R.
This means that
Y (t)Φ+(0)c = M(t)Y (t)Φ+(0)c (2.11)
for any c ∈ Rm and, hence, for unit vectors of the space Rm. It follows from (2.11) that
Y (t)Φ+(0) = M(t)Y (t)Φ+(0) (2.12)
for any t ∈ R.
Let Xt denote the matrix
Xt = Φ(t)Y (t)Φ+(0). (2.13)
We rewrite (2.12) in the form of the relation
Y (t)Φ+(0) = Φ+(t)Xt, (2.14)
which is true for any t ∈ R. Differentiating (2.14) with regard for (2.12) and (2.13), we
obtain
Q(t)Y (t)Φ+(0) =
dΦ+(t)
dt
Xt + Φ+(t)
dXt
dt
, (2.15)
Q(t)Φ+(t)Xt =
dΦ+(t)
dt
Xt + Φ+(t)
(
dΦ(t)
dt
XtΦ
+(0) + Φ(t)Q(t)Y (t)Φ+(0)
)
=
=
dΦ+(t)
dt
Xt + Φ+(t)
(
dΦ(t)
dt
+ Φ(t)Q(t)
)
Φ+(t)Xt. (2.16)
Subtracting (2.15) from (2.16), we get
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
244 A. M. SAMOILENKO
Φ+(t)
dXt
dt
= Φ+(t)
(
dΦ(t)
dt
+ Φ(t)Q(t)
)
Φ+(t)Xt.
This proves that
Φ+(t)
[
dXt
dt
−
(
dΦ(t)
dt
+ Φ(t)Q(t)
)
Φ+(t)Xt
]
= 0 (2.17)
for any t ∈ R. Since ker Φ+(t) = 0, equality (2.17) is possible only if
dXt
dt
= P (t)Xt, (2.18)
where
P (t) =
(
dΦ(t)
dt
+ Φ(t)Q(t)
)
Φ+(t). (2.19)
By definition, we have
X0 = Φ(0)Y (0)Φ+(0) = E, E ∈Mn.
Therefore,
Xt = X(t) ∀t ∈ R.
Thus, if Mn(t) is an invariant manifold of Eq. (2.1), then (2.14) takes the form
Y (t)Φ+(0) = Φ+(t)X(t), (2.20)
where X(t), X(0) = E, is the fundamental matrix of solutions of Eq. (2.18).
Multiplying (2.20) from the left by Φ(t), we get
X(t) = Φ(t)Y (t)Φ+(0).
Differentiating (2.12), we obtain
QY (t)Φ+(0) =
(
dM(t)
dt
+M(t)Q(t)−Q(t)M(t)
)
Y (t)Φ+(0) +QM(t)Y (t)Φ+(0).
(2.21)
In view of (2.12), relation (2.21) yields
L(M(t), Q(t))Y (t)Φ+(0) = 0. (2.22)
Using (2.22) and (2.14), we get
L(M(t), Q(t))Φ+(t)X(t) = 0.
Multiplying this equality from the right by X−1(t), we obtain the final result
L(M(t), Q(t))Φ+(t) = 0
for any t ∈ R.
Thus, the fact that the subspace Mn(t) is integral implies that all conditions of the
theorem related to this case are satisfied; to this end, it suffices to rewrite equality (2.20)
as an equation of the subspace Mn(t) in the parametric form:
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SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 245
y = Φ+(t)x, ∈ Rn, t ∈ R.
Assume that the condition
L
(
M(t), Q(t)
)
Φ+(t) = 0 ∀t ∈ R, (2.23)
is satisfied. Multiplying (2.23) from the right by the matrix Φ(t)X(t), where X(t) is the
fundamental matrix of solutions of Eq. (2.18) with coefficient matrix (2.19), X(0) = E,
we obtain
L
(
M(t), Q(t)
)
M(t)X(t) = 0 ∀t ∈ R.
Now consider the function
r = y(t)− Φ+(t)X(t)c, (2.24)
where y = y(t) is the solution of Eq. (2.23) such that
y(t0) = Φ+(t0)X(t0)c (2.25)
and X(t), X(0) = E, is the fundamental matrix of solutions of the equation
dx
dt
= P (t)x
with coefficient matrix (2.19). Differentiating function (2.24), we obtain
dr
dt
= Q(t)y(t)−
(
dΦ+(t)
dt
X(t)c+ Φ+(t)P (t)X(t)c
)
=
= Q(t)
[
(y(t)− Φ+(t)X(t)c)
]
+
+ Q(t)Φ+(t)X(t)c−
(
dΦ+(t)
dt
X(t)c+ Φ+(t)P (t)X(t)c
)
=
= Q(t)r −
(
dΦ+(t)
dt
+ Φ+(t)P (t)−Q(t)Φ+(t)
)
X(t)c . (2.26)
Let us prove that the second term in (2.26) is 0 ∈ Mmn. Indeed, taking (2.19) into
account, we get
dΦ+(t)
dt
+ Φ+(t)
(
dΦ(t)
dt
+ Φ(t)Q(t)
)
Φ+(t)−Q(t)Φ+(t) =
=
dΦ+(t)
dt
+ Φ+(t)
dΦ(t)
dt
Φ+(t) +M(t)Q(t)Φ+(t)−Q(t)Φ+(t) =
=
(
dΦ+(t)
dt
Φ(t) + Φ+(t)
dΦ(t)
dt
)
Φ+(t),
M(t)Q(t)Φ+(t)−Q(t)M(t)Φ+(t) +Q(t)M(t)Φ+(t)−Q(t)Φ+(t) =
=
(
dM(t)
dt
+M(t)Q(t)−Q(t)M(t)
)
Φ+(t) +
+ Q(t)Φ+(t)Φ(t)Φ+(t)−Q(t)Φ+(t) = 0.
(2.27)
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246 A. M. SAMOILENKO
With regard for (2.27), equality (2.26) takes the form
dr
dt
= Q(t)r. (2.28)
For t = t0, according to (2.25), function (2.24) is equal to zero:
r(t0) = y(t0)− Φ+(t0)X(t0)c = 0.
Therefore, it follows from (2.28) that
r(t) = 0 ∀t ∈ R,
and, hence,
y(t) = Φ+(t)X(t)c ∀t ∈ R, (2.29)
where c is an arbitrary point of the space Rn.
Since (2.29) is the parametric representation of the equation of the subspace Mn(t),
it follows from (2.29) that the condition that
y(t) ∈Mn(t) (2.30)
for t = t0 implies that inclusion (2.30) holds for any t ∈ R. This proves that condition (2.23)
is not only necessary but also sufficient for the subspace Mn(t) to be an invariant
manifold of Eq. (2.1).
This completes the proof of the second assertion of Theorem 1.
Let the subspace Mm−n(t) be an invariant manifold of Eq. (2.1). Consider the
solutions y = y(t) of Eq. (2.1) defined by the relation
y(t) = Y (t)(E −M(0))c,
where c is an arbitrary point of Rm.
It follows from the relation
(0)y(0) = M(0)Y (0)(E −M(0))c = M(0)(E −M(0))c = 0
that y(0) ∈Mm−n(0) and, hence, y(t) ∈Mm−n(t) ∀t ∈ R. This proves that
M(t)Y (t)(E −M(0))c = 0 ∀t ∈ R.
Differentiating this equality, we get(
dM(t)
dt
+M(t)Q(t)−Q(t)M(t)
)
Y (t)(E −M(0))c +
+ Q(t)M(t)Y (t)(E −M(0))c = L(M(t), Q(t))Y (t)(E −M(0))c = 0. (2.31)
The points y = (E −M(0))c define the subspace Mm−n(0). Therefore, the equation
y = Y (t)(E −M(0))c, c ∈ Rm, t ∈ R,
defines the subspace Mm−n(t) in the parametric form. Therefore, equality (2.31) means
that
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SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 247
kerL(M(t), Q(t)) ⊃Mm−n(t) ∀t ∈ R. (2.32)
Thus, inclusion (2.32) is a necessary condition for the subspace Mm−n(t) to be an
invariant manifold of Eq. (2.1).
