Abstract and Applied Analysis

Volume 2013 (2013), Article ID 481501, 12 pages

http://dx.doi.org/10.1155/2013/481501

Research Article

Critical Periods of Perturbations of Reversible Rigidly Isochronous Centers

Jiamei Zhou,¹ Na Li,1,2 and Maoan Han¹

¹The Institute of Mathematics, Shanghai Normal University, Shanghai 200234, China

²Shanghai University of Engineering Science, Shanghai 201620, China

Received 29 March 2013; Accepted 10 May 2013

Academic Editor: Valery G. Romanovski

Copyright © 2013 Jiamei Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the problem of bifurcation of critical periods of a time-reversible polynomial system of degree . We first present a new method to find the number of zeros of the period function. Then applying our results, we study the number of critical periods for some polynomial systems and obtain new results.

1. Introduction

Consider a two-dimensional analytic real differential system of the form

where we suppose there is a nondegenerate center at the origin . Let denote the orbit passing through the point with of (1) and denote its period. As we all know, the isolated zeros of derivative of are named critical periods. An interesting problem is to investigate the number of the critical periods. This is an important problem in the research of period functions and a lot of results have been obtained for polynomial differential systems, for example, monotonicity [1–8], finiteness of critical periods [9, 10] and isochronicity [11–13], and local bifurcation of critical periods [14]. The number of critical periods was also discussed in [15–17] for perturbations of isochronous vector fields.

Recently, the authors in [18] studied the following system:

where , ,

Under the conditions above, system (2) is time-reversible and hence has a center at the origin.

As shown in [15–20], for , the period function of system (2) can be written as

where is a positive constant. The authors [18] gave expressions of and . It is also proved in [18] that one critical period can appear for the case and two critical periods can be found for the case .

In this paper, we consider the following system:

where

, , , , and both and are small parameters. From the above expressions, we know that the system (5) satisfies the following properties: (i) it is a time-reversible system, and the perturbations with are of general form; (ii) for all , the origin is a center, and for , the origin is isochronous since there exist polynomials and such that , , , .

Let denote the period of the periodic orbit of system (5) passing through the point . Then it has the following expansion:

where

and since (5) becomes (2) as , we have .

Next section, we give some preliminary lemmas.

2. Preliminary Lemmas

In this section, we cite some results obtained in [18].

Lemma 1 (see [18]). For system (5),

where

moreover, if is odd, , and

Lemma 2 (see [18]). Let and in (5). Then

where and are constants depending on the coefficients appearing in , , , and . Furthermore, for sufficiently small ,

(i)the center preserves the isochronicity when ; period function is increasing (resp., decreasing) for when and (resp., 0);(ii)there is at most one critical period in when . Moreover, there is exactly one critical period in if and only if

Lemma 3 (see [18]). Let and in (5). Then

where and , are constants depending on the coefficients appearing in , , , and . Furthermore, for sufficiently small , one has the following results.

(i)The center preserves the isochronicity when

(ii)If (20) do not hold, there are at most two critical periods in and the maximum is achievable.

3. Main Results

As in [18], one can make the polar coordinates , , so that system (5) becomes

where and are given by (13) and (14),

Obviously, when , which implies that the unperturbed system (5) has an isochronous center at which is called a rigidly or uniformly isochronous center. We have the following fundamental result.

Theorem 4. Let (8) and (9) hold. Then, for system (5), one has

where

Proof. We follow the idea of proving Lemma 1 which is Theorem 2.1 given in [18].

From (21), we have

Let be the solution of the above equation satisfying the initial condition . It can be written as a series of

where , , .

Substituting (30) into differential equation (29) and comparing the coefficients of , we have

We can write , where , , . Then, substituting into (31), then comparing the coefficients of and , we can obtain

Solving the above ODEs associated with the initial values and , we can get (27) and (28). By (21) we have

Therefore,

It implies that

Thus, (26) is proved. This ends the proof of Theorem 4.

With the same method as for , we can compute for and give an expression of for from the proof of Theorem 4, which are omitted here.

Define

Another fundamental result is as follows.

Theorem 5. Let be defined as before with . Then for , one has the following.

 (i) The period function of system (5) is increasing (resp., decreasing) in if (resp., 0) for all . (ii) If is not identically zero, the number of critical periods of   in is not more than the number of zeros (take the multiplicity into consideration) of in . And there are exactly critical periods if has exactly simple positive zeros.  (iii) If , the number of critical periods of in is not larger than the number of zeros of in . And critical periods can appear if has simple zeros. Similarly, the period function of (5) is increasing (resp., decreasing) in if (resp., < 0).

Proof. From (8) and (9)

If , , then there exists a such that for , has the same sign with . For , the sign of the function is the same as . Then the conclusion (i) is proved.

Further, suppose that are the zeros of with the multiplicity , respectively. We only need to prove that the number of critical periods of is less than or equal to . For the purpose, it suffices to prove that has at most roots. Thus, we only need to prove that has at most zeros near , . At first, we prove that has at most zeros near for small. If it is not the case, then has at least zeros near ; that is, there exists such that has at least zeros    near , where   . By Rolls theorem, has at least zeros near , and   has at least one zero near . Thus, . Letting , we have . This is a contradiction. Thus, has at most zeros near . Using the same method, for sufficiently small , or has at most zeros near . Thus, has at most critical periods for each . Thus, we have proved the first part of (ii).

For the second part of (ii), assume that are the simple zeros of ; that is, and , . Then

By the Implicit Function Theorem, there is a unique function such that and , . Therefore are simple zeros of . By the same method, there is a unique function such that , and . The second part of (ii) is proved.

When , then

Then, we can prove conclusion (iii) in the same way as proving conclusion (ii). The proof is completed.

