Research on the Poincaré center-focus problem of some cubic differential systems by using a new method
- Zhengxin Zhou^{1}Email authorView ORCID ID profile,
- Fangfang Mao^{1} and
- Yuexin Yan^{1}
DOI: 10.1186/s13662-017-1107-4
© The Author(s) 2017
Received: 19 July 2016
Accepted: 6 February 2017
Published: 16 February 2017
Abstract
In this article, we discuss the Poincaré center-focus problem of some cubic differential systems by using a new method (Mironenko’s method) and obtain some new sufficient conditions for a critical point to be a center. By using this method we not only solve a center-focus problem, but also at the same time, we open a class of differential systems, which do not have to be polynomial differential systems, with the same qualitative behavior at the critical point.
Keywords
cubic system Poincaré center-focus problem Mironenko’s method reflecting functionMSC
34A12 34A34 34C141 Introduction and preliminaries
As usually, we often apply the method of Lyapunov and Poincaré to study center-focus problem. A new technique recently developed by Christopher and Sadowskii and others. Polynomial ideals and invariant algebraic curves are sought and appropriate Dulac functions constructed [1–6]. But for high-order polynomial systems and general planar systems, to give the center conditions is very difficult. Necessary and sufficient conditions are known for very few classes of systems. There are well-known conditions for quadratic systems and the problem has been resolved for systems in which P and Q are cubic polynomials without quadratic terms and other some particular forms. In this paper, we will apply the Mironenko’s (reflecting function method) method to study the Poincaré center-problem of the certain cubic differential systems which are in the general form.
Now, we simply introduce the concept of the reflecting function, which will be used throughout the rest of this article.
Theorem 1.2
[7]
If system (1.3) is 2ω-periodic with respect to t, and \(F(t,x)\) is its reflecting function, then \(T(x):=F(-\omega,x)\) is the Poincaré mapping of (1.3) over the period \([-\omega,\omega]\), and the solution \(x=\phi(t;-\omega,x_{0})\) of (1.3) defined on \([-\omega,\omega]\) is 2ω-periodic if and only if \(x_{0}\) is a fixed point of \(T(x)\).
From this theorem, we know that if system (1.3) and its reflecting function \(F(t,x)\) are 2ω-periodic with respect to t, then all the solutions of (1.3) defined on \([-\omega,\omega]\) are 2ω-periodic.
Definition 1.3
If the reflecting functions of two differential systems coincide in their common domain, then these systems are said to be equivalent [7].
Theorem 1.4
[7]
All the equivalent 2ω-periodic systems have a common Poincaré mapping over the period \([-\omega, \omega]\), and the qualitative behavior of the periodic solutions of these systems are the same. By this one can study the qualitative behavior of the solutions of a complicated system by using a simple differential system [7].
There are many papers which are also devoted to investigations of qualitative behavior of solutions of differential systems by help of reflecting functions [7–16].
Mironenko’s method [7] was introduced and applied to the study of Poincaré center-focus problem in [13], in which the author has stated in detail how to apply this method to calculate the focus quantities and derive the center conditions and use the reflecting function of a particular differential system to study the qualitative behavior of the periodic solutions of its equivalent systems. In this paper we observe that any differential equation that takes the form of (1.2) has an analytic reflecting function and the calculation can be carried out easily. Thus by Mironenko’s method, the center-focus problem of a wide range of systems that are equivalent to (1.2) can be solved automatically. To be more specific, we will study the center-focus problem of some cubic differential systems (1.1) by looking for the rational reflecting function of (1.2). We will give a set of new sufficient conditions under which the critical point of (1.1) is a center. Meanwhile, we discover a class of differential systems, which is equivalent to (1.1) but not necessarily polynomial, with the same character at its critical point.
Lemma 1.5
- (1)
If \(F(t,x)=\sum_{i=0}^{n}f_{i}(t)x^{i}\) is a reflecting function of one differential system, then \(n=1\), i.e., \(F=f_{0}(t)+f_{1}(t)x\).
- (2)
If \(F(t,x)=\frac{Q(t,x)}{P(t,x)}\) is a reflecting function of one differential system, \(P, Q\) are relatively prime polynomials (\((P,Q)=1\)), \(P=\sum_{i=0}^{n}p_{i}(t)x^{i}\) (\(n\geq1\), \(p_{n}p_{0}\neq0\)), \(Q=\sum_{j=0}^{m} q_{j}(t)x^{j}\), \(p_{i}(t)\), \(q_{j}(t)\) are continuously differentiable functions.
