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Research on the Poincaré centerfocus problem of some cubic differential systems by using a new method
Advances in Difference Equations volume 2017, Article number: 52 (2017)
Abstract
In this article, we discuss the Poincaré centerfocus 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 centerfocus 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.
Introduction and preliminaries
Consider the cubic system
where \(p_{i}(x,y)\) and \(q_{i}(x,y)\) are homogeneous polynomials in x and y of degree i (\(i=1,2,3\)). Taking \(x=r\cos\theta\), \(y=r\sin\theta\), then (1.1) becomes
where \(a_{i}(\theta)\) and \(b_{i}(\theta)\) (\(i=0,1,2,\ldots,n\)) are polynomials in cosθ and sinθ. By [1–5], we know that the origin \((0,0)\) of (1.1) is a center, if and only if equation (1.2) has a center at \(r=0\), i.e., all the solutions nearby \(r=0\) are 2πperiodic.
As usually, we often apply the method of Lyapunov and Poincaré to study centerfocus 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 highorder 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 wellknown 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 centerproblem 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.
Consider the differential system
which has a continuously differentiable righthand side and general solution \(\phi(t; t_{0}, x_{0})\).
Definition 1.1
For system (1.3), \(F(t,x):=\phi(t, t, x)\) is called its reflecting function [7].
By this, for any solution \(x(t)\) of (1.3), we have \(F(t,x(t))=x(t)\), \(F(0,x)=x\) and \(F(t,x)\) is a reflecting function of system (1.3) if and only if it is a solution of the Cauchy problem
Theorem 1.2
[7]
If system (1.3) is 2ωperiodic 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 2ωperiodic 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 2ωperiodic with respect to t, then all the solutions of (1.3) defined on \([\omega,\omega]\) are 2ωperiodic.
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]
If \(F(t,x)\) is the reflecting function of (1.3), then one system is equivalent to (1.3) if and only if this system can be expressed as follows:
where \(G(t,x)\) is an arbitrary vector function such that the every solution of the system (1.5) is uniquely determined by its initial conditions.
All the equivalent 2ωperiodic 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 centerfocus 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 centerfocus 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 centerfocus 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.
Then \(m=n=1\), i.e., \(F=\frac{q_{0}(t)+q_{1}(t)x}{p_{0}(t)+p_{1}(t)x}\).
Proof
(1) If \(F=\sum_{i=0}^{n}f_{i}(t)x^{i}\) is a reflecting function of one differential system, by [7], we have \(F(t,F(t,x))\equiv x\), i.e.,
it implies
from this we get \(F(t,x)x{f}_{0}(t) \) (\(xf_{0}(t)\) is divisible by \(F(t,x)\)), thus \(n=1\), i.e., \(F(t,x)=f_{0}(t)+f_{1}(t)x\).
(2) If \(F(t,x)=\frac{Q(t,x)}{P(t,x)}\) is a reflecting function of one differential system, by [7], we get \(F(t,F(t,x))\equiv x\), i.e.,
where \(\bar{p}_{i}=p_{i}(t)\), \(\bar{q}_{i}=q_{i}(t)\).
Case 1. If \(n>m\), from (1.6) we get
it implies \(P\bar{p}_{n} xQ^{n}\). Since \((P,Q)=1\) and \(p_{0}\neq0\), so \(P\bar{p}_{n}\), i.e., \(n=0\), this is contradiction with \(n\geq1\). Therefore, \(n\leq m\).
Case 2. If \(n< m\), from (1.6) we have
which yields \(P\bar{q}_{m}Q^{m}\), as \((P,Q)=1\), so \(P\bar{q}_{m}\), i.e., \(n=0\), this is contradiction with \(n\geq1\). Thus \(m=n\).
Case 3. If \(m=n\), from (1.6), we get
For the coefficients of the degree \(n^{2}+1\) of x on both sides of (1.7) we have
i.e.,
it implies \(P=(x\frac{\bar{q}_{n}}{\bar{p}_{n}})\tilde{P}\), P̃ is a polynomial of degree \(n1\) with respect to x.
From (1.7) follows
by this we get
which implies \(P(\bar{p}_{n}x\bar{q}_{n})Q^{n}\), and so \(\tilde{P}Q^{n}\). As, \((P,Q)=1\), thus \(\partial\tilde{P}=0\), and \(\partial{P}=1\), i.e., \(n=1\).
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 centrefocus 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 righthand side and have a unique solution for their initial value problem in a neighborhood of the origin.
Main results
Now, let us consider the general cubic system
where \(a_{ij}\), \(b_{ij}\) (\(i,j=0,1,2,3\)) are constants.
