# The Research of Periodic Solutions of Time-Varying Differential Models

Authors: Wenjun Liu; Yingxin Pan; Zhengxin Zhou
Journal: International Journal of Differential Equations
 (2014)

---

## Abstract

We have studied the periodicity of solutions of some
nonlinear time-varying differential models by using the theory of reflecting functions. We have established a new relationship between the linear differential system and the Riccati equations and applied the obtained results to discuss the behavior of periodic solutions of the Riccati equations.


---

## Body

## 1. Introduction

From the book [1] we know that a lot of biological models can be expressed by the following differential system:
(1)
x
′
=
P
(
t
,
x
)
,
y
′
=
q
0
(
t
,
x
)
+
q
1
(
t
,
x
)
y
+
q
2
(
t
,
x
)
y
2
=
Q
(
t
,
x
,
y
)
which has a continuous differentiable right-hand side, and have an unique solution for their initial value problem.Because of this nonlinear and time-varying differential system, to discuss its qualitative behavior is very difficult. Now, we use the method of reflecting function [2, 3] to study the property of solutions of (1) and got some good results.In the present section, we introduce the concept of the reflecting function, which will be used throughout the rest of this paper.Consider differential system(2)
x
′
=
X
(
t
,
x
)
,
t
∈
R
,
x
∈
R
n
,
which has a continuous differentiable right-hand side, with a general solution φ
(
t
;
t
0
,
x
0
). For each such system, thereflecting function [2, 3]is defined as F
(
t
,
x
)
:
=
φ
(
-
t
,
t
,
x
). Therefore, for any solution x
(
t
) of (2), we haveF
(
t
,
x
(
t
)
)
=
x
(
-
t
).If system (2) is2
ω-periodic with respect tot, then T
(
x
)
:
=
F
(
-
ω
,
x
)
=
φ
(
ω
;
-
ω
,
x
)is the Poincaré mapping of (2) over the period[
-
ω
,
ω
]
.Thus, the solution x
=
φ
(
t
;
-
ω
,
x
0
)of (2) defined on[
-
ω
,
ω
]is2
ω-periodic if and only ifx
0is a fixed point ofT
(
x
). The stability of this periodic solution is equivalent to the stability of the fixed pointx
0.A differentiable functionF
(
t
,
x
)is a reflecting function of system (2) if and only if it is a solution of the Cauchy problem
(3)
F
t
′
+
F
x
′
X
(
t
,
x
)
+
X
(
-
t
,
F
)
=
0
,
F
(
0
,
x
)
=
x
.If the reflecting function of (2) has the form ofF
(
t
,
x
)
=
F
(
t
)
x, then the matrixF
(
t
)is calledreflecting matrix. Thus, this matrix is a reflecting matrix of linear systemx
′
=
A
(
t
)
x, if and only if
(4)
F
′
(
t
)
+
F
(
t
)
A
(
t
)
+
A
(
-
t
)
F
(
t
)
=
0
,
F
(
0
)
=
E
.There are many papers which are devoted to investigations of qualitative behavior of solutions of differential systems by help of reflecting functions. V. I. Mironenko and V. V. Mironenko [2–6] combined the theory of reflecting function with the integral manifold to discuss the symmetry and other geometric properties of solutions of (2), and obtained a lot of excellent new conclusions. Alisevich [7] has discussed when a linear system has triangular reflecting function. Musafirov [8] has studied when a linear system has reflecting function which can be expressed as a product of three-exponential matrix. Veresovich [9] has researched when the nonautonomous two-dimensional quadric systems are equivalent to a linear system. Maĭorovskaya [10] has established the sufficient conditions under which the quadratic systems have linear reflecting function. Zhou [11–13] has discussed the structure of reflecting function of quadratic systems and applied the obtained conclusions to study the qualitative behavior of solutions of such differential systems.In this paper we will discuss when the system (1) has the reflecting function in the form of
(5)
F
(
t
,
x
,
y
)
=
(
F
1
,
F
2
)
T
=
(
x
,
β
(
t
,
x
)
+
ς
(
t
,
x
)
y
γ
(
t
,
x
)
+
α
(
t
,
x
)
y
)
T
,
whereα
(
t
,
x
), β
(
t
,
x
), γ
(
t
,
x
), and ς
(
t
,
x
)are continuously differentiable functions inR
2. We will give the sufficient conditions for system (1) which has reflecting function of the form (5). We establish the relationship between the Riccati equation
(6)
d
y
d
t
=
a
0
(
t
)
+
a
1
(
t
)
y
+
a
2
(
t
)
y
2
and linear system
(7)
d
z
d
t
=
(
-
1
2
a
1
(
t
)
-
a
2
(
t
)
a
0
(
t
)
1
2
a
1
(
t
)
)
z
.