Assume that the subspaces Mn(t) and Mm−n(t), taken together, are invariant
manifolds of Eq. (2.1). Then equality (2.22) yields
ker L(M(t), Q(t)) ⊃ Y (t)M(0)c ∀t ∈ R, (2.33)
where c is an arbitrary point of Rm. Moreover, since the equation
y = Y (t)M(0)c, c ∈ Rm, t ∈ R,
defines the subspace Mn(t) in the parametric form, it follows from (2.33) that
kerL(M(t), Q(t)) ⊃Mn(t) ∀t ∈ R.
Thus, if the subspaces Mn(t) and Mm−n(t), taken together, are invariant manifolds of
Eq. (2.1), then
kerL
(
M(t), Q(t)
)
⊃ (Mn(t) ∪Mm−n(t)). (2.34)
Since rankMn(t) = n, rankMm−n(t) = m − n, and Mn(t) ∩Mm−n(t) = {0}, we
conclude that the union on the right-hand side of expression (2.34) contains a basis of
the space Rm. Therefore, relation (2.34) yields
kerL(M(t), Q(t)) ⊃ Rm ∀t ∈ R. (2.35)
Since kerL(M(t), Q(t)) ∈Mm(R) ∀t ∈ R, inclusion (2.35) is possible only if
kerL(M(t), Q(t)) = 0. (2.36)
Thus, condition (2.36) is not only sufficient but also necessary for the subspaces Mn(t)
and Mm−n(t), taken together, to be invariant manifolds of Eq. (2.1).
3. Equivalence of linear differential equations of different orders. We prove the
following theorem.
Theorem 2. If the subspaces Mn(t) and Mm−n(t), taken together, are invariant
manifolds of the differential equation
dy
dt
= Q(t)y, (I)
then the change of variables
y = Φ+
1 (t)x+ Φ+
2 (t)z (II)
reduces this equation to the system of differential equations
dx
dt
= P (t)x,
dz
dt
= G(t)z (III)
with coefficient matrices
P (t) =
(
dΦ1(t)
dt
+ Φ1(t)Q(t)
)
Φ+
1 (t), (IV)
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248 A. M. SAMOILENKO
G(t) =
(
dΦ2(t)
dt
+ Φ2(t)Q(t)
)
Φ+
2 (t), (V)
and vice versa, if the differential equation (I) can be reduced by the change of variables (II)
to the system of differential equations (III), then the subspaces Mn(t) and Mm−n(t),
taken together, are invariant manifolds of Eq. (I), and the coefficient matrices of
system (III) are defined by relations (IV) and (V).
Indeed, let the subspacesMn(t) andMm−n(t), taken together, be invariant manifolds
of Eq. (I). Then, according to assertion 2 of Theorem 1, Eq. (I) is equivalent on Mn(t)
and Mm−n(t) to the corresponding first and second equations of system (III) with the
coefficient matrices defined by relations (IV) and (V), respectively. Let Y (t), X(t), and
Z(t), where Y (0) = E, X(0) = E, and Z(0) = E, be the fundamental matrices of
solutions of Eqs. (I) and (III) and let E be the identity matrices of the corresponding
orders. According to assertion 2 of Theorem 1, we have
Y (t)Φ+
1 (0) = Φ+
1 (t)X(t), Y (t)Φ+
2 (0) = Φ+
2 (t)Z(t) (3.1)
for all t ∈ R. Thus, according to (3.1),
Y (t)(Φ+
1 (0),Φ+
2 (0)) = (Φ+
1 (t),Φ+
2 (t))
(
X(t) 0
0 Z(t)
)
(3.2)
for all t ∈ R. The equality
Y (t) = Φ+
1 (t)X(t)Φ1(0) + Φ+
2 (t)Z(t)Φ2(0)
for all t ∈ R follows from (3.2). Thus, for an arbitrary y0 ∈ Rm, we have
Y (t)y0 = Φ+
1 (t)X(t)x0 + Φ+
2 (t)Z(t)z0 (3.3)
for all t ∈ R and x0 and z0 chosen according to the condition
x0 = Φ1(0)y0, z0 = Φ2(0)y0.
Equality (3.3) means that the change of variables (II) reduces the differential equation (I)
to the system of differential equations (III).
Now assume that the differential equation (I) can be reduced to the system of
differential equations (III) by the change of variables (II). Taking into account that the
subspaces z = 0 and x = 0 are invariant manifolds of system (III) and using (II), we
obtain relations (3.1), which yield
X(t) = Φ1(t)Y (t)Φ+
1 (0), Z(t) = Φ2(t)Y (t)Φ+
2 (0) (3.4)
for all t ∈ R.
Substituting (3.4) into relations (3.1), we obtain
Y (t)Φ+
1 (0) = M1(t)Y (t)Φ+
1 (0), Y (t)Φ+
2 (0) = M2(t)Y (t)Φ+
2 (0) (3.5)
for all t ∈ R.
It follows from the first relation in (3.5) that
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SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 249
y(t) = M1(t)y(t) (3.6)
for any solution y(t) of Eq. (I) that satisfies the condition
y(0) = Φ+
1 (0)c, (3.7)
where c is an arbitrary constant from Rn. Since points (3.7) fill the subspace Mn(0), we
conclude that, according to (3.6), the integral curves (t, y(t)) of Eq. (I) that pass through
points of the subspace Mn(0) for t = 0 belong to the subspace Mn(t) for any t ∈ R.
This is sufficient for the subspace Mn(t) to be an invariant manifold of Eq. (I).
It follows from the second relation in (3.5) that
y(t) = M2(t)y(t)
for any solution y(t) of Eq. (I) that satisfies the condition
y(0) = Φ+
2 (0)c,
where c is an arbitrary constant from Rm−n.
By analogy, we prove that the subspace
Mm−n
2 (t) =
{
y ∈ Rm−n : y = M2(t)y
}
is an invariant manifold of Eq. (I).
According to Lemma 1, the equality
Mm−n
2 (t) = Mm−n(t)
holds for any t ∈ R. This proves that the subspace Mm−n(t) is an invariant manifold of
the differential equation (I). Thus, the subspaces Mn(t) and Mm−n(t), taken together,
are invariant manifolds of Eq. (I). According to assertion 2 of Theorem 1, this is sufficient
for relations (IV) and (V) to be true.
Let F (t) ∈ Mpn(R), n > p, F+(t) ∈ Mpn(R), rankF (t) = p, and let F (t) and
F+(t) be continuously differentiable functions for all t ∈ R. Also assume that F+(t)
is a matrix pseudoinverse to the matrix F (t) and K(t) = F+(t)F (t). Finally, let the
subspace
Kp(t) =
{
x ∈ Rn : x = K(t)x
}
be an invariant manifold of the differential equation
dx
dt
= P (t)x, (VI)
which is equivalent on Kp(t) to the differential equation
dz
dt
= R(t)z. (VII)
The system of differential equations (III) is called a decomposition of the differential
equation (I) if the change of variables (II) reduces this equation to the system of
differential equations (III).
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250 A. M. SAMOILENKO
The differential equation (VII) is called a restriction of the differential equation (VI)
to the subspace Kp(t) if the subspace Kp(t) is an invariant manifold of Eq. (VI), and
this equation is equivalent to Eq. (VII) on Kp(t).
We say that the differential equations (I) and (VI) are equivalent if Eq. (VI), together
with its restriction to Km−n(t) (VII), is a decomposition of Eq. (I).
By definition, the fundamental matrices of solutions of equivalent differential equations
are expressed in terms of one another via the matrices that define the invariant subspaces
of these differential equations. Indeed, using the definitions presented above and taking
into account that
G(t) = R(t)
for all t ∈ R, we conclude that relation (3.1) and the relation
X(t)F+(0) = F+(t)Z(t) (3.8)
for the fundamental matrices of the solutions Y (t), X(t), and Z(t) of the differential
equations (I), (VI), and (VII) are true.