Setting \(x=r\cos\theta\), \(y=r\sin\theta\) in (2.1), we obtain
where
where \(C:=\cos\theta\), \(S:=\sin\theta\).
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
\(F=f_{0}(\theta)+f_{1}(\theta)r\) is the reflecting function of (2.2), if and only if
and \(f_{0}=0\), \(f_{1}=1\). Therefore, \(r=0\) is a center of (2.2).
Proof
By equation (1.4), we know \(F=f_{0}(\theta)+f_{1}(\theta )r\) is the reflecting function of (2.2), if and only if
Taking \(r=0\), it implies that
By the uniqueness of solutions for initial value problems of the above differential equation, it yields \(f_{0}(\theta)=0\). Substituting it into (2.5) we have
Equating the coefficients of the same power of r on both sides of (2.6), we obtain (2.4). So, \(F=r\) is the reflecting function of (2.2). By Theorem 1.2, all the solutions of (2.2) nearby \(r=0\) are 2πperiodic, thus, the \(r=0\) is a center. □
Applying Theorem 1.4 we have the following.
Corollary 2.2
If all the conditions of Theorem 2.1 are satisfied, then for the equation
\(r=0\) is a center. Here \(G(\theta,r)\) is an arbitrary continuously differentiable vector function and such that \(G(\theta,0)G(\theta,0)=0\).
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
If
then the cubic system (2.1) has a center at \((0,0)\).
Proof
Since the even part of an odd function is equal to zero, using (2.3) and the first relation of (2.4) and Lemma 2.3 we get
Using the second relation of (2.4) and simplifying we obtain
Applying the third relation of (2.4) and Lemma 2.3 we have
By the fourth relation of (2.4) and simplifying, we get
It follows from (2.8)(2.11) that equation (2.7) is true. Thus, it follows that equation (2.7) holds. Due to Theorem 2.1, \(F=r\) is the reflecting function of (2.2), by Theorem 1.2, the origin point \((0,0)\) of (2.1) is a center. Thus, the proof is finished. □
Simplifying equation (2.7), we can get the equivalent theorem as following.
Theorem 2.4′
The origin point \((0,0)\) of the cubic system (2.1) is a center, if \(a_{20}=a_{30}=0\) and one of the following conditions is satisfied:
From Theorem 2.4′, the origin point \((0,0)\) is a center of system (2.1), if it can be expressed in one of the following forms:
Theorem 2.5
The fractional function \(F=\frac{\beta_{0}(\theta )+\beta_{1}(\theta)r}{1+\alpha(\theta)r}\) (\(\alpha(0)=0\), \(\beta _{0}(0)=0\), \(\beta_{1}(0)=1\)) is a reflecting function of (2.2), if and only if,
and \(\beta_{0}=0\), \(\beta_{1}=1\), \(\alpha=\int_{0}^{\theta}(b_{1}+\bar {b_{1}})\, d\theta\). Besides, if \(\int_{0}^{\pi}(b_{1}+\bar{b_{1}})\, d\theta= 0\), then \(r=0\) is a center of (2.2); If \(\int_{0}^{\pi}(b_{1}+\bar{b_{1}})\, d\theta\neq0\), then equation (2.2) has only one 2πperiodic solution, i.e., \(r=0\).
Proof
By (1.4), \(F=\frac{\beta_{0}(\theta)+\beta_{1}(\theta )r}{1+\alpha(\theta)r}\) is the reflecting function of (2.2), if and only if
Taking \(r=0\) in (2.13) we get
By the uniqueness of solutions for initial value problems of the above differential equation, it yields \(\beta_{0}(\theta)=0\). Substituting it into (2.13) and equating the coefficients of the same power of r on both sides of (2.13), we obtain
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 2πperiodic solution of (2.2).
Thus, the proof is completed. □
Applying Theorem 1.4, we get the following corollary.
Corollary 2.6
If all the conditions of Theorem 2.5 are satisfied, then for the equation
\(r=0\) is a center. Here \(G(\theta,r)\) is an arbitrary continuously differentiable vector function and such that \(G(\theta,0)G(\theta,0)=0\).
Theorem 2.7
Suppose that \(a_{20}\neq0\) and one of the following conditions is satisfied
Then the origin point \((0,0)\) is a center of the cubic system (2.1).