We apply the obtained conclusions to study the behavior of solutions of system (1) and the above Riccati equations.In the following, we will denotea
=
a
(
t
,
x
), a
-
=
a
(
-
t
,
x
), F
=
F
(
t
,
x
,
y
), and so forth.


## 2. Main Results

Lemma 1.
If the function (5) is the reflecting function of any system, then for small|
t
|, it can be written in the formF
2
=
(
β
^
+
γ
-
e
δ
y
)
/
(
γ
e
-
δ
+
α
^
y
), in which
(8)
α
^
+
α
^
-
=
0
,
β
^
+
β
^
-
=
0
,
δ
+
δ
-
=
0
.Proof.
As the function (5) is a reflecting function, so F
2
(
-
t
,
F
1
,
F
2
)
=
y, F
2
(
0
,
x
,
y
)
=
y; that is,
(9)
y
(
γ
γ
-
+
α
-
β
+
(
α
γ
-
+
α
-
ς
)
y
)
=
γ
β
-
+
β
ς
-
+
(
α
β
-
+
ς
ς
-
)
y
.
Equating the coefficients of the same power ofy, we get
(10)
β
-
γ
+
β
ς
-
=
0
;
(11)
γ
γ
-
+
α
-
β
=
α
β
-
+
ς
ς
-
;
(12)
α
γ
-
+
α
-
ς
=
0
,
(13)
α
(
0
,
x
)
=
β
(
0
,
x
)
=
0
,
γ
(
0
,
x
)
=
ς
(
0
,
x
)
=
1
.
The relation (11) implies that
(14)
γ
γ
-
=
ς
ς
-
,
α
β
-
=
β
α
-
.
Thus,
(15)
γ
=
ς
-
e
2
δ
,
δ
+
δ
-
=
0
.
Substituting it into (10) and (12), we get
(16)
β
=
β
^
e
δ
,
α
=
α
^
e
δ
,
β
^
+
β
^
-
=
0
,
α
^
+
α
^
-
=
0
.
So
(17)
F
2
=
β
^
e
δ
+
γ
-
e
2
δ
y
γ
+
α
^
e
δ
y
=
β
^
+
γ
-
e
δ
y
γ
e
-
δ
+
α
^
y
.
The proof is finished.By Lemma1, in the following, we all supposeF
2
=
(
β
+
γ
-
y
)
/
(
γ
+
α
y
), in whichα
+
α
-
=
β
+
β
-
=
0.Theorem 2.
Suppose thatm, n, α, and β are the solutions of the Cauchy problem
(18)
m
t
′
+
P
m
x
′
=
q
1
e
n
+
q
2
e
β
-
q
0
e
α
,
m
(
0
,
x
)
=
1
;
n
t
′
+
P
n
x
′
=
-
q
0
o
α
+
q
1
e
m
-
q
2
o
β
,
n
(
0
,
x
)
=
0
;
α
t
′
+
P
α
x
′
=
-
q
1
o
α
+
2
q
2
e
m
+
2
q
2
o
n
,
α
(
0
,
x
)
=
0
;
β
t
′
+
P
α
x
′
=
2
q
0
o
n
+
β
q
1
o
-
2
q
0
e
m
,
β
(
0
,
x
)
=
0
,
whereP
+
P
-
=
0and q
i
e
=
(
q
i
+
q
-
i
)
/
2, q
i
o
=
(
q
i
-
q
-
i
)
/
2, i
=
0,1
,
2.