It follows from (3.1) and (3.8) that
Y (t) = (Φ+
1 (t)X(t) + Φ+
2 (t)F (t)X(t)F+(0))
(
Φ1(0)
Φ2(0)
)
=
= Φ+
1 (t)X(t)Φ1(0) + Φ+
2 (t)F (t)X(t)F+(0)Φ2(0), (3.9)
X(t) = Φ1(t)Y (t)Φ+
1 (0) (3.10)
for all t ∈ R. Relations (3.9) and (3.10) describe the relationship between the fundamental
matrices of solutions of the equivalent differential equations (I) and (VI).
The notion of equivalence of differential equations of orders m and n defined above
for
m > n > m− n
can easily be generalized to the case
m = 2n. (3.11)
Indeed, since the space Rn is an invariant manifold of the differential equation (VI), and
Eq. (VI) is equivalent on it to the differential equation (VII) with the same coefficient
matrix, we conclude that, in case (3.11), the equivalence of the differential equations (I)
and (VI) is determined by the decomposition of Eq. (I) into the system of equations
dx
dt
= P (t)x,
dz
dt
= P (t)z.
The results presented above yield the following statement:
Corollary. The differential equations (I) and (VI) are equivalent if and only if
L(M(t), Q(t)) = 0, L(K(t), P (t))K(t) = 0, (VIII)
P (t) =
(
dΦ1(t)
dt
+ Φ1(t)Q(t)
)
Φ+
1 (t), (IX)
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SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 251(
dΦ2(t)
dt
+ Φ2(t)Q(t)
)
Φ+
2 (t) =
(
dF (t)
dt
+ F (t)P (t)
)
F+(t) (X)
for all t ∈ R.
Indeed, assume that the differential equations (I) and (VI) are equivalent. Then we
have the decomposition of Eq. (I) into the system of equations (III) the second equation
of which is the restriction of the differential equation (VI) to Km−n(t). It follows from
the definition of decomposition and Theorem 2 that the subspaces Mn(t) and Mm−n(t)
are invariant manifolds of the differential equation (I). It follows from the definition of
the restriction of the differential equation (VI) to the subspace Km−n(t) that Km−n(t)
is an invariant manifold of this equation. According to assertions 1 and 2 of Theorem
1, this is sufficient for relations (VIII) and (IX) to be true. Moreover, this is sufficient
for the coefficient matrices of the differential equations (I), (III), and (VII) to satisfy the
relations
G(t) =
(
dΦ2(t)
dt
+ Φ2(t)Q(t)
)
Φ+
2 (t), (3.12)
R(t) =
(
dF (t)
dt
+ F (t)P (t)
)
F+(t), (3.13)
and
G(t) = R(t) (3.14)
for all t ∈ R.
The last relation proves equality (X).
Let relations (VIII) – (X) be true. Then, according to assertions 1 and 2 of Theorem 1,
the subspacesMn(t) andMm−n(t) are invariant manifolds of the differential equation (I),
and the subspace Km−n(t) is an invariant manifold of the differential equation (VI);
furthermore, the coefficient matrices of the corresponding differential equationsG(t) and
R(t) are defined by relations (3.12) and (3.13), and, hence, according to condition (X),
they satisfy equality (3.14). According to Theorem 2, this implies that the system
of differential equations (III) the second equation of which is the restriction of the
differential equation (VI) to the subspace Km−n(t) is a decomposition of the differential
equation (I). This proves that relations (VIII) – (X) yield the equivalence of the differential
equations (I) and (VI).
Note that, for m = 2n, conditions (VIII) – (X) are simplified because, in this case,
F (t) and K(t) are the identity matrices. In this case, these conditions take the form
L(M(t), Q(t)) = 0,
P (t) =
(
dΦ1(t)
dt
+ Φ1(t)Q(t)
)
Φ+
1 (t) =
(
dΦ2(t)
dt
+ Φ2(t)Q(t)
)
Φ+
2 (t)
for any t ∈ R.
Also note that the equivalence of the differential equations (I) and (VI) means that
the relations
Y (t)Φ+
1 (0) = Φ+
1 (t)X(t), Y (t)Φ+
2 (0) = Φ+
2 (t)F (t)X(t)F+(0) (3.15)
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252 A. M. SAMOILENKO
for the fundamental matrices of solutions of Eqs. (I) and (VI) Y (t) and X(t), as well as
the other relations that can be obtained from (3.15) by the corresponding transformations,
are true.
4. Addition to the Floquet – Lyapunov theory. Consider the linear differential
equation
dx
dt
= P (t)x, (I)
where x ∈ Rn, P (t) ∈Mn(R), and P (t) is a continuous periodic matrix with period T.
According to the well-known Floquet theorem [ 5 ], the fundamental matrix of
solutions of Eq. (I) X(t), X(0) = E, can be represented in the form
X(t) = Φ(t)eHt, (II)
where Φ(t) is a matrix periodic in t with period T, and H is the constant matrix defined
by the monodromy matrix X(T ) of Eq. (I) according to the formula
H =
1
T
lnX(T ). (III)
The logarithm is a multivalued function whose real value does not always exist. Thus,
relation (I) with matrix (III) such that
H ∈Mn(R) (IV)
is not always true. According to the theory of matrices [ 6 ], condition (IV) is satisfied
if and only if every elementary divisor corresponding to the negative eigenvalues of
the matrix X(T ) is repeated an even number of times. Thus, only in this case does
equality (II) hold with matrices Φ(t) and H from the space of real matrices Mn(R).
In the case where condition (IV) cannot be satisfied, the Floquet representation (II)
exists only with matrices Φ(t) and H from the space Mn(C), where C is the plane of
complex numbers, or this representation transforms into equality (II) with real matrices
Φ(t) and H, the first of which is periodic with period 2T and the second is defined by
the relation
H =
1
2T
lnX(2T ). (V)
The Floquet representation (II) with matrix (V) is a consequence of the presence of
negative eigenvalues of the monodromy matrix of Eq. (I).
We consider in detail the differential equation (I) whose monodromy matrix possesses
this property and prove several previously unknown statements for this equation.
Theorem 3. Suppose that the coefficient matrix of the differential equation (I)
P (t) belongs to Mn(R) for any t ∈ R and is continuous on R and periodic in t with
period T.
Then the following assertions are true:
1. The algebraic number p of negative eigenvalues of the monodromy matrix X(T )
of Eq. (I) is even.
2. Equality (II) holds for the matrix
H =
1
T
ln(X(T )I), (VI)
where I is the real matrix defined by the conditions
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SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 253
I2 = E, ln(X(T )I) ∈Mn(R),
and for the periodic matrix Φ(t) such that
Φ(t+ T )I1 = Φ(t)I1, Φ(t+ T )I2 = −Φ(t)I2 (VII)
for all t ∈ R, where
I1 =
E + I
2
, I2 =
E − I
2
.
3. There exists a nonsingular matrix (U(t), V (t)) continuously differentiable and
real for all t ∈ R, periodic with period T, and such that the change of variables
x = U(t)z1 + V (t)z2
reduces the differential equation (I) to the system of differential equations
dz1
dt
= H1z1,
dz2
dt
= G(t)z2, (VIII)
where H1 is a constant matrix, G(t) is a periodic matrix with period T, and the set
of eigenvalues of the monodromy matrix Z2(T ) of the second equation of the system is
either the set of all negative eigenvalues of the matrix X(T ) or its subset.
To prove the theorem, we use the representation of the matrix X(T ) in terms of its
Jordan form J(λ), namely
X(T ) = SJ(λ)S−1,
and obtain the equality
detX(T ) =
n∏
ν=1
λν , (4.1)
which associates the determinant of the matrix X(T ) with its eigenvalues λν , ν = 1, n.
