305
Iern. J. Math. & Math. Sci.
Vol. 5 No. 2 (1982) 305-309
ON THE PERIODIC SOLUTIONS OF LINEAR HOMOGENEOUS
SYSTEMS OF DIFFERENTIAL EQUATIONS
A.K. BOSE
Department of Mathematical Sciences
Clemson University
29613 U.S.A.
Clemson, South Carolina
(Received October 13, 1981)
ABSTRACT.
Given a fundamental matrix
Y’
geneous differential equations
(x) of
an n-th order system of linear homo-
A(x)Y, a necessary and sufficient condition for
the existence of a k-dimensional (k < n) periodic sub-space (of period T) of the
solution space of the above system is obtained in terms of the rank of the scalar
matrix
(T)
(0).
KEY WORDS AND PHRASES.
me marrY,
sca rix,
Linear homogeneous system
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
1.
of differenti equations, Fundaof (period T), Rank of the
Perdic solutions, perdic sub-spaces
Linearly independent veers.
34A20.
INTRODUCT ION.
Consider the n-th order system of linear homogeneous differential equations
Y’
where Y
col
(Yl(X)’ Y2(x)
A(x)Y,
Yn(X))’
Y’
is a square matrix of order n, each element
continuous on the real line R.
Let S
n
col
aij (x)
(i.i)
(Yi(x),
12
(x)
(x)
(x)
((aij(x)))
be the solution space of the system of equa-
Y21 (x)
Y22 (x)
Let
Ynl (x)
Yn2 (x)
."
$
A(x)
of A(x) is a real-valued function
tions (i.i) on the real line R and T > 0 be a real number.
Yll
Yn’(x))
(1.2)
in
(x)
Y2n (x)
Ynn (x)
A.K. BOSE
306
be a fundamental matrix of the system (l.1).
The column vectors of (x) are lln-
early independent solutions of (i.i).
The purpose of this note is to deduce a necessary and sufficient condition
for the existence of periodic sub-spaces (of period T) of the solution space S
n
of
the system (1.1) and to show that the existence and dimensions of these periodic
sub-spaces depend not on any prior assumption about the periodicity (of period T)
of the elements a..(x) of the coefficient matrix A(x) of the system (I.i) (that is,
all the elements a..(x) of A(x) need not be periodic of period T), but precisely
on the rank of the scalar matrix
(T)
2.
(1.3)
(0).
MAIN RESULTS.
The
(1.3)
condition
is stated more explicitly in the following theorem:
Let k be a non-negative integer, 0 < k < n.
THEOREM.
sional sub-space S
k
of the solution space
such that each member of S
Sn
There exists a k-dimen-
of the linear homogeneous system (i.i)
is periodic of period T and no member of
k
Sn
periodic of period T if and only if the rank of the scalar matrix (T)
S
is
k
(0) is
n-ko
The above theorem can also be phrased is terms of the eigen values of the scafar matrix
-i
(0)(T)
COROLLARY.
as follows.
Let k be a non-negative integer, 0 < k < n.
mensional sub-space S
k
of the solution space
Sn
There exists a k-di-
of the linear homogeneous system
(i.i) such that each member of S k is periodic of period T and no member of S n
of multiplicity k.
Let k be a non-negative integer, 0 -< k -< n and rank of
PROOF OF THE THEOREM.
(T)
k
i is an eigen value of the scalar matrix
is periodic of period T if and only if
-I(0)(T)
S
(0) is n-k.
Then the dimension of the kernel of (T)
(0)
is k.
Hence
there exists k linearly independent vectors
v.
n
belonging to R
col
Cin ),
Cil ci2
i
1,2,
k
such that
((T)
#(0))v
i
0,
i
1,2
k
(2.1)
307
PERIODIC SOLUTIONS OF SYSTEMS OF DIFFERENTIAL EQUATIONS
Let
fi(x)
(x)vi,
1,2,...,k.
i
independence of the k solution vectors
v implies the linear
k
vl, v2,...
fl(x)’ f2(x)
fk(x)
fi(T)
The linear independence of the vectors
of the system (i.i).
((T)
fi(0)
Also,
(0))v i
0,
(2.2)
k,
1,2
i
implies by the uniqueness of the solutions of initial value problem that
fi(x+T)
period T) sub-space of
member of
Sn
trivial and
S
n
Let S
k, is periodic of period T.
1,2
i
S
k
Sn
generated by
k
Let
’fk(x)’ gl(x)
f’l(x)’
be the k-dimensional periodic (of
fl(x)"’’’fk(x)"
is periodic of period T.
fl(x), f2(x),
Hence each solution vector
k.
1,2
0 for all x, i
fi(x)
gl(x)
We need to show that no
g
Sn
S
k.