Then(19)
m
=
m
-
,
n
+
n
-
=
α
+
α
-
=
β
+
β
-
=
0
,
m
2
-
n
2
-
α
β
=
1
(20)
F
=
(
x
,
β
+
γ
-
y
γ
+
α
y
)
T
is the reflecting function of system (1). Therefore, nearby t
=
0, γ
+
α
y
>
0, whereγ
=
m
+
n.Proof.
Putting(21)
y
1
=
m
-
m
-
;
y
2
=
n
+
n
-
,
y
3
=
α
+
α
-
,
y
4
=
β
+
β
-
,
then by (18), we have
(22)
y
1
t
′
+
y
1
x
′
P
=
q
1
e
y
2
+
q
2
e
y
4
-
q
0
e
y
3
,
y
1
(
0
,
x
)
=
0
;
y
2
t
′
+
y
2
x
′
P
=
-
q
0
o
y
3
+
q
1
e
y
1
-
q
2
o
y
4
,
y
2
(
0
,
x
)
=
0
;
y
3
t
′
+
y
3
x
′
P
=
-
q
1
o
y
3
+
2
q
2
e
y
1
+
2
q
2
o
y
2
,
y
3
(
0
,
x
)
=
0
;
y
4
t
′
+
y
4
x
′
P
=
2
q
0
o
y
2
+
q
1
o
y
4
-
2
q
0
e
y
1
,
y
4
(
0
,
x
)
=
0
.
By the uniqueness of the solution of linear partial differential equations, we get(23)
y
1
=
y
2
=
y
3
=
y
4
≡
0
;
that is,
(24)
m
=
m
-
,
n
+
n
-
=
α
+
α
-
=
β
+
β
-
=
0
.
Obviously,γ
γ
-
-
α
β
=
Cis the first integral of system (18); by the initial conditions, we haveγ
γ
-
-
α
β
≡
1
.Thus, the functions γandαhave not the common zeros. Asγ
(
0
,
x
)
+
α
(
0
,
x
)
y
=
1
>
0, so in the nearby area oft
=
0, γ
+
α
y
>
0.
Using the conditions of the present theorem, it is not difficult to check that function (20) satisfies the relation (3) with respect to X
=
(
P
,
Q
)
T
.So, the function (20) is the reflecting function of (1). The proof is completed.Corollary 3.
If all the conditions of Theorem2 are satisfied, and the system (1) is2
ω-periodic with respect tot, then its Poincaré mapping can be expressed by
(25)
T
(
x
,
y
)
=
(
x
,
-
β
(
ω
,
x
)
+
γ
(
ω
,
x
)
y
γ
(
-
ω
,
x
)
-
α
(
ω
,
x
)
y
)
T
.
So, the solution(
x
(
t
)
,
y
(
t
)
)of (1) defined on[
-
ω
,
ω
]
is
2
ω-periodic, if and only if(
x
(
-
ω
)
,
y
(
-
ω
)
)is the solution of equations
(26)
α
(
ω
,
x
)
y
2
+
2
γ
o
(
ω
,
x
)
y
-
β
(
ω
,
x
)
=
0
,
x
(
-
ω
)
=
x
(
ω
)
.Corollary 4.
Suppose that(27)
q
0
+
q
-
0
=
0
,
P
+
P
-
=
0
,
lim
⁡
t
→
0
q
1
e
q
0
=
0
,
andα
=
q
1
e
/
q
0is continuously differentiable and satisfies
(28)
α
t
′
+
α
x
′
P
=
-
q
1
o
α
+
2
q
2
e
.