We now use the Liouville – Ostrogradskii – Jacobi formula and represent the determinant
of the matrix X(T ) in terms of the trace of the coefficient matrix of Eq. (I):
detX(T ) = exp {trP (t)dt} . (4.2)
Equating the right-hand sides of relations (4.1) and (4.2), we obtain an equality that
proves that
n∏
ν=1
λν > 0. (4.3)
Since each pair of complex conjugate eigenvalues of the matrix X(T ) in the product
of all its eigenvalues gives a positive number, it follows from relation (4.3) that the
product of all negative eigenvalues of the matrix X(T ) also gives a positive number.
Thus, the algebraic number of negative eigenvalues of the matrix X(T ), i.e., the sum of
multiplicities of the roots of characteristic equations for all different negative eigenvalues
of the matrix X(T ), is an even number.
Prior to the proof of assertion 2 of Theorem 3, note that, in the case where the
logarithm of the matrix X(T ) is real, by setting I = E one can reduce equalities (II) and
(VI) to the Floquet relations (II) and (III) with a matrix Φ(T ) that possesses properties
that follow from these relations and are indicated in assertion 2 of Theorem 3.
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254 A. M. SAMOILENKO
It remains to consider the case where the matrix X(T ) has negative eigenvalues and
does not have a real logarithm. In this case, the real canonical form of the matrix X(T )
can be represented in the form of decomposition into two blocks A and B, where A
either is empty or has a real logarithm, and B has only negative eigenvalues and does
not have a real logarithm.
Let B ∈Md(R), where
n > d.
Then the following equality is true:
X(T ) = S
(
A 0
0 B
)
S−1 , (4.4)
where S, A, and B are real matrices with properties indicated above for A and B.
We set
Y (t) = S−1X(t)S, B1 = −B. (4.5)
According to properties of the fundamental matrix of solutions of Eq. (I), we have
X(t+ T ) = X(t)X(T ) . (4.6)
Therefore, it follows from (4.4), (4.5), and (4.6) that
Y (t+ kT ) = S−1X(t)Xk(T )S = S−1X(t)SS−1Xk(T )S =
= Y (t)
(
Ak 0
0 Bk
)
= Y (t)
(
Ak 0
0 (−1)kBk1
)
(4.7)
for any integer k.
We represent Y (t) in the block form
Y (t) = (Y1(t), Y2(t)) (4.8)
consistent with decomposition (4.4) of the matrix X(T ) into the blocks A and B. Using
relations (4.7), we get
Y1(t+ kT ) = Y1(t)Ak, Y2(t+ kT ) = (−1)kY2(t)Bk1 (4.9)
for any integer k.
Since the eigenvalues of the matrix B1 are positive by virtue of the definition (4.5)
of this matrix, both matrices A and B1 have real logarithms lnA and lnB1.
In view of the arguments presented above, relation (4.9) yields
Y1(t) = Y1
(
t−
[
t
T
]
T +
[
t
T
]
T
)
= Y1
(
t−
[
t
T
]
T
)
A[t/T ] =
= Y1
(
t−
[
t
T
]
T
)
exp
{([
t
T
]
T − t
)
lnA
T
}
exp
{
t
T
lnA
}
, (4.10)
Y2(t) = Y2
(
t−
[
t
T
]
T +
[
t
T
]
T
)
= Y2 (t− [t/T ]T ) (−1)[t/T ]B
[t/T ]
1 =
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SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 255
= (−1)[t/T ]Y2
(
t−
[
t
T
]
T
)
exp
{([
t
T
]
T − t
)
lnB1
T
}
exp
{
t
T
lnB1
}
(4.11)
for all t ∈ R; here, [ t ] denotes the integer part of the number t.
Let Φ1(t) and Φ2(t) denote the coefficients of exp
{
t
T
lnA
}
and exp
{
t
T
lnB1
}
in relations (4.10) and (4.11), respectively. Then, using (4.8), (4.10), and (4.11), we
obtain
Y (t) =
(
Φ1(t),Φ2(t)
)exp
{
t
T
lnA
}
0
0 exp
{
t
T
lnB1
}
(4.12)
for all t ∈ R. This equality implies that the matrices Φ1(t) and Φ2(t) are continuously
differentiable on R. Furthermore, it follows from the introduced notation that the matrix
Φ1(t) is periodic with period T, and the matrix Φ2(t), which is the product of the
function (−1)[t/T ] and a periodic matrix with period T, satisfies the condition
Φ2(t+ T ) = −Φ2(t)
for all t ∈ R.
Let I0 denote the matrix (
E1 0
0 −E2
)
,
where E1 and E2 are the identity matrices from Mn−d(R) and Md(R), respectively.
Then
Y (T )I0 =
(
A 0
0 B1
)
and relation (4.12) takes the form
Y (t) =
(
Φ1(t),Φ2(t)
)
exp
{
t
T
ln(Y (T )I0)
}
. (4.13)
Using (4.13) and the first equality in (4.5), we obtain
X(t) = S
(
Φ1(t),Φ2(t)
)
S−1 exp
{
t
T
S(ln(Y (T )I0))S−1
}
. (4.14)
Since
S
(
ln(Y (T )I0)
)
S−1 = ln(SY (T )S−1SI0S
−1) = ln(X(T )I),
where
I = SI0S
−1, (4.15)
relation (4.14) takes the form of the required representation (II) under the condition that
H =
1
T
ln(X(T )I),
Φ(t) = S(Φ1(t),Φ2(t))S−1.
Taking (4.15) into account, we get
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256 A. M. SAMOILENKO
I2 = E, I1 = S
(
E1 0
0 0
)
S−1, I2 = S
(
0 0
0 E2
)
S−1,
where E1 and E2 are the identity matrices of the corresponding orders.
Using the expressions for I1 and I2, we obtain
Φ(t)I1 = S(Φ1(t), 0)S−1, Φ(t)I2 = S(0,Φ2(t))S−1 (4.16)
for all t ∈ R. In view of properties of the matrices Φ1(t) and Φ2(t), relation (4.16)
yields
Φ(t+ T )I1 = Φ(t)I1, Φ(t+ T )I2 = −Φ(t)I2
for all t ∈ R, which completes the proof of assertion 2 of Theorem 3 in the case
considered.
Let d = n. In this case, we obtain the equality
X(T ) = SBS−1
instead of (4.4), the equality
Y (t+ kT ) = (−1)kY (t)Bk1
instead of (4.7), and the equality
Y (t) = Φ2(t) exp
{
t
T
lnB1
}
and condition
Φ2(t+ T ) = −Φ2(t)
for all t ∈ R instead of (4.12).
We set
I0 = −E.
Using the last two formulas, we obtain equality (II) of the form
Y (t) = Φ2(t) exp
{
t
T
ln(−Y (t))
}
,
where
H =
1
T
ln(−X(T )), H ∈MnR,
Φ(t) = SΦ2(t)S−1, Φ(t+ T ) = −Φ(t), Φ(t) ∈MnR,
for all t ∈ R, which completes the proof of assertion 2 of Theorem 3.
We now pass to the proof of assertion 3 of Theorem 3. In this assertion, we separate
two limiting cases, namely, the case where the matrix X(t) has a real logarithm and the
second case where all eigenvalues of the matrix X(t) are negative and their elementary
divisors are different.
In the first case, assertion 3 of Theorem 3 follows from the Floquet relations (II) and
(III), according to which the change of variables
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x = Φ(t)z
reduces the differential equation (I) to the differential equation
dz
dt
= Hz
and guarantees the properties of the matrices H and Φ(t) indicated in Theorem 3.
In the second case, assertion 3 of Theorem 3 is trivial: the change of variables
x = z
reduces the differential equation (I) to a differential equation with the same coefficient
matrix:
dz
dt
= P (t)z.
Associating these limiting cases with the representation of the matrix X(T ) via its
real canonical form (4.4), we establish that the first case corresponds to
X(T ) = SAS−1
and the second case corresponds to
X(T ) = SBS−1.