Then
gl(x)
is non-
are k+l linearly independent members of
Let
g!(x)
If possible, let
(X)Vk+l,
gl(x)
where
col(Ck+1 iCk+l 2,...,Ck+l
Vk+1
be periodic of period T.
gl(x
+ T)
gl(x)
0
n
).
That is
for all x.
Then
gl(r)
gl(0)
(2.3)
0.
Since any set of linearly independent members of S
Sn,
n
form a part of a basis of
fl(x),f2(x),...,fk (x)’gl (x)’g2 (x)’’’’’gn-k(x) be a basis of Sn and
,n-k,
2,3
gi(x) (X)Vk+i, i
where
n-k. The linear independence
Ck+i n i 2,3,
Vk+i col(Ck+i 1 Ck+i 2
of the basis vectors fl(x)’ f2(x)"’’fk(x)’gl(x)
’gn-k(X) implies that the
let
matrix
cll
c21
Cnl
12
c22
Cn2
In
C2n
Cnn
C
is non-singular and hence
rank of ((T)
(0))
rank of ((T)
(0))C,
[see, 2],
(2.4)
A.K. BOSE
308
But, by actual multiplication and using (2.2) and (2.3), we see that the first k+l
#(0))C are zero-vectors and hence the rank of
column vectors of (#(T)
@(O))C
((T)
rank of ((T)
n-k
implying a contradiction.
member of S
S
n
k
T.
k
rank of ((T)
#(0))
Hence
gl(x)
cannot be periodic of period T.
k
is periodic of period T and no member of
fk(x)
fl(x),f2(x)
gn_k(X) be
gi(x), i
a basis of
Sn.
(0)
gl(r)
be a basis of S
gi(x)
Clearly
gl(0)’ g2(r)
are linearly independent.
1(gl(T)
That is no
g
Sn
S
k
such that every
is periodic of period
is n-k.
and
k
Sn
S k, i
1,2,...,n-k.
Hence each
Again the n-k vectors
gn-k(0)
gn-k(T)
g2(0)
fk(x),gl(x)
fl(x),f2(x)
n-k, is not periodic of period T.
1,2,
Sn
be a k-dimensional sub-space of
We need to show that the rank of (T)
Let
< n-k-i
#(0))C
is periodic of period T.
Conversely, suppose that S
member of S
Therefore, from (2.4)
is at most n-k-l.
For
gl(0)) + 2(g2(T)
+
g2(0)) +
n-k(gn-k(T)
gn-k(0))
0
implies by the uniqueness of the solutions of initial value problem that
1g l(x)
g(x)
+
+
n_kgn_ k(x)
is a periodic solution of the system (i.i) of period T.
g(x)
g
S
k
and therefore
1gl(x) +
where bl,b
b are
2
k
fk(x),gl(x),
fl(x)
g(x)
for all x,
Since
Hence by our hypothesis
+
n_kgn_ k
blfl(x)
+
+
bkfk(x),
real constants.
gn_k(X) form
a basis of
i
0,
i
1,2,...,n-k
b.
0,
j
1,2,...,k.
Sn,
it follows that
Hence the n-k vectors
gl(r)
gl(0),g2(T)
g2(O)
gn_k(r)
gn_k(0)
are linearly independent.
Let H(x) be the fundamental matrix of the linear system (i.i) whose column
vectors are
309
PERIODIC SOLUTIONS OF SYSTEMS OF DIFFERENTIAL EQUATIONS
gn-k(X)
fk(x)’ gl(x)
fl (x), f2 (x)
and C be a non-singular scalar matrix such that
H(x)
(x)C.
H(0)
((T)
Th en
H(T)
(0))C
(2.5)
Since C is non-singular,
(0))
rank of ((T)
But, the first k columns of H(T)
fl(x), f2(x)
fk(x)
gl(T)
of H(T)
H(T)
(0))C
rank of ((T)
rank of (H(T)
H(0) are zero vectors by the periodicity of
and the last n-k column vectors
gn-k(r)
g2(0)
gl(0)’ g2(r)
H(0) are linearly independent as proved before.
H(O) is n-k.
H(0)).
gn-k(0)
Hence the rank of
(0) is n-k.
That is, the rank of (T)
This completes the
proof of the theorem.
To prove the corollary, we see, from (2.1) that
(0)(T)v i
v
1,2
i
i
k.
That is,
(-l (0)(T)where I is the identity matrix.
the scalar matrix
-I(0)(T)
l)v
i
0,
i
This means that
of multiplicity k.
1,2
k.
i must be an eigen value of
Hence arguing similarly as in the
proof of the theorem one can prove the corollary easily.
REFERENCES
I.
CODDINGTON, EARL A. and LEVINSON, NORMAN. Theory of Ordinary Differential
Equations, McGraw-Hill Book Company, New York.
2.
BIRKHOFF, GARRETT and MACLANE, SAUNDERS.
Macmillan Company, New York.
A Survey of Modern Algebra,