Then F
=
(
x
,
y
/
(
1
+
α
y
)
)
T is the reflecting function of system (1). In addition, if the system (1) is a2
ω-periodic system with respect tot, then all the solutions of (1) defined on[
-
ω
,
ω
]are2
ω-periodic.Proof.
It is not difficult to check that, under the conditions of above corollary, the functionF
=
(
x
,
y
/
(
1
+
α
y
)
)
Tis the solution of the Cauchy problem (3), so it is a reflecting function of (1). In view ofα
=
q
1
e
/
q
0, soα
(
-
ω
,
x
)
≡
0andF
(
-
ω
,
x
,
y
)
≡
(
x
,
y
)
T. Thus, the conclusion of above corollary is true.Example 5.
Differential system(29)
x
′
=
ϕ
(
t
,
x
)
sin
t
cos
⁡
x
,
y
′
=
sin
⁡
t
(
1
+
cos
⁡
2
x
)
+
y
(
ϕ
(
t
,
x
)
sin
t
sin
x
+
sin
2
t
(
1
+
cos
⁡
2
x
)
cos
⁡
x
)
+
y
2
(
1
2
cos
⁡
t
cos
⁡
x
+
ψ
(
t
,
x
)
)
has reflecting functionF
=
(
x
,
y
/
(
1
+
y
sin
t
cos
⁡
x
)
)
T. Where ϕ
(
t
,
x
)
=
ϕ
(
-
t
,
x
), ψ
(
t
,
x
)
+
ψ
(
-
t
,
x
)
=
0
.When ϕ
(
t
+
2
π
,
x
)
=
ϕ
(
t
,
x
), ψ
(
t
+
2
π
,
x
)
=
ψ
(
t
,
x
), all the solutions of above system defined on[
-
π
,
π
]are2
π-periodic.Example 6.
Differential system(30)
x
′
=
λ
(
t
,
x
)
cos
⁡
x
,
y
′
=
μ
(
t
,
x
)
sin
t
cos
⁡
x
+
y
(
μ
sin
t
cos
⁡
2
x
+
λ
(
t
,
x
)
sin
x
)
+
y
2
(
1
2
cos
⁡
t
cos
⁡
x
+
κ
(
t
,
x
)
)
,
has reflecting function F
=
(
x
,
y
/
(
1
+
y
sin
t
cos
⁡
x
)
)
T. Where λ
(
t
,
x
)
+
λ
(
-
t
,
x
)
=
0, μ
(
t
,
x
)
=
μ
(
-
t
,
x
), κ
(
t
,
x
)
+
κ
(
-
t
,
x
)
=
0.
If the functionsλ
(
t
,
x
), μ
(
t
,
x
), κ
(
t
,
x
) are2
π-periodic with respect to t, then all the solutions of above system defined on [
-
π
,
π
] are 2
π-periodic.
Similar to Corollary4, we get the following.Corollary 7.
If(31)
P
+
P
-
=
0
,
q
2
+
q
-
2
=
0
,
lim
⁡
t
→
0
q
1
e
q
2
o
=
0
,
β
=
q
1
e
/
q
2
o is continuously differentiable and satisfies
(32)
β
t
′
+
β
x
′
P
=
β
q
1
o
-
2
q
0
e
.
Then F
=
(
x
,
β
+
y
)
T is the reflecting function of system (1). In addition, if the system (1) is a2
ω-periodic system, then all the solutions of (1) defined on [
-
ω
,
ω
] are2
ω-periodic.Example 8.
Differential system(33)
x
′
=
ρ
(
t
,
x
)
cos
⁡
x
,
y
′
=
ξ
(
t
,
x
)
-
1
2
cos
⁡
t
cos
⁡
x
+
(
-
ρ
(
t
,
x
)
sin
x
+
η
(
t
,
x
)
sin
2
t
cos
⁡
x
)
y
+
η
(
t
,
x
)
sin
t
y
2
has reflecting function F
=
(
x
,
sin
t
cos
⁡
x
+
y
)
T.
