It turns out that to obtain a stability result, one needs take
, a periodic matrix [2]. Indeed, this allows using the Floquet theory for linear periodic system (1.7).
We need the following three well-known theorems [3–5].
Theorem 3.1.
Let
be the fundamental matrix of (1.1) with 
The zero solution of (1.1) is
-
(i)
stable if and only if there exists a positive constant M such that
-
(ii)
asymptotically stable if and only if
where
is a norm in
.
Theorem 3.2.
Consider system (1.1) with
a constant regular matrix. Then its zero solution is
-
(i)
stable if and only if
and the eigenvalues of unit modulus are semisimple;
-
(ii)
asymptotically stable if and only if
, where
is an eigenvalue of
is the spectral radius of 
Consider the linear periodic system
where
,
for some positive integer N.
From the literature, we know that if
with
is a fundamental matrix of (3.3), then there exists a constant
matrix, whose eigenvalues are called the Floquet exponents, and periodic matrix
with period N such that 
Theorem 3.3.
The zero solution of (3.3) is
-
(i)
stable if and only if the Floquet exponents have modulus less than or equal to one; those with modulus of one are semisimple;
-
(ii)
asymptotically stable if and only if all the Floquet exponents lie inside the unit disk.
In view of Theorems 3.1, 3.2, and 3.3, we obtain from Corollary 2.2 the following new stability criteria for (1.1).
Theorem 3.4.
The zero solution of (1.1) is stable if and only if there exists a
regular periodic matrix
satisfying (2.8) such that
-
(i)
the Floquet exponents of
have modulus less than or equal to one; those with modulus of one are semisimple;
-
(ii)
; those eigenvalues of
of unit modulus are semisimple.
Theorem 3.5.
The zero solution of (1.1) is asymptotically stable if and only if there exists a
regular periodic matrix
satisfying (2.8) such that either
-
(i)
all the Floquet exponents of
lie inside the unit disk and
; those eigenvalues of
of unit modulus are semisimple; or
-
(ii)
the Floquet exponents of
have modulus less than or equal to one; those with modulus of one are semisimple; and 
Remark 3.6.
Let
be periodic with period N. The Floquet exponents mentioned in Theorem 3.3 are the eigenvalues of
where 
Example 3.7.
Consider the system
Note that the conditions of Example 2.3 are all satisfied. It follows that
Now,
for which the eigenvalues are 
On the other hand, for
if
and
if
.
Applying Theorems 3.4 and 3.5, we see that the zero solution of (3.4) is asymptotically stable if
and is stable if 
In fact, the unique solution of (3.4) satisfying
is
where
,
,
,
, and
.
It is easy to see that
if
and
is bounded if 
Remark 3.8.
In the computation of
,
is calculated by using Example 2.3, and
is obtained by the method given in [6, 7].