Thus, the only nonlimiting case in assertion 3 of Theorem 3 is the case where
A ∈Mn−d(R), B ∈Md(R), n > d > 1.
Assume that these conditions are satisfied. Then it follows from the proof of assertion 2
of Theorem 3 that the matrix Y (t) associated with the matrix X(t) by relation (4.5) has
the form (4.12). Denoting
U(t) = Φ1(t), V (t) = Φ2(t), H1 =
lnA
T
, H2 =
lnB1
T
,
we represent (4.12) in the form
Y (t) = (U(t), V (t))
(
eH1t 0
0 eH2t
)
. (4.17)
It follows from (4.17) that
Y (t)
(
E1
0
)
= (U(t), V (t))
(
eH1t
0
)
= U(t)eH1t , (4.18)
where E1 is the identity matrix of order n− d.
Differentiating equality (4.18) with regard for the first relation in (4.5), we get
S−1P (t)SY (t)
(
E1
0
)
= S−1P (t)SU(t)eH1t =
dU(t)
dt
eH1t + U(t)H1e
H1t.
Thus,
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258 A. M. SAMOILENKO
dU(t)
dt
+ U(t)H1 = S−1P (t)SU(t) (4.19)
for all t ∈ R.
The matrix Y (t) is the fundamental matrix of solutions of the differential equation
dy
dt
= S−1P (t)Sy. (4.20)
Let W (t) ∈ Mnd(R) for all t ∈ R and let this matrix be continuously differentiable
on R, periodic with period T, and such that
det
(
U(t),W (t)
)
6= 0
for all t ∈ R.
The existence of this matrix follows from the theorem on a quasiperiodic basis in
Rn presented in [ 7 ].
In the differential equation (4.20), we perform the change of variables according to
the formula
y = U(t)y1 +W (t)y2 . (4.21)
Using equality (4.19), we obtain the differential equation
U(t)
(
dy1
dt
−H1y1
)
+W (t)
dy2
dt
=
(
S−1P (t)SW (t)− dW (t)
dt
)
y2 .
Solving this equation with the use of the matrix(
L1(t)
L2(t)
)
(4.22)
that is inverse to the matrix (U(t),W (t)), we obtain the following system of differential
equations for
dy1
dt
and
dy2
dt
:
dy1
dt
= H1y1 + L1(t)
(
S−1P (t)SW (t)− dW (t)
dt
)
y2 , (4.23)
dy2
dt
= L2(t)
(
S−1P (t)SW (t)− dW (t)
dt
)
y2. (4.24)
Since the coefficient matrix of system (4.23), (4.24) has a block-triangular form, the
fundamental matrix of solutions of this system is the matrix(
eH1t Y1(t)
0 Y2(t)
)
(4.25)
the second column of which is formed by solutions of the system of differential
equations (4.23), (4.24) with given initial values y1 = Y1(0) and y2 = Y2(0) such
that
det Y2(0) 6= 0.
In view of (4.21), the matrix
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(U(t),W (t))
(
eH1t Y1(t)
0 Y2(t)
)
is a fundamental matrix of solutions of Eq. (4.20). Moreover, relation (4.17) also determi-
nes a fundamental matrix of solutions of Eq. (4.20). According to the theory of linear
differential equations, there exists the following relation between these two fundamental
matrices of solutions:
(
U(t), V (t)
)(eH1t 0
0 eH2t
)
C =
(
U(t),W (t)
)(eH1t Y1(t)
0 Y2(t)
)
(4.26)
for all t ∈ R, where C is a nonsingular constant matrix. Substituting t = 0 into (4.26),
we obtain the following algebraic equation for the determination of the matrix C:
(U(0), V (0))C = (U(0),W (0))
(
E1 Y1(0)
0 Y2(0)
)
. (4.27)
Multiplying (4.27) by the matrix
(
L1(0)
L2(0)
)
, we obtain
(
E1 L1(0)V (0)
0 L2(0)V (0)
)
C =
(
E1 Y1(0)
0 Y2(0)
)
.
This equality implies that, first,
det
(
L2(0), V (0)
)
6= 0
and, second, for
Y1(0) = L1(0)V (0), Y2(0) = L2(0)V (0), (4.28)
we have
C = E. (4.29)
Thus, determining the solutions
(
Y1(t)
Y2(t)
)
of the system of differential equations (4.23),
(4.24) with initial values (4.28), we obtain the following equality from (4.26) and (4.29):
(
U(t), V (t)
)(eH1t 0
0 eH2t
)
=
(
U(t),W (t)
)(eH1t Y1(t)
0 Y2(t)
)
(4.30)
for all t ∈ R.
Multiplying (4.30) by matrix (4.22), we get(
E1 L1(t)V (t)
0 L2(t)V (t)
)(
eH1t 0
0 eH2t
)
=
(
eH1t Y1(t)
0 Y2(t)
)
.
Thus,
Y1(t) = L1(t)V (t)eH2t, (4.31)
Y2(t) = L2(t)V (t)eH2t (4.32)
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260 A. M. SAMOILENKO
for all t ∈ R. Since the matrix Y2(t) is nonsingular, we can determine the value of eH2t
from (4.32). Substituting this value into (4.31), we establish that
Y1(t) = L1(t)V (t)(L2(t)V (t))−1Y2(t) (4.33)
for all t ∈ R.
We rewrite the system of differential equations (4.23), (4.24) in the form of the
system
dy1
dt
= H1y1 +R1(t)y2, (4.34)
dy2
dt
= G(t)y2, (4.35)
where
R1(t) = L1(t)
(
S−1P (t)SW (t)− dW (t)
dt
)
,
G(t) = L2(t)
(
S−1P (t)SW (t)− dW (t)
dt
)
.
Using the matrix
F (t) = L1(t)V (t)(L2(t)V (t))−1, (4.36)
we rewrite equality (4.33) in the form
Y1(t) = F (t)Y2(t). (4.37)
Differentiating equality (4.37) and taking into account that the matrix
(
Y1(t)
Y2(t)
)
is a block
of the fundamental matrix (4.25) of solutions of the system of differential equations
(4.23), (4.24) [and, hence, of system (4.34), (4.35)], we get
dF (t)
dt
+ F (t)G(t) = H1F (t) +R1(t) (4.38)
for all t ∈ R. Finally, performing the change of variables
y1 = z1 + F (t)z2, y2 = z2,
we obtain the system
dz1(t)
dt
+
dF (t)
dt
z2 + F (t)G(t)z2 = H1z1 +H1F (t)z2 +R1(t)z2,
dz2(t)
dt
= G(t)z2
instead of the system differential equations (4.34), (4.35). By virtue of (4.38), this system
takes the form
dz1(t)
dt
= H1z1,
dz2(t)
dt
= G(t)z2. (4.39)
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SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 261
Since the second equation of system (4.39) coincides (to within notation) with Eq. (4.35),
the matrix Y2(t) is a fundamental matrix of solutions of the second equation of sys-
tem (4.39). Then, according to relation (4.32), the matrix
L2(t)V (t)eH2t(L2(0)V (0))−1
is a fundamental matrix of solutions of the second equation of system (4.39) and is equal
to the identity matrix for t = 0. Thus, the matrix
(L2(T )V (T ))eH2T (L2(0)V (0))−1 (4.40)
is the monodromy matrix of the second equation of system (4.39).
By definition, the matrix L2(t) is periodic with period T, the matrix V (t) satisfies
the condition
V (t+ T ) = −V (t), (4.41)
and the matrix H2 has the form
H2 =
1
T
ln(−B).
Taking into account the properties of the matrices L2(t), V (t), and H2 presented above,
we conclude that matrix (4.40) has the form
(−L2(0)V (0))(−B)(L2(0)V (0))−1 = L2(0)V (0)B(L2(0)V (0))−1.