Whereρ
(
t
,
x
)
+
ρ
(
-
t
,
x
)
=
0, ξ
(
t
,
x
)
+
ξ
(
-
t
,
x
)
=
0, η
(
t
,
x
)
=
η
(
-
t
,
x
). Besides, if functions ρ
(
t
,
x
), ξ
(
t
,
x
), η
(
t
,
x
) are2
π-periodic with respect to t, then all the solutions of above system defined on[
-
π
,
π
]are2
π-periodic.Corollary 9.
Suppose thatP
+
P
-
=
0
,
(34)
δ
t
′
+
δ
x
′
P
=
q
2
e
+
q
0
e
,
δ
(
0
,
x
)
=
0
,
(
q
0
e
-
q
2
e
)
cos
⁡
δ
+
q
1
o
sin
δ
=
0
;
q
1
e
cos
⁡
δ
+
(
q
2
o
-
q
0
o
)
sin
δ
=
0
.
and then F
=
(
x
,
(
-
sin
δ
+
y
cos
⁡
δ
)
/
(
cos
⁡
δ
+
y
sin
δ
)
)
T is the reflecting function of (1). In addition, if the system (1) is a 2
ω-periodic system, then the solution (
x
(
t
)
,
y
(
t
)
) of (1) defined on [
-
ω
,
ω
] is2
ω-periodic, and if and only if δ
(
ω
,
x
(
-
ω
)
)
=
k
π, k is a natural number.Proof.
In Theorem2 taking n
=
0, β
=
-
α
=
-
sin
δ, m
=
cos
⁡
δ, under the above hypothesis, the function F
=
(
x
,
(
-
sin
δ
+
y
cos
⁡
δ
)
/
(
cos
⁡
δ
+
y
sin
δ
)
)
T is the reflecting function of (1). Thus, when the system (1) is 2
ω-periodic, its Poincaré mapping can be expressed byT
(
x
,
y
)
=
F
(
-
ω
,
x
,
y
), its solution(
x
(
t
)
,
y
(
t
)
) defined on [
-
ω
,
ω
] is2
ω-periodic, and if and only if (
y
(
-
ω
)
2
+
1
)
sin
δ
(
ω
,
x
(
-
ω
)
)
=
0, it implies the conclusion of the present corollary.Example 10.
Differential system(35)
x
′
=
P
(
t
,
x
)
,
y
′
=
q
(
t
,
x
)
+
1
+
sin
2
t
2
+
(
sin
2
t
-
sin
t
cos
⁡
t
)
y
+
(
q
(
t
,
x
)
-
sin
t
cos
⁡
t
+
1
-
sin
2
t
2
)
y
2
has reflecting function
(36)
F
=
(
x
,
-
sin
t
+
y
cos
⁡
t
cos
⁡
t
+
y
sin
t
)
T
and all the solutions of above system defined on[
-
π
,
π
]are2
π-periodic, where functions P
(
t
,
x
) and q
(
t
,
x
) are arbitrary continuously differentiable odd and2
π-periodic with respect to t.Theorem 11.
Let the reflecting function of linear system(37)
d
z
d
t
=
A
(
t
)
z
,
A
(
t
)
=
(
-
1
2
a
1
(
t
)
-
a
2
(
t
)
a
0
(
t
)
1
2
a
1
(
t
)
)
,
have the form
(38)
G
(
t
,
z
)
=
(
g
1
(
t
)
g
2
(
t
)
g
3
(
t
)
g
1
(
-
t
)
)
z
.
