ON THE REFLECTING FUNCTION AND THE QUALITATIVE BEHAVIOR OF SOLUTIONS OF SOME NON-AUTONOMOUS DIFFERENTIAL EQUATIONS∗

In this article, we use the Mironenko’s method to discuss the qualitative behavior of some non-autonomous differential equations. We study the structure of the reflecting functions of the simplest differential equations, and obtain some sufficient conditions under which these equations have the rational reflecting functions. We apply the obtained results to discuss the numbers of periodic solutions of the non-autonomous differential systems and derive some sufficient conditions for a critical point of theirs to be a center.


Introduction
By [1] we know, for the polynomial differential system, where a ij and b ij are real constants, there has been a longstanding problem, called the Poincaré center-focus problem, for the system (1.1) find explicit conditions of a ij and b ij under which (1.1) has a center at the origin (0, 0), i.e., all the orbits nearby are closed. The problem is equivalent to an analogue for a corresponding periodic equation To see this let us note that the phase curves of (1.1) near the origin (0, 0) in polar coordinates x = r cos θ , y = r sin θ are determined by (1.2), where p i (θ) and q i (θ) (i = 0, 1, 2, . . . , n) are polynomials in cos θ and sin θ.
In this paper, we apply the theory of reflecting function to study directly the qualitative behavior of the solutions of equation (1.2), and obtain the sufficient conditions for r = 0 to be a center.
First of all, we study under which conditions the scalar differential equation (1.2) is the simplest equation with reflecting function F (θ, r) and discuss the structure of F (θ, r). Secondly, we find out the sufficient conditions under which these equations have the rational reflecting functions. Finally, we apply the obtained results to research the numbers of the periodic solutions of (1.2) and obtain the center conditions.
In the present section, we introduce the concept of the reflecting function, which will be used throughout the rest of this article.
Consider differential system which has a continuously differentiable right-hand side and with a general solution ϕ(t; t 0 , x 0 ).
By this, for any solution x(t) of (1.4), we have F (t, x(t)) = x(−t), F (0, x) = x. If system (1.4) is 2ω-periodic with respect to t, and F (t, x) is its reflecting function, then T (x) := F (−ω, x) = ϕ(ω; −ω, x) is the Poincaré mapping of (1. (1.5) Each continuously differentiable function F (t, x) that satisfies F (−t, F (t, x)) = x, F (0, x) = x, is a reflecting function of the whole class of system of the form is called the Simplest System with reflecting function F (t, x).
Thus, to study the behavior of the solutions of (1.6), only need to discuss the property of the solutions of the simplest system (1.7).
In the following, we will denote p i = p i (θ),p i = p i (−θ), P = P (θ, r),P = P (−θ, F ), etc. The notation "δ = 0" means that in some deleted neighborhood of (0, 0) and θ 2 + r 2 being small enough δ is different from zero. We always assume that all equations in this paper have a continuously differentiable right-hand side and have a unique solution for their initial value problem.

Main Results
Let us consider differential equation (1.2), in which P and Q are coprime polynomials of degree n ( n is a positive integer number) with respect to r.
Firstly, we will discuss the structure of the reflecting function F when the equation (1.2) is the simplest equation.
If system (1.2) is the simplest with reflecting function F , by Lemma 1.2 we have R(θ, r) = R(−θ, F ), i.e., Sm is a solution of (2.1), then l = m or l = m + 1 (m ≤ n). Where R l , S m are coprime polynomials with respect to r of degree l, m, respectively, l and m are nonnegative integers.
Sm is the solution of (2.1), Since R l , S m are coprime polynomials, so from (2.2) implies that A n+1 is divisible by S m and m ≤ n. According to (P, Q) = 1 and equating the same powers of r of equation (2.2) follows l = m or l = m + 1. Proof. Without loss of generality, we may assume that q n = 0. Otherwise, similarly, we can get the same conclusion.
Applying these relations and simply computing we get Using relations (2.3) and (2.8), we have F = ν n−1 −λ n = Bn−1 Bn − An An+1 . Substituting it into (2.1), we confirm that the conclusion of the present theorem is true.
Therefore, the proof is finished. 1) The equation (1.2) has at most n + 1 periodic solutions.

3) The equation (1.2) does not have any periodic solution.
Proof. By Theorem 2.1, we know the reflecting function of (1.2) is in the form of F = Rm Sm or F = Rm+1 Sm (m ≤ n). Then the Poincaré mapping of periodic equation (1.2) is T (r) = F (−π, r). The number of 2π-periodic solutions of (1.2) is equal to the number of roots of the fixed point equation F (−π, r) = r. From this follows the present conclusions.
On the other hand, if F = f 0 + f 1 x or F = β0+β1x α0+α1x is the reflecting function of (1.2), by Lemma 1.1 and the uniqueness of the solutions of the initial value problem of differential equation (1.5) implies f 0 ≡ 0, β 0 ≡ 0. So, in the following, we only discuss when the equation (1.2) has the reflecting function in the form of F = f 1 r and F = βr 1+αr . Theorem 2.2. Suppose that p 0 = 0, Then F = f 1 r is the reflecting function of (1.2). Moreover, if p i (θ + 2π) = p i (θ), q i (θ + 2π) = q i (θ) (i = 0, 1, 2, . . . , n), then one of the following conclusions is correct.
Proof. By the present conditions, it is not difficult to check that F = f 1 r is a solution of the Cauchy problem: Thus, F = f 1 r is the reflecting function of (1.2). If the equation (1.2) is 2π-periodic, then the Poincaré mapping of (1.2) is T (r) = p 0 (θ) dθ . By this and [6] yields the present conclusions.
Proof. By Lemma 1.1, we see F = βr 1+αr is the reflecting function of (1.2) if and only if Equating the coefficients of the same power of r implies β p 0p0 + βq 0q0 + βp 0q0 = 0, β(0) = 1, which implies the relation (2.13). Substituting these relations into the third relation of the above we can obtain (2.11). Therefore, under the assumption of the present theorem, F = βr 1+αr is the reflecting function of (1.2).
(2.15) As this equation is the simplest equation with reflecting function F = r 1+r sin θ and F (−π, r) ≡ r, thus the critical point (0, 0) of (2.14) is a center point.
From the previous introduction and (1.6), we know the equation (2.15) is equivalent to equation where G(θ, r) is an arbitrary continuously differentiable function, s := sin θ, c := cos θ.
Thus, the origin (0, 0) of the above systems is a center. As G(θ, r) is an arbitrary continuously differentiable function, similarly, we can write infinitely many polynomial differential systems, their origin point (0,0) is a center.
From this example, we see that using the method of Mironenko (reflecting function) we not only solve a center-focus problem, but also at the same time, we open a class of differential equations with the same character of point r = 0. So, we can say, sometimes, the method of Mironeneko is more effective than Lyapunov's method. Therefore, if we can find out the reflecting function of a differential system, then the qualitative behavior of periodic solutions and the center-focus problem are solved. Unfortunately, looking for reflecting function is also very difficult task, so we need further study it.