Proof
The proof is completed. □
By Lemma 1.5, to seek the analytic rational reflecting function of (1.2), only need to search the reflecting function in the form of \(F=f_{0}(t)+f_{1}(t)x\) and \(F=\frac{q_{0}(t)+q_{1}(t)x}{p_{0}(t)+p_{1}(t)x}\). Thus, in this paper, we are interested in when the differential equation (1.2) has such reflecting function, and how to apply these reflecting functions to study the centre-focus problem of the cubic system (1.1) and their equivalent systems.
In the following, all the differential systems have been discussed which have a continuously differentiable right-hand side and have a unique solution for their initial value problem in a neighborhood of the origin.
2 Main results
In the following, we will denote \(\bar{a}_{i}=a_{i}(-\theta)\), \(\bar{b}_{i}=b_{i}(-\theta)\), \(\overline{\phi (\theta)}=\phi(-\theta)\).
Theorem 2.1
Proof
Applying Theorem 1.4 we have the following.
Corollary 2.2
It is not difficult to prove the following lemma by using the method of mathematical induction.
Lemma 2.3
If \(\sum_{i+j=n}a_{ij}\cos^{i}\theta\sin^{j}\theta \equiv 0\), \(\theta\in R\), then \(a_{ij}=0\) (\(i,j=0,1,2,\ldots,n\)). Here \(a_{ij}\) are constants.
Theorem 2.4
Proof
Simplifying equation (2.7), we can get the equivalent theorem as following.
Theorem 2.4′
Theorem 2.5
Proof
From (2.14) follows \(\beta_{1}(\theta)=1\). By (2.15) we get \(\alpha(\theta)=\int_{0}^{\theta}(b_{1}+\bar{b}_{1})\, d\theta\). Substituting \(\alpha'=b_{1}+\bar{b}_{1}\) into (2.16)-(2.19) and simplifying, it follows equation (2.12) is true. From above we know \(F=\frac{r}{1+\alpha(\theta)r}\) is the reflecting function of (2.2), so \(F(-\pi,r)=r\) is equivalent to \(\alpha(-\pi )r=0\), it implies if \(\alpha(-\pi)=0\), \(r=0\) is a center, if \(\alpha (-\pi)\neq0\), \(r=0\) is an unique 2π-periodic solution of (2.2).
Thus, the proof is completed. □
Applying Theorem 1.4, we get the following corollary.
Corollary 2.6
Theorem 2.7
Proof
Remark 1
Remark 2
Remark 3
In the literature [1], the authors have studied the center-focus problem of the cubic system in the Kukles form, i.e., \(a_{ij}=0\) (\(i,j=0,1,2,3\)) and in the literature [2], they have discussed the cubic system with \(a_{03}=0\) and \(b_{30}=0\). In the other literature [3–5], the center-focus problem of some special cubic system has been studied. In this paper, we use the Mironenko’s method to give the sufficient conditions of the center of the cubic system in the general form (2.1). Just this advantage is not enough, by the equivalence [7] of differential system, if the reflecting function of one differential system is given, at the same time, we know the reflecting function of its infinite equivalent differential systems. Thus, if we solve the center-focus problem of one system, at the same time, we open a class of differential systems with the same character at the critical point. The following example will illustrate this advantage.
Example 1
By this, the system (2.62) does not exist a limit cycle nearby \((0,0)\) (\(|x|<1\)).
Remark 4
If in the system (2.1) taking \(a_{20}=\frac{1}{2}\), then under the assumption of Theorem 2.7, equation (2.2) has a center at \(r=0\) which can be expressed as (2.58).
Remark 5
From the above, we see that using the method of Mironenko we not only solve a center-focus problem, but also at the same time, we open a class of differential systems, which do not have to be polynomial differential systems, with the same character at the critical point \((0,0)\). Therefore, we can say, sometimes, the method of Mironenko is more effective than Poincaré and Lyapunov’s method.
Declarations
Acknowledgments
This work is supported by the National Natural Science Foundation of China (61374010, 11571301), and the National Natural Science Foundation of province Jiangsu (BK20161327), and the advantage subject Fund Project of High Education Institutions of Jiangsu Province (PPZY2015B109).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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