Then the function
(39)
F
=
g
3
(
t
)
+
g
1
(
-
t
)
y
g
1
(
t
)
+
g
2
(
t
)
y
is the reflecting function of the Riccati equation
(40)
d
y
d
t
=
a
0
(
t
)
+
a
1
(
t
)
y
+
a
2
(
t
)
y
2
.Proof.
AsG
(
t
)is the reflecting matrix of the linear system (37), thus,
(41)
G
′
(
t
)
+
G
(
t
)
A
(
t
)
+
A
(
-
t
)
G
(
t
)
=
0
,
k
k
k
k
k
k
k
k
k
k
G
(
0
)
=
E
,
|
G
(
t
)
|
≡
1
,
and it implies
(42)
g
1
e
′
=
a
1
e
g
1
o
+
a
2
e
g
3
-
a
0
e
g
2
,
g
1
e
(
0
)
=
1
;
g
1
o
′
=
-
a
0
o
g
2
+
a
1
e
g
1
e
-
a
2
o
g
3
,
g
1
o
(
0
)
=
0
;
g
2
′
=
-
a
1
o
g
2
+
2
a
2
e
g
1
e
+
2
a
2
o
g
1
o
,
g
2
(
0
)
=
0
;
g
3
′
=
2
a
0
o
g
1
o
+
g
3
a
1
o
-
2
a
0
e
g
1
e
,
g
3
(
0
)
=
0
,
where a
i
e
=
(
a
(
t
)
+
a
(
-
t
)
)
/
2, a
i
o
=
(
a
(
t
)
-
a
(
-
t
)
)
/
2
(
i
=
0,1
,
2
), g
1
e
=
(
g
1
(
t
)
+
g
1
(
-
t
)
)
/
2, and g
1
o
=
(
g
1
(
t
)
-
g
1
(
-
t
)
)
/
2. Similar to Theorem 2, we get function (39) is the reflecting function of the Riccati equation (40).Corollary 12.
Ifa
i
(
t
+
2
ω
)
=
a
i
(
t
)
(
i
=
0,1
,
2
) and the conditions of Theorem 11 are satisfied, then(1)
ifg
2
(
ω
)
≠
0 and |
g
1
e
(
ω
)
|
>
1, then the Riccati (40) has two2
ω-periodic solutions. Ifg
2
(
ω
)
≠
0and|
g
1
e
(
ω
)
|
<
1, (40) has no one2
ω-periodic solution. Ifg
2
(
ω
)
≠
0and|
g
1
e
(
ω
)
|
=
1, (40) has a unique2
ω-periodic solution;
(2)
ifg
2
(
ω
)
=
0andg
1
o
(
ω
)
≠
0, then (40) has a unique2
ω-periodic solution. Ifg
2
(
ω
)
=
0 andg
1
o
(
ω
)
=
0, g
3
(
ω
)
=
0, all the solutions of (40) defined on[
-
ω
,
ω
]are2
ω-periodic; Ifg
2
(
ω
)
=
0andg
1
o
(
ω
)
=
0, g
3
(
ω
)
≠
0, (40) has no one2
ω-periodic solution.Proof.
As (39) is the reflecting function of (40) and g
1
(
t
)
g
1
(
-
t
)
-
g
2
(
t
)
g
3
(
t
)
≡
1. Thus, the Poincaré mapping can be expressed by T
(
y
)
=
F
(
-
ω
,
y
), and the solutions y
(
t
) of (40) defined on [
-
ω
,
ω
] are2
ω-periodic, if and only if y
(
-
ω
) is a solution of equation: F
(
-
ω
,
y
)
=
y
;
that
is,
(43)
g
2
(
ω
)
y
2
+
2
g
1
o
(
ω
)
y
-
g
3
(
ω
)
=
0
,
and from this relation andg
1
e
2
(
t
)
-
1
=
g
1
o
2
(
t
)
+
g
2
(
t
)
g
3
(
t
), it is easy to deduce the present conclusion.Theorem 13.