Proof
Applying Theorem 2.5 and equation (2.3), we know
As the even part of the odd function is equal to zero, using (2.3) and Lemma 2.3, from the first relation of (2.12) we get
Using the second relation of (2.12) and Lemma 2.3, we obtain
Substituting (2.23) into (2.27), we have
Equation (2.29) plus (2.31) minus (2.28) minus (2.30) gives us
which implies
Thus, \(\alpha(\theta)=2a_{20}\sin\theta\). As \(a_{20}\neq0\), by (2.32) we get \(b_{30}=0\). Substituting \(b_{30}=0\) and (2.33) into (2.23), (2.24) and (2.25) we obtain
Substituting (2.33) and (2.26) into (2.35), we obtain (2.35), too. Using the above relations and simplifying (2.28)(2.31) we have
Computing the third relation of (2.12) and using the above relations we get
Applying the fourth relation of (2.12) and Lemma 2.3, we get
Using (2.23) and (2.48) we obtain
Obviously, equation (2.47) is the same as (2.45). Using (2.49) and (2.26) and (2.34), (2.46) becomes
Thus, from the fourth relation of (2.12) we have
Using (2.35) and (2.23) and (2.49) we derive
Substituting (2.52) and (2.51) into (2.41),(2.42) and (2.43), it shows that these equations are identical. Substituting (2.51) and (2.52) into (2.36) we have
Substituting (2.51) and (2.52) into (2.39) we obtain
Substituting (2.33), (2.53), (2.35), (2.52) into (2.37) and (2.38), it shows that these equations are identical. Thus from above discussion we know equation (2.12) is equivalent to the following relations:
Considering (2.55) and using Theorem 2.5 and Theorem 1.2, the conclusion of the present theorem is true and the proof is finished. □
Remark 1
Under the conditions (2.20), the system (2.1) with \(a_{20}\neq0\) can be expressed as follows:
and this cubic system has a center at \((0,0)\). Here \(a_{11}\), \(a_{30}\), \(a_{21}\), \(a_{12}\), \(a_{03}\) are arbitrary constants.
Remark 2
Under the conditions (2.21), the system (2.1) with \(a_{20}\neq0\) can be expressed as follows:
where
and this cubic system has a center at \((0,0)\).
Remark 3
In the literature [1], the authors have studied the centerfocus 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 centerfocus 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 centerfocus 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
Consider the cubic system
It is easy to check that, for this system the condition (2.20) is satisfied, so the point \((0,0)\) is a center.
Taking \(x=r\cos\theta\), \(y=r\sin\theta\), (2.56) becomes
and it has a reflecting function \(F=\frac{r}{1+r\sin\theta}\).
By Theorem 1.4, equation (2.57) is equivalent to the equation
and which has a center at \(r=0\). Here \(G(\theta,r)\) nearby \(r=0\) is an arbitrary continuously differentiable function and such that \(G(\theta ,0)G(\theta,0)=0\).
By [17], the first integral of (2.58) is \(H(x,y)=\Phi(\theta, u)\), where \(u=\frac{2+r\sin\theta}{1+r\sin\theta}r\) and \(\Phi(\theta,u)\) is the first integral of the following first order equation
where \(\alpha(\theta, u)=(1+(1+r\sin\theta)^{2})(G(\theta,r)\frac {1}{(1+r\sin\theta)^{2}}G(\theta,\frac{r}{1+r\sin\theta}))\).
Taking
where \(k\geq1\) is an arbitrary constant, then equation (2.58) becomes
by Corollary 2.6, its equivalent system,
has a center at \((0,0)\), too.
Taking \(k=1\) in (2.60), which shows the cubic system
has a center at \((0,0)\). On the other hand, it is easy to check that system (2.61) satisfies the condition (2.20) of Theorem 2.7.
The Darboux first integral [18] of (2.61) is \(H(x,y)=\Phi(\theta,u) \) (\(u=\frac{2+r\sin\theta}{1+r\sin \theta}r\)), \(\Phi(\theta,u)\) is the first integral of the following first order equation:
Taking \(G=\frac{r^{2}\sin\theta(2+r\sin\theta)^{2}}{2(1+r\sin\theta )^{2}(1+(1+r\sin\theta)^{2})}\), then (2.58) becomes
and its equivalent system
has a center at \((0,0)\).
Solving the equation \(\frac{du}{d\theta}=u^{2}\sin\theta\), we get the Darboux first integral of (2.62),
and its inverse integrating factor is
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 centerfocus 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.
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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).
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Authors’ contributions
ZZ gave the proof of all theorems, finishing the article. FM gave the calculations and verifications of some theorems. YY gave some constructive comments. All authors have read and approved the final manuscript.
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DOI
MSC
 34A12
 34A34
 34C14
Keywords
 cubic system
 Poincaré centerfocus problem
 Mironenko’s method
 reflecting function