Thus, it follows from the results presented above and the definition of the matrix B that
the set of eigenvalues of matrix (4.40) is either the set of all negative eigenvalues of the
matrix X(T ) or its subset.
Consider the matrix F (t). The definition of this matrix [see (4.36)] and the fact that
the matrices L1(t) and L2(t) are periodic with period T and the matrix V (t) satisfies
condition (4.41) imply that
F (t+ T ) = (−L1(t)V (t))(−L2(t)V (t))−1 = F (t)
for all t ∈ R.
Thus, the matrix F (t) is periodic with period T.
To complete the proof of assertion 3 of Theorem 3, it remains to take into account
that the change of variables
x = Sy (4.42)
transforms the differential equation (I) into the differential equation (4.20). Therefore, the
superposition of changes (4.42), (4.21), and (4.20) transforms the differential equation (I)
into the system of differential equations (4.39), and both the change of variables and
the differential equations of system (4.39) themselves possess the properties indicated in
Theorem 3.
We now make several remarks on assertions 2 and 3 of Theorem 3.
The first remark deals with relation (VI), which defines the matrix H. It follows
from the proof of Theorem 3 that H is not always uniquely defined. This nonuniqueness
is caused by the condition of decomposition of the canonical form of the matrix X(T )
into blocks A and B according to which the matrix B can be either a block of the
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262 A. M. SAMOILENKO
Jordan form of the matrix X(T ) formed by all its Jordan cells corresponding to its
negative eigenvalues or a block of this form obtained from the block indicated above by
elimination of an arbitrary number of pairs of identical Jordan cells.
The second remark deals with the minimum possible value of the order of the second
differential equation of system (VIII). It follows from the proof of Theorem 3 that this
order is also related to the condition of decomposition of the real canonical form of
the matrix X(T ) into blocks A and B and is equal to the minimum possible order of
the matrix B of this decomposition. It follows from the first remark that the minimum
possible value of the order of the second equation of system (VIII) is equal to the
order of the matrix obtained from the Jordan form of the matrix X(T ) by elimination
of all Jordan cells corresponding to negative eigenvalues of the matrix X(T ) and the
maximum possible even number of identical Jordan cells of this matrix that correspond
to its negative eigenvalues.
Also note that, according to the proof of Theorem 3, the matrix B is the Jordan form
of the monodromy matrix Z2(T ) of the second equation of system (VIII), and, hence, the
fundamental matrix of solutions Z2(t) of this equation possesses all the corresponding
properties.
Finally, note that, by virtue of Theorem 2 and assertion 3 of Theorem 3, the
differential equation (I) has the invariant manifolds
Kn−d(t) =
{
x ∈ Rn : U(t)L1(t)x = x
}
,
Kd(t) =
{
x ∈ Rn : V (t)L2(t)x = x
}
periodic with period T ,
Kν(t+ T ) = Kν(t), ν ∈ {(n− d) ∨ d},
for all t ∈ R. Moreover, Eq. (I) is equivalent on Kn−d(t) to the first differential equation
of system (VIII) and on Kd(t) to the second differential equation of this system.
Corollary. The fundamental matrix of solutions of the differential equation (I) X(t)
admits the representation
X(t) = Φ(t)eHtΦ+(0), (IX)
where
H =
1
T
ln
(
X(T ) 0
0 Z(T )
)
, (X)
H ∈ Mm(R), Z(T ) is the monodromy matrix of the restriction of (I) to its periodic
invariant manifold Kd(t), Φ(t) is a periodic matrix with period T that satisfies the
equation
dΦ
dt
+ ΦH = P (t)Φ, (XI)
Φ(t) ∈ Mnm(R) for all t ∈ R, Φ+(0) is a matrix pseudoinverse to the matrix Φ(0),
and m = n+ d, n ≥ d ≥ 0.
Indeed, according to the last remark, the differential equation (I) has the periodic
invariant manifoldKd(t) on which Eq. (I) is equivalent to the second differential equation
of system (VIII). Consider the system of differential equations
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SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 263
dx
dt
= P (t)x,
dz
dt
= G(t)z, (4.43)
which is formed of Eq. (I) and the second equation of system (VIII). According to the
proof of assertion 3 of Theorem 3, the real canonical form of the monodromy matrix of
this system (
X(T ) 0
0 Z(T )
)
(4.44)
is the matrix
A 0 0
0 B 0
0 0 B
, (4.45)
where A and B are the blocks of decomposition of the real canonical form of the matrix
X(T ) such that the matrix A has a real logarithm. Since the matrix(
B 0
0 B
)
is formed by pairwise identical Jordan cells, it has a real logarithm. Thus, the logarithm
of matrix (4.45) can be chosen real. Therefore, we can choose the real logarithm of
matrix (4.44) and define the matrix H according to relation (X) so that it satisfies
the condition H ∈ Mn+d(R), where d is the order of the matrix B. Applying the
Floquet formula (II) to the fundamental matrix of solutions of the system of differential
equations (4.43) (
X(t) 0
0 Z(t)
)
,
we obtain (
X(t) 0
0 Z(t)
)
=
(
Φ1(t)
Φ2(t)
)
eHt, (4.46)
where H is matrix (X) from the space Mn+d(R), Φ1(t) and Φ2(t) are periodic matrices
with period T, and Φ1(t) ∈ Mn n+d(R) and Φ2(t) ∈ Md n+d(R) for all t ∈ R.
Differentiating equality (4.46), we obtain the following matrix differential equation for
the matrix Φ(t) =
(
Φ1(t)
Φ2(t)
)
:
dΦ
dt
+ ΦH =
(
P (t) 0
0 G(t)
)
Φ.
This equation implies that the matrix Φ1(t) satisfies the differential equation (XI). Finally,
multiplying equality (4.46) by the matrix Φ+
1 (0), which is pseudoinverse to the matrix
Φ1(0), we obtain the equality
X(t) = Φ1(t)eHtΦ+
1 (0),
which coincides (to within notation) with (IX).
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264 A. M. SAMOILENKO
5. Two applications of obtained results. 5.1. Let x ∈ Rn, let P (t) be a continuous
matrix periodic with period T, let P (t) ∈ Mn(R) for all t ∈ R, and let X(t, x) be a
function of variables t ∈ R and x ∈ Rn that takes values in Rn and is continuous for all
t ∈ R and x ∈ Rn.
Consider the differential equation
dx
dt
= P (t)x+X(t, x). (5.1)
Let
X(t, x) ≡ 0.
Then the differential equation (5.1) has a fundamental matrix of solutions X(t), which
can be represented in the form
X(t) = Φ(t)eHtΦ+(0), (5.2)
and, moreover, the properties of the matrices Φ(t) and H are determined in the corollary
in the last section.
To simplify the differential equation (5.1), we use relation (5.2). To this end, we
change the variables in (5.1) by introducing a variable y ∈ Rm instead of x ∈ Rn
according to the relation
x = Φ(t)y. (5.3)
Taking into account that the matrix Φ(t) is a solution of the differential equation (XI),
we obtain the following equality from (5.1) and (5.3):
Φ(t)
(
dy
dt
−Hy
)
= X(t,Φ(t)y).
We represent this equality in the form
dy
dt
−Hy = Φ+(t)X(t,Φ(t)y), (5.4)
where Φ+(t) is a matrix pseudoinverse to Φ(t) that has the same smoothness and period
as Φ(t). In particular, as Φ+(t), we can take the first block of the matrix
(
Φ+
1 (t), Φ+
2 (t)
)
,
which is inverse to the matrix
(
Φ1(t)
Φ2(t)
)
defined by relation (4.46) of the previous section.
Solving Eq. (5.4) with respect to
dy
dt
−Hy, we get
dy
dt
= Hy + Φ+(t)X(t,Φ(t)y). (5.5)
The selected linear part of Eq. (5.5) has a constant coefficient matrix, and the general
part preserves the properties of the corresponding part of the original equation (5.1).