Suppose that(1)
λ
1
=
(
a
0
e
+
a
2
e
)
/
a
1
o, λ
0
=
(
a
2
o
-
a
0
o
)
/
a
1
o are continuous andlim
⁡
t
→
0
λ
1
=
0;
(2)
μ
1
/
μ
0is continuously differentiable and lim
⁡
t
→
0
(
μ
1
/
μ
0
)
=
0and satisfies
(44)
(
μ
1
μ
0
)
′
=
(
a
1
e
+
(
a
2
e
-
a
0
e
)
λ
0
)
(
μ
1
μ
0
)
2
+
(
(
a
0
e
-
a
2
e
)
λ
1
-
(
a
0
o
+
a
2
o
)
λ
0
)
μ
1
μ
0
+
(
a
0
o
+
a
2
o
)
λ
1
-
a
1
e
.Then the function (39) is the reflecting function of the Riccati equation (40). In addition, ifa
i
(
t
+
2
ω
)
=
a
i
(
t
)
(
i
=
0,1
,
2
), then all the solutions of (40) defined on[
-
ω
,
ω
]are2
ω-periodic, where
(45)
g
1
e
=
exp
⁡
∫
0
t
‍
(
(
a
2
e
-
a
0
e
)
λ
1
-
μ
1
μ
0
(
a
1
e
+
λ
0
a
2
e
-
λ
0
a
0
e
)
)
d
t
,
g
1
o
=
-
μ
1
μ
0
g
1
e
,
g
2
=
g
3
=
λ
1
g
1
e
+
λ
0
g
1
o
;
μ
0
=
λ
0
′
+
a
1
e
λ
1
+
a
1
o
λ
0
-
2
a
2
o
+
(
a
2
e
-
a
0
e
)
λ
1
λ
0
-
(
a
0
o
+
a
2
o
)
λ
0
2
;
μ
1
=
λ
1
′
+
λ
1
2
(
a
2
e
-
a
0
e
)
-
λ
1
λ
0
(
a
0
o
+
a
2
o
)
-
λ
0
a
1
e
+
λ
1
a
1
o
-
2
a
2
e
.Proof.
It is not difficult to check that under the hypothesis of Theorem13 the functions g
1
e, g
1
o, g
2
=
g
3
=
λ
1
g
1
e
+
λ
0
g
1
o are the solution of the Cauchy problem (42), so, the function (39) is2
ω-periodic reflecting function of (40); by [1] it implies the result of the present theorem.From a similar discussion we can get the following results.Theorem 14.
Suppose that(1)
k
1
=
a
1
e
/
a
0
o, k
3
=
-
a
2
o
/
a
0
o are continuous and lim
⁡
t
→
0
k
1
=
0;
(2)
ρ
1
/
ρ
3is continuously differentiable and lim
⁡
t
→
0
(
ρ
1
/
ρ
3
)
=
0and satisfies
(46)
(
ρ
1
ρ
3
)
′
=
(
a
2
e
-
a
0
e
k
3
)
(
ρ
1
ρ
3
)
2
+
(
a
1
o
+
a
0
e
k
1
)
ρ
1
ρ
3
+
2
a
0
e
.Then the function (39) is the reflecting function of the Riccati equation (40). In addition, if a
i
(
t
+
2
ω
)
=
a
i
(
t
)
(
i
=
0,1
,
2
), then all the solutions of (40) defined on[
-
ω
,
ω
]are2
ω-periodic, where
(47)
g
1
=
exp
⁡
∫
0
t
‍
(
(
a
0
e
k
3
-
a
2
e
)
ρ
1
ρ
3
-
a
0
e
k
1
)
d
t
;
g
3
=
-
ρ
1
ρ
3
g
1
,
g
2
=
k
1
g
1
+
k
3
g
3
;
ρ
1
=
k
1
′
-
a
0
e
k
1
2
-
2
a
0
e
k
3
+
a
1
o
k
1
-
a
2
e
;
ρ
3
=
k
3
′
+
a
2
e
k
1
-
a
0
e
k
1
k
3
+
2
a
1
o
k
3
.Theorem 15.
