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 wellknown 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].