5.2. Consider the differential equation
dx
dt
= X(x) +X1(t, x), (5.6)
where X(x) is a continuously differentiable function of x and X1(t, x) is a continuous
function of t and x that takes values in Rn for all t ∈ R and x ∈ Rn, n ≥ 2.
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SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 265
Assume that, under the condition
X1(t, x) ≡ 0, (5.7)
Eq. (5.6) has a T -periodic solution
x(t) = ξ(ωt), (5.8)
where ξ(ϕ) is a function periodic in ϕ with period 2π and ω =
2π
T
is the frequency of
the periodic solution.
The variational equation corresponding to the periodic solution (5.8) of the differential
equation (5.6) with condition (5.7) has the form
dδξ
dt
=
∂X(ξ(ωt))
∂x
δξ. (5.9)
This equation has the solution
δξ = ξ′(ωt),
where “ ′ ” stands for the derivative with respect to the variable ϕ.
Indeed, by definition, we have
ξ′(ϕ)ω = X(ξ(ϕ)). (5.10)
Thus,
ξ′′(ϕ)ω =
∂X(ξ(ϕ))
∂x
ξ′(ϕ) (5.11)
for all ϕ ∈ R. Substituting ωt for ϕ in (5.11), we obtain the identity
d
dt
ξ′(ωt) =
∂X(ξ(ωt))
∂x
ξ′(ωt),
which proves the required statement.
Let B(ϕ) be a continuously differentiable periodic matrix with period 2π, let B(ϕ) ∈
∈Mnn−1(R), and let
det
(
ξ′(ϕ), B(ϕ)
)
6= 0
for all ϕ ∈ R.
Using the change of variables
δξ = ξ′(ωt)c+B(ωt)g, (5.12)
we reduce the variational equation (5.9) to the differential equation
ξ′′(ωt)ωc+ ξ′(ωt)
dc
dt
+B′(ωt)ωg +B(ωt)
dg
dt
=
∂X(ξ(ωt))
∂x
(ξ′(ωt)c+B(ωt)g),
or, with regard for (5.11), to the equation
ξ′(ωt)
dc
dt
+B(ωt)
dg
dt
=
(
∂X(ξ(ωt))
∂x
B(ωt)−B ′(ωt)ω
)
g.
Solving this equation with respect to the derivatives
dc
dt
and
dg
dt
with the use of the
matrix
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266 A. M. SAMOILENKO(
(ξ′(ωt))+
B+(ωt)
)
, (5.13)
which is inverse to (ξ′(ωt), B(ωt)), we reduce (5.9) to the system of differential
equations
dc
dt
= (ξ′(ωt))+
(
∂X(ξ(ωt))
∂x
B(ωt)−B ′(ωt)ω
)
g,
(5.14)
dg
dt
= B+(ωt)
(
∂X(ξ(ωt))
∂x
B(ωt)−B ′(ωt)ω
)
g.
According to the change of variables (5.12), the monodromy matrix of the system
of differential equations (5.14) is similar to the monodromy matrix of the variational
equation (5.9). Thus, the eigenvalues of both monodromy matrices coincide.
It follows from system (5.14) that one of the eigenvalues of its monodromy matrix
is equal to 1, whereas the other eigenvalues are eigenvalues of the monodromy matrix
of the second differential equation of system (5.14). Thus, the same is true for the
eigenvalues of the monodromy matrix of the variational equation (5.9).
We denote the coefficient matrix of the second differential equation of system (5.14)
by Q(ωt), where Q(ϕ) is a periodic matrix with period 2π, and consider the differential
equation
dg
dt
= Q(ωt)g. (5.15)
By virtue of the corollary in the previous section, the fundamental matrix of solutions
of Eq. (5.15) G(t) admits the representation
G(t) = Φ(ωt)eHtΦ+(0), (5.16)
where
H =
1
T
ln
(
G(T ) 0
0 Z(T )
)
∈Mm(R), (5.17)
2(n − 1) ≥ m ≥ (n − 1), Z(t) is the fundamental matrix of the restriction of the
differential equation (5.15) to its periodic invariant manifold Km−(n−1)(t), Φ(ϕ) is a
periodic matrix with period 2π, Φ(ϕ) ∈ Mn−1m(R) for all ϕ ∈ R, Φ(ϕ) satisfies the
differential equation
dΦ
dϕ
ω + ΦH = Q(ϕ)Φ, (5.18)
and Φ+(0) is a matrix pseudoinverse to Φ(0).
We now use the results obtained above for the introduction of amplitude – phase
coordinates in the neighborhood of the closed curve
x = ξ(ϕ), ϕ ∈ R, (5.19)
and for the reduction of the differential equation (5.6) in the neighborhood of this curve
to a simpler amplitude – phase system of differential equations.
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SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 267
To this end, we change the variables in Eq. (5.6) according to the relation
x = ξ(ϕ) +B(ϕ)g, (5.20)
where B(ϕ) is the matrix defined above.
Using equality (5.10), we obtain the following differential equation instead of (5.6):
(ξ′(ϕ) +B′(ϕ)g)
(
dϕ
dt
− ω
)
+B(ϕ)
dg
dt
=
= X(ξ(ϕ) +B(ϕ)g)−X(ξ(ϕ))−B′(ϕ)ωg +X1(t, ξ(ϕ) +B(ϕ)g). (5.21)
We solve Eq. (5.21) with respect to
dϕ
dt
− ω and
dg
dt
by using the matrix
(
L1(ϕ, g)
L2(ϕ, g)
)
(5.22)
that is inverse to (ξ′(ϕ) +B′(ϕ)g,B(ϕ)).
Choosing a sufficiently small δ > 0, one can easily construct matrix (5.22) for all
ϕ ∈ R, ‖g‖ ≤ δ,
on the basis of matrix (5.13) by setting(
L1(ϕ, 0)
L2(ϕ, 0)
)
=
(
(ξ′(ϕ))+
B+(ϕ)
)
.
Using (5.21), we obtain the system of differential equations
dϕ
dt
= ω + L1(ϕ, g)
[
X(ξ(ϕ) +B(ϕ)g)−X(ξ(ϕ)) +X1(t, ξ(ϕ) +B(ϕ)g)
]
, (5.23)
dg
dt
= L2(ϕ, g)
[
X(ξ(ϕ) +B(ϕ)g)−X(ξ(ϕ)) +X1(t, ξ(ϕ) +B(ϕ)g)
]
. (5.24)
We rewrite the differential equation (5.24) in the form
dg
dt
= B+(ϕ)
∂X(ξ(ϕ))
∂x
B(ϕ)g +G(ϕ, g) + L2(ϕ, g)X1(t, ξ(ϕ) +B(ϕ)g), (5.25)
where G(ϕ, g) denotes the function
L2(ϕ, g)(X(ξ(ϕ) +B(ϕ)g)−X(ξ(ϕ))− ∂X(ξ(ϕ))
∂x
B(ϕ)g) +
+ (L2(ϕ, g)− L2(ϕ, 0))
∂X(ξ(ϕ))
∂x
B(ϕ)g,
which satisfies the conditions
G(ϕ, 0) = 0,
∂G(ϕ, 0)
∂g
= 0.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
268 A. M. SAMOILENKO
According to the definition of the matrix Q(ωt), the coefficient matrix of the selected
linear part of the differential equation (5.25) coincides with the matrix Q(ϕ). Thus,
Eq. (5.25) takes the form
dg
dt
= Q(ϕ)g +G(ϕ, g) + L2(ϕ, g)X1(t, ξ(ϕ) +B(ϕ)g). (5.26)
Let Φ(ϕ) and H be the matrices determined from representation (5.16) of a fundamental
matrix of solutions of the differential equation (5.15). With the use of these matrices, we
transform the system of differential equations (5.23), (5.24) by setting
g = Φ(ϕ)h. (5.27)
As a result, instead of (5.26), we obtain
Φ′(ϕ)h+ Φ(ϕ)
(
dh
dt
−Hh
)
+ Φ(ϕ)Hh = Q(ϕ)Φ(ϕ)h+G(ϕ,Φ(ϕ)h) +
+ L2(ϕ,Φ(ϕ)h)X1(t, ξ(ϕ) +B(ϕ)Φ(ϕ)h),
or, taking into account that Φ(ϕ) is a solution of the differential equation (5.18),
Φ(ϕ)
(
dh
dt
−Hh
)
= G(ϕ,Φ(ϕ)h)+L2(ϕ,Φ(ϕ)h)X1(t, ξ(ϕ)+B(ϕ)Φ(ϕ)h). (5.28)
Solving Eq. (5.28) with respect to
dh
dt
− Hh with the use of the matrix Φ+(ϕ) that is
pseudoinverse to Φ(ϕ), we obtain
dh
dt
= Hh+ Φ+(ϕ)
[
G(ϕ,Φ(ϕ)h) + L2(ϕ,Φ(ϕ)h)X1(t, ξ(ϕ) +B(ϕ)Φ(ϕ)h)
]
.