4. Application

In this section, we apply Theorems 4 and 5 to the cases , ; , , and , , respectively.

Case 1 (,  ). In this case, system (5) becomes

where . For system (40), is a center because of the symmetry.

We only need to consider the case of in (40); otherwise, we can use the transformation to change (40) into the same form with opposite signs to the coefficients. Further, suppose , otherwise, system (40) can be simplified as the form of system (41) by the transformation , . In this case, system (40) becomes

It is easy to conclude that system (41) has a first integral of the form

Theorem 6. For system (41), one has

 (i)

 where . For sufficiently small and , there is at most one critical period if is not identically zero and ; there is exactly one critical period if and only if . Otherwise, the period function is increasing for (resp., decreasing) if (resp., < 0); (ii) if , then for ,

 where

There can appear three critical periods in if is not identically zero.

Proof. For system (41), from (13), (14), (22)–(25), we have

From (27), we have

From (10), we have

Then, from [18], we have

where and

Support that . By (50) we have four zeros of as follows:

where is given by . It is not difficult to examine that is not in and is not in . In addition, if and only if . Thus, there is at most one critical period in . If , has no zero in . From the expression of , we can easily check that (resp., 0) when (resp., 0). Therefore, for sufficiently small and , (resp., 0) if (resp., 0). The proof of conclusion (i) is completed.

For conclusion (ii), let , which gives or . From (26), we have that

Then we expand at by letting

By (52), we have

Note that

and that

Therefore, can be taken as free parameters. Hence, we can change the sign of , satisfying

which ensures that has three positive zeros in near . Thus, conclusion (ii) is proved.

We remark that for cubic system (40), we obtain three critical periods using , which is one more than the results in Lemma 2.3 obtained in [18].

Case 2 (, ). In this case, system (40) becomes

where we suppose . This is a new system which is not studied in [18]. For the case , by the rescaling

system (58) can be written as

where

If , system (58) can be simplified as (60) similarly by the change , .

Theorem 7. For system (60), one has

 (i)

 where . If is not identically zero, there are at most two critical periods in for sufficiently small and , and the maximum can be achievable; (ii) if , one has

 and there are at most two critical periods in if is not identically zero.

Proof. From (13), (14), (22)–(25), we have that for system (60)

From (27) and (28), we have

Hence, from (10),

Thus, , where , . If , , we can control the coefficients , , , which gives two zeros of in . If , according to the expression of , we suppose that are four zeros of . From the relationship of root and coefficients, we have ; thus not all of the zeros are in . Then from [18], and it is impossible for to have 3 zeros in . Thus, the first part of conclusion (i) is proved.

Next, we give an example to show that the zeros in can be achievable. We can choose , , , then, , so, has two simple zeros and in . Thus, has exactly two critical periods in . The proof of conclusion (i) is completed.

For the conclusion (ii), if , we get or .

Hence, from (26), we have

So, , where , . From the proof above, we can get result (ii). This ends the proof of Theorem 7.

Corollary 8. For system (58) with , one obtains

where . For sufficiently small and , there are at most two critical periods for in and the maximum number can be achievable.

Proof. Let denote the period function of (60) and denote the period function of (58). Then we have , and . Hence by Theorem 7 and (61), we have for system (58),

which gives the formula of . Other conclusions in the corollary can also be gotten from results in Theorem 7 similarly. This ends the proof of Corollary 8.

Case 3 (, ). In this case, system (5) becomes

Without loss of generality, we suppose , otherwise, system (70) can be simplified as the form of system (71) by the transformation , . In this case, system (70) becomes

Theorem 9. For system (71) and sufficiently small and , one has

(i)

where

There can exist three critical periods for system (71) if is not identically zero;(ii)if , one has

where

There can appear three critical periods for system (71) if is not identically zero.

Proof. From (13), (14), (22)–(25), we have

From (10), we can get

where

By the same method as the proof of Theorem 6, we have conclusion (i).

For conclusion (ii), if , giving , thus from (26),

where

We can get conclusion (ii) with the same method as the proof of Theorem 6. This ends the proof of Theorem 9.

Acknowledgment

The project was supported by National Natural Science Foundation of China (11271261).

References

L. P. Bonorino, E. H. M. Brietzke, J. P. Lukaszczyk, and C. A. Taschetto, “Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation,” Journal of Differential Equations, vol. 214, no. 1, pp. 156–175, 2005.

C. Chicone, “The monotonicity of the period function for planar Hamiltonian vector fields,” Journal of Differential Equations, vol. 69, no. 3, pp. 310–321, 1987.

S.-N. Chow and J. A. Sanders, “On the number of critical points of the period,” Journal of Differential Equations, vol. 64, no. 1, pp. 51–66, 1986.

S.-N. Chow and D. Wang, “On the monotonicity of the period function of some second order equations,” Časopis Pro Pěstování Matematiky, vol. 111, no. 1, pp. 14–25, 1986.

A. Cima, A. Gasull, and F. Mañosas, “Period function for a class of Hamiltonian systems,” Journal of Differential Equations, vol. 168, no. 1, pp. 180–199, 2000.

E. Freire, A. Gasull, and A. Guillamon, “First derivative of the period function with applications,” Journal of Differential Equations, vol. 204, no. 1, pp. 139–162, 2004.

J. Smoller and A. Wasserman, “Global bifurcation of steady-state solutions,” Journal of Differential Equations, vol. 39, no. 2, pp. 269–290, 1981.

J. Waldvogel, “The period in the Lotka-Volterra system is monotonic,” Journal of Mathematical Analysis and Applications, vol. 114, no. 1, pp. 178–184, 1986.

C. Chicone and F. Dumortier, “Finiteness for critical periods of planar analytic vector fields,” Nonlinear Analysis: Theory, Methods & Applications, vol. 20, no. 4, pp. 315