If there is a continuously differentiable even functionh
(
t
)and(1)
η
1
=
(
a
0
e
h
(
t
)
+
a
2
e
)
/
(
h
(
t
)
a
0
o
-
a
2
o
) and η
3
=
(
h
′
(
t
)
+
2
a
1
o
h
(
t
)
)
/
(
2
a
2
o
-
2
h
(
t
)
a
0
o
)are continuous and lim
⁡
t
→
0
η
1
=
0;
(2)
ζ
1
/
ζ
3 is continuously differentiable and lim
⁡
t
→
0
(
ζ
1
/
ζ
3
)
=
0 and satisfies
(48)
(
ζ
1
ζ
3
)
′
=
(
a
2
e
-
a
0
e
h
(
t
)
+
a
1
e
η
3
)
(
ζ
1
ζ
3
)
2
+
(
a
1
o
+
+
2
a
0
o
η
3
-
a
1
e
η
1
)
ζ
1
ζ
3
+
2
a
0
e
-
2
a
0
o
η
1
.Then the function (39) is the reflecting function of the Riccati equation (40). In addition, if a
i
(
t
+
2
ω
)
=
a
i
(
t
)
(
i
=
0,1
,
2
), h
(
t
+
2
ω
)
=
h
(
t
), then all the solutions of (40) defined on[
-
ω
,
ω
]are2
ω-periodic, where(49)
g
1
e
=
exp
⁡
∫
0
t
‍
(
a
1
e
η
1
-
ζ
1
ζ
3
(
η
3
a
1
e
+
a
2
e
-
a
0
e
h
(
t
)
)
)
d
t
;
g
1
o
=
(
η
1
-
ζ
1
ζ
3
η
3
)
g
1
e
,
g
3
=
-
ζ
1
ζ
3
g
1
e
,
g
2
=
h
(
t
)
g
3
;
ζ
1
=
η
1
′
+
2
a
0
o
η
1
2
-
2
a
0
e
η
1
+
a
1
e
η
1
η
3
-
a
1
e
;
ζ
3
=
η
3
′
+
(
a
1
o
+
2
a
0
o
η
3
)
η
1
+
a
1
e
η
3
2
+
(
a
2
e
-
a
0
e
h
(
t
)
)
η
3
+
a
2
o
+
a
0
o
h
(
t
)
.Example 16.
The Riccati equation(50)
y
′
=
1
2
cos
⁡
t
(
sin
4
t
+
sin
2
t
+
2
)
-
2
sin
t
+
(
2
sin
2
t
-
sin
3
t
cos
⁡
t
)
y
+
(
1
2
cos
⁡
3
t
+
2
sin
t
)
y
2
has reflecting function
(51)
F
=
sin
t
(
2
+
sin
2
t
)
+
(
1
+
sin
2
t
)
y
1
+
sin
2
t
+
y
sin
t
.
So, all the solutions of the equation above defined on[
-
π
,
π
]are2
π-periodic.Example 17.
The Riccati equation(52)
y
′
=
-
1
2
C
e
S
(
2
-
2
S
+
S
2
-
3
S
3
+
S
4
-
S
5
)
+
C
(
1
+
2
S
-
S
3
+
S
4
)
y
+
1
2
C
e
-
S
(
1
+
S
2
-
3
S
3
+
S
4
-
S
5
)
y
2
has reflecting function
(53)
F
=
S
(
2
+
S
2
)
+
(
1
+
S
2
)
e
-
S
y
(
1
+
S
2
)
e
S
+
y
S
,
whereS
:
=
sin
t and C
=
cos
⁡
t. And all the solutions of the equation above defined on[
-
π
,
π
]are2
π-periodic.



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*Source: 430951-2014-04-16.xml*