Abstract
We study the number of limit cycles for the quadratic polynomial differential systems , having an isochronous center with continuous and discontinuous cubic polynomial perturbations. Using the averaging theory of first order, we obtain that 3 limit cycles bifurcate from the periodic orbits of the isochronous center with continuous perturbations and at least 7 limit cycles bifurcate from the periodic orbits of the isochronous center with discontinuous perturbations. Moreover, this work shows that the discontinuous systems have at least 4 more limit cycles surrounding the origin than the continuous ones.
1. Introduction
In 1900, Hilbert [1] proposed 23 famous mathematical problems in the second International Congress of Mathematicians, and the second part of the sixteenth problem asks for the maximum of the number of limit cycles and the relative positions for all planar polynomial differential systems of degree . Mathematicians have done a lot of effective works to research on the numbers of limit cycles for the continuous polynomial system (especially the quadratic polynomial system); see for instance the books [2, 3] and the hundreds of references quoted therein. One of the methods for studying on the number of the limit cycles is averaging method. In [4], Buică and Llibre introduced the averaging method for finding limit cycles of continuous differential systems via Brouwer degree. Further, in [5], Llibre et al. used the averaging method for studying the periodic orbits of discontinuous differential systems. Since most phenomena in real life are discontinuous, this created a great interest for the mathematicians to study the limit cycle of discontinuous differential system, especially for the numbers of limit cycles of the discontinuous quadratic polynomial system; see, for instance, [5–9]. For the discontinuous Liénard equations, in [10], Martins and Mereu showed that for any there are differential equations of the form , with and being polynomials of degree and 1, respectively, having limit cycles. In [11], Libre and Teixeira provided lower bounds for the maximum number of limit cycles for -piecewise discontinuous polynomial differential equations. For the switching systems, Han and Sheng [12] discussed the bifurcation of limit cycles in piecewise smooth systems via Melnikov functions. Ten limit cycles are found around a center in switching quadratic systems in [13].
Consider a quadratic polynomial system: Chicone and Jacobs proved in [14] that at most 2 limit cycles bifurcate from the periodic orbits of the isochronous center of system (1). Their study is based on the displacement function using some results of Bautin [15]. In [4], Buică and Llibre easily reproved, using the averaging method, by continuous quadratic polynomial perturbing system (1), the existence of at least 2 limit cycles bifurcating from the periodic orbit of the center of system (1) when this is perturbed inside the class of all quadratic polynomial differential systems. Recently, Llibre and Mereu [7] proved, using the averaging method, by discontinuous quadratic polynomial perturbing system (1), the existence of at least 5 limit cycles bifurcating from the periodic orbit of the center of system (1) when this is perturbed inside the class of all quadratic polynomial differential systems, and the discontinuous systems have at least 3 more limit cycles surrounding the origin than the continuous systems.
In this paper, consider the following systems: where is a small parameter, and Using averaging method and some results established in [12], we obtain our main results as follows.
Theorem 1. For sufficiently small there are continuous cubic polynomial differential systems (2) having exactly 3 limit cycles bifurcating from the periodic orbits of isochronous center (1).
Theorem 2. For sufficiently small there are discontinuous cubic polynomial differential systems (3) having at least 7 limit cycles bifurcating from the periodic orbits of isochronous center (1).
By Theorems 1 and 2, we can get the following.
Corollary 3. Using the averaging method of first order, the discontinuous systems have at least 4 more limit cycles surrounding the origin than the continuous systems when we perturbed the center (1).
In some sense, we extend the work by Llibre and Mereu with the difference of number of limit cycles of discontinuous and continuous differential systems.
2. Preliminary Results
In this section, we introduce some preliminary results on the averaging theory which will be applied to study the cubic continuous and discontinuous polynomial systems (2) and (3).
The following theorems are the first-order averaging theory for continuous and discontinuous differential systems. For the proof, we refer the reader to [4, 5].
Theorem 4 (see [4]). Consider the following differential system: where are continuous functions, T-periodic in the first variable, and is an open subset of . Define : and assume that(i)and are locally Lipschitz with respect to x.(ii)For with , there exists a neighborhood of a such that for all and .Then, for sufficiently small, there exists a T-periodic solution of system (5) such that as .
Theorem 5 (see [5]). Consider the following discontinuous differential system: with where are continuous functions, T-periodic in the first variable, and is an open subset of . We also suppose that is a function having 0 as a regular value. Denote by , , , and its elements by .
Define the averaged function : Assume the following three conditions:(i), and are locally L-Lipschitz with respect to .(ii)For with , there exists a neighborhood of a such that for all and (i.e., the Brouwer degree of f at a is not zero).(iii)If for some , then .Then, for sufficiently small, there exists a T-periodic solution of system (7) such that as .
Remark 6 (see [4]). Let be a function, with , where is an open subset of and . Whenever , there exists a neighborhood of a such that for all . Then .
Consider a integrable planar differential system: where are continuous functions under the assumption
(A1) System (10) has a period annulus around the singular point (0, 0): where is a first integral of (10), is the critical level of corresponding to the center , and denotes the value of for which the period annulus terminates at a separatrix polycycle. Without loss of generality we can assume that . We denote by the integrating factor of system (10) corresponding to the first integral .
Perturbed systems (10) are as follows: where are continuous functions.
In order to apply the averaging method for studying limit cycles of (12) for sufficiently small, we need write system (12) into the standard forms (5) and (7). The following result by [4] provides a way to fulfill it.
Theorem 7 (see [4]). Consider system (10) and its first integral H. Assume that (A1) holds for system (10) and that for all in the period annulus formed by the ovals . Let be a continuous function such that for all and . Then the differential equation which describes the dependence between the square root of energy, , and the angle for system (12) is where .