By virtue of the results presented above, the change of variables (5.27) reduces the
system of differential equations (5.23), (5.24) to the system
dϕ
dt
= ω + f(t, ϕ,Φ(ϕ)h), (5.29)
dh
dt
= Hh+ Φ+(ϕ)F (t, ϕ,Φ(ϕ)h), (5.30)
where H is a matrix of the form (5.17),
f(t, ϕ, g) = L1(ϕ, g)
[
X(ξ(ϕ) +B(ϕ)g)−X(ξ(ϕ)) +X1(t, ξ(ϕ) +B(ϕ)g)
]
,
and
F (t, ϕ, g) = L2(ϕ, g)
[
X(ξ(ϕ) +B(ϕ)g)−X(ξ(ϕ))− ∂X(ξ(ϕ))
∂x
B(ϕ)g
]
+
+ (L2(ϕ, g)− L2(ϕ, 0))
∂X(ξ(ϕ))
∂x
B(ϕ)g.
The system of differential equations (5.29), (5.30) is the required one.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
SOME PROBLEMS OF THE LINEAR THEORY OF SYSTEMS OF ORDINARY DIFFERENTIAL . . . 269
Thus, by using the superposition of changes (5.20) and (5.27), and, hence, the change
of variables
x = ξ(ϕ) +B(ϕ)Φ(ϕ)h,
the original differential equation (5.6) can be reduced in the neighborhood of the closed
curve (5.19) to the system of differential equations (5.29), (5.30), where the functions
f(t, ϕ, g) and F (t, ϕ, g) are continuous in the variables t, ϕ, and g for t ∈ R, ϕ ∈ R,
and g ∈ Rn−1, ‖g‖ ≤ δ, take values in R and Rn−1, respectively, and are periodic in ϕ
with period 2π, the matrix Φ(ϕ) belongs to Mn−1m(R) for all ϕ ∈ R and is periodic
with period 2π, the matrix H belongs to Mm(R), its eigenvalues are the numbers
1
T
lnλj , j = 1, n− 1,
and their p-fold repetitions, 1 ≥ pj ≥ 0,
∑n−1
j=1
pj = m − (n − 1), 2(n − 1) ≥ m ≥
≥ (n− 1), and 1 and λ1, . . . , λn−1 are the eigenvalues of the monodromy matrix of the
variational equation (5.9).
The reduction of the differential equations considered above to equations with the
constant matrix of coefficients in their separated linear part is essential for the subsequent
investigation of these equations. A confirmation of this statement can be found, e.g., in
[4, 8], where, however, the problem of this reduction was only partially solved.
1. Samoilenko А. М. On invariant manifolds of linear differential equations (in Ukrainian). – Kyiv, 2009.
– 10 p. – (Preprint / Nat. Acad. Sci. Ukraine. Inst. Math.; № 2009.7).
2. Samoilenko А. М. On invariant manifolds of linear differential equations. II (in Ukrainian). – Kyiv, 2010.
– 43 p. – (Preprint / Nat. Acad. Sci. Ukraine. Inst. Math.; № 2010.3).
3. Bogolyubov N. N. On some statistical methods in mathematical physics (in Russian). – Kiev: Acad. Sci.
Ukr.SSR, 1945. – (Collection of Scientific Works (in Russian) / N. N. Bogolyubov. – Moscow: Nauka,
2006. – Vol. 4. – 432 p.).
4. Bogolyubov N. N., Mitropol’skii Yu. A. Asymptotic methods in the theory of nonlinear oscillations (in
Russian). – Moscow: Fizmatgiz, 1963. – (Collection of Scientific Works (in Russian) / N. N. Bogolyubov.
– Moscow: Nauka, 2005. – Vol. 3. – 605 p.).
5. Floquet G. Sur les ’equations diff’erentielles lin’eaires ‘a coefficients p’eriodiques // Ann. ’Ecole Norm.
Super. – 1883. – № 12. – P. 47 – 88.
6. Gantmakher F. R. Theory of matrices (in Russian). – Moscow: Nauka, 1988. – 552 p.
7. Samoilenko А. М. Elements of the mathematical theory of multifrequency oscillations. Invariant tori (in
Russian). – Moscow: Nauka, 1987. – 304 p.
8. Samoilenko А. М., Recke L. Conditions for synchronization of one oscillating system // Ukr. Mat. Zh. –
2005. – 57, № 7. – P. 922 – 945.
Received 13.12.10
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
|
| id | umjimathkievua-article-2714 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:28:54Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3b/0fd30e960460b346908b11ccc682d23b.pdf |
| spelling | umjimathkievua-article-27142020-03-18T19:34:23Z Some problems of the linear theory of systems of ordinary differential equations Деякi проблеми лiнiйної теорiї систем звичайних диференцiальних рiвнянь Samoilenko, A. M. Самойленко, А. М. We consider problems of the linear theory of systems of ordinary differential equations related to the investigation of invariant hyperplanes of these systems, the notion of equivalence for these systems, and the Floquet &#8211; Lyapunov theory for periodic systems of linear equations. In particular, we introduce the notion of equivalence of systems of linear differential equations of different orders, propose a new formula of the Floquet form for periodic systems, and present the application of this formula to the introduction of amplitude-phase coordinates in a neighborhood of a periodic trajectory of a dynamical system. Розглянуто проблеми лiнiйної теорiї систем звичайних диференцiальних рiвнянь, пов’язанi з дослiдженням iнварiантних гiперплощин таких систем, поняттям еквiвалентностi для вказаних систем та теорiєю Флоке – Ляпунова для перiодичних систем лiнiйних рiвнянь. Зокрема, введено поняття еквiвалентностi систем лiнiйних диференцiальних рiвнянь рiзних порядкiв, запропоновано нову формулу вигляду Флоке для перiодичних систем, наведено застосування цiєї формули для введення амплiтудно-фазових координат в околi перiодичної траєкторiї динамiчної системи. Institute of Mathematics, NAS of Ukraine 2011-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2714 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 2 (2011); 237-269 Український математичний журнал; Том 63 № 2 (2011); 237-269 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2714/2184 https://umj.imath.kiev.ua/index.php/umj/article/view/2714/2185 Copyright (c) 2011 Samoilenko A. M. |
| spellingShingle | Samoilenko, A. M. Самойленко, А. М. Some problems of the linear theory of systems of ordinary differential equations |
| title | Some problems of the linear theory of systems of ordinary differential equations |
| title_alt | Деякi проблеми лiнiйної теорiї систем звичайних диференцiальних рiвнянь |
| title_full | Some problems of the linear theory of systems of ordinary differential equations |
| title_fullStr | Some problems of the linear theory of systems of ordinary differential equations |
| title_full_unstemmed | Some problems of the linear theory of systems of ordinary differential equations |
| title_short | Some problems of the linear theory of systems of ordinary differential equations |
| title_sort | some problems of the linear theory of systems of ordinary differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2714 |
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