On the Generalized Reflective Function of the Three-Degree Polynomial Differential System

Abstract

The structure of the generalized reflective function of three-degree polynomial differential systems is considered in this paper. The generated results are used for discussing the existence of periodic solutions of these systems.

1. Introduction

As we know, it is very important to study the properties of the solution of differential system for both the theory and application of an ordinary differential equation.

If ( is a positive constant), we can use the Poincaré mapping introduced in [1] to study the behavior of the solutions of (1). But it is very difficult to find the Poincaré mapping for many systems which cannot be integrated in quadratures. In the 1980s, the Russian mathematician Mironenko first established the theory of reflective functions (RF) in [2, 3]. Since then a quite new method to study (1) has been found. In recent years, more and more experts and scholars have achieved many good results; refer to [4–9] in this direction.

The aim of this article is to use the theory of the generalized reflective function to study the behavior of the solutions of differential systems. The obtained results will provide a new theoretical basis and criterion to further explain the laws of the movement of objects.

In the present section, we first recall some basic notions and results of the generalized reflective function (GRF), which will be used throughout the rest of this article.

Now consider system (1) with a continuous differentiable right-hand side and with a general solution . For each such system, the GRF of system (1) is defined as , , where is a continuous differentiable function such that , . Then for any solution of (1), we have and . By the definition in [3], a continuous differentiable vector function on is called GRF if and only if it is a solution of the Cauchy problemRelation (2) is called a basic relation (BR).

Besides this, suppose that system (1) is periodic with respect to the variable , and is its GRF; if there exists a number on such that , then is the Poincaré mapping of (1) over the period . So, for any solution of (1) defined on , it will be periodic if and only if which is called a basic lemma (BL).

Now, we consider the higher dimensional polynomial differential systemwhere , , and (; ) are continuously differentiable functions in , (in some deleted neighborhood of and being small enough but different from zero), and there exists a unique solution for the initial value problem of (3). And suppose thatis the GRF of (3).

In this article, we discuss the structure of () when and obtain the good results () which are useful for the research of the existence of periodic solutions and establishing the sufficient conditions of system (3) with the form of GRF.

In the following, we denote(; ). The notation “” means that, in some deleted neighborhood of , is small enough but different from zero.

2. Main Results

Without loss of generality, we suppose that . Otherwise, we can take the transformation , , .

Now, we consider system (3).

Lemma 1. For system (3), let , ; then

Proof. Using relation (2), we can geti.e.,Putting , we have , , which implies that relation (6) is valid.

In the following discussion, we always assume (6) holds without further mentioning.

Case I (). From relation (8), we getwhere Differentiating relation (9) with respect to implieswhere

Lemma 2. For system (3), suppose , and the limit () exist; then

Proof. Using relation (11), we have , i.e.by , which implies that the results of Lemma 2 are true.

Theorem 3. For system (3), suppose that and all the conditions of Lemmas 1 and 2 are satisfied; then

Proof. As , by relation (11), we haveDifferentiating relation (11) with respect to , we can obtainSubstituting (9) into the above, we getwhereSubstituting (16) into (18), we have in which(1) If , we get by relation (20). Through the expression of , (; ), we know that is a quadratic polynomial with respect to , and is a cubic polynomial with respect to , . Substituting into (11), we obtain which implies that or and . Substituting the relations into (11) and equating the coefficients of the same powers of and , we have , . Finally, we arrive at(2) If , it follows from (20). By simple computation, we obtainLet ; we haveby (23) and (24).
Sincewherewe can obtain , , and by taking advantage of Lemma 2. Thus . It is easy to verify thatUsing identity (25), we get By the uniqueness of solutions of the initial problem of linear partial differential equations, we have . ThereforeSo, using relation (11), we get Finally, we get by relation (9).
Summarizing the above, the proof is finished.

Theorem 4. For system (3), let , , , and (); then

Case II (, ). From relation (8), we havewhere Differentiating this identity with respect to giveswhere

Likewise, we have the following conclusions.

Lemma 5. Suppose , , , , and the limits () and exist; then

Theorem 6. Suppose that all the conditions of Lemmas 1 and 5 are satisfied; then

Theorem 7. Suppose that , , . (); then

Theorem 8. Function is the GRF of system (3), if the following conditions are satisfied:

Proof. By checkout of the BR, it can be proved that the functionis the GRF of system (3).

Theorem 9. Let the hypotheses of Theorem 8 be satisfied, system (3) is periodic with respect to , and there exists a number in such that ; then all the solutions of system (3) defined on the interval are periodic if and only if , , , and .

Proof. By Theorem 8, the Poincaré mapping of system (3) is .
According to the previous introduction, the solutions of system (3) are periodic if and only if , which implies that the assertions of the present theorem hold. The proof is finished.

Example 10. The differential systemhas GRF  , in whichSince this system is a periodic system, and there exists a number in such that , then ; by Theorem 9, all the considered solutions defined on are periodic.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.