Remark 8. Further, system (14) can become where and .
The following lemma presents the version of the formula of the first-order Melnikov function associated with system (12) in the polar coordinates [4].
Lemma 9 (see [4]). Under the conditions of Theorem 7, define for system (12), where is the integrating factor of system (10) corresponding to the first integral H and ; . Then and expressed by (16) are the displacement function and the first-order Melnikov function of system (12), respectively.
Under the assumption (A1), the assumptions (I) and (II) hold as in [12]. So, we can establish the function as follows: since is the first-order Melnikov function of system (12). Based on Theorems 4 and 7, Lemma 9, Theorem 1.1 in [12], and (18), we have the following.
Lemma 10. Under the assumption (A1), let be the averaged function of system (14); then the following relation holds: where is defined by (6) and is defined by (18).
In order to study the number of zeros of the averaged functions (6) and (9), we will use the following result proved in [16].
Let be a set and let . We say that are functions if and only if we have that
Proposition 11 (see [16]). If are linearly independent, then there exist and such that for every
3. Proof of Theorem 1
In this section, we will prove Theorem 1 by using Theorem 4 for the continuous case. We recall that the period annulus of a center is the topological annulus formed by all the periodic orbits surrounding the center which is the only singular point of the system.
A first integral and an integrating factor in the period annulus of the center of the quadratic differential system (1) have the expressions and , respectively. For this system we note that . By (13), the function satisfying the hypotheses of Theorem 7 is given by , . It is easy to know that assumption (A1) holds.
Using Theorem 7, we transform system (2) into the form where and .
The continuous differential system (22) is under the assumptions of Theorem 4. So we just need to study the zeros of the averaged function : Using Mathematica (Mathematica software), we compute the above integral and obtain where
We have the equalities Thus the function can be written as The Wronskian of the functions in variable is We have for all . In fact, if there exist such that , then and it is not difficult to obtain that Obviously, it is impossible for all . Then, the functions in are linearly independent. By Proposition 11, there exist , such that the linear combination of four functions has at least 3 zeros ; that is, for all , the following equations hold: Comparing (28) and (32), we get a set of equations about variables : The rank of coefficients matrix of (33) is 4; then the solutions of these equations exist. Thus there exist , such that the averaged function has at least 3 zeros , where are the solutions of (33); other ones are 0.
On the other hand, let ; the averaged function becomes where As a result of the symmetry of coefficients of , we know that if is one root of , so is , but only one of and is in interval . Hence, the fact that has at most 3 zeros in implies that there exist at most 3 zeros for in .
Thus, the averaged function has exactly 3 zeros. By Theorem 4, Lemmas 9 and 10, and Theorem 1.1 in [12], we get that there are exactly 3 limit cycles bifurcate from the period annulus around the isochronous center for (2) with sufficiently small . This completes the proof of Theorem 1.
4. Example
In this section, we not only provide some examples satisfying the property of Theorem 1, but also introduce a method to construct such systems.
Suppose that Take the constants From (35) and (37), we have where and other ones are any real constants.
Hence, in system (2), for the sufficiently small , we obtain a family of systems: The averaged function of systems (39) is Apparently, has exactly 3 positive zeros in , denoted by corresponding to , and in . Moreover, we have That is to say, , ; it follows from Theorem 4 and Lemmas 9 and 10 that, for the sufficiently small , system (39) has exactly 3 limit cycles emerging from the period annulus of the corresponding unperturbed system.
5. Proof of Theorem 2
In this section, we will prove Theorem 2 by using the first-order averaging method for the continuous case. The unperturbed system of system (3) is also system (1); then using Theorem 7 and Remark 8 we transform system (3) into the form whereand , and .
The discontinuous differential system (43) is under the assumptions of Theorem 5. So we just need to study the zeros of the averaged function : The above integral is calculated by the mathematical software Mathematica obtaining where
The following equalities hold: So we can rewrite the function as Through calculation, the Wronskian of the eight functions in variable is valued as at , so the eight functions are linearly independent. By Proposition 11, there exist , and at least seven , such that the equality holds, for all .
From (49) and (50), we obtain linear equations about variables , and it is easy to know that these linear equations have solutions. Hence, there exist (where are the solutions of these equations, and other ones are 0), such that the averaged function has at least 7 zeros .
In short, there are discontinuous cubic polynomial systems (3) having at least 7 limit cycles bifurcating from the periodic orbits of the isochronous center of system (1) by using Theorem 5. This completes the proof of Theorem 2.
6. Example
In this section, we will introduce a method to construct some examples satisfying the property of Theorem 2; moreover, we will provide such systems.
Firstly, we take , in (50), respectively. Thus, we get linear equations about variable . Solve these equations, and we choose one of the solutions of these equations: From (49) and (50), we obtain the following linear equations about variables , , , :
Second, solve (52), in (3), and we choose and all other coefficients are any real constants; then system (3) becomes where The averaged function of system (3) is The function has at lest 7 zeros: that is, . is shown in Figure 1. Hence, by Theorem 5, it follows that, for sufficiently small, system (54) has at least 7 limit cycles surround the origin.
Competing Interests
The author declares that there are no competing interests.
Acknowledgments
The author was supported by Sichuan Provincial Scientific Research and Innovation Team in Universities (Grant no. 14TD0040), the Natural Science Foundation of Education Department of Sichuan Province (Grant no. 15ZA0337), the Major Projects of Aba Teachers University (Grant no. ASA15-09), and Provincial Discipline Open Platform Project of Xihua University (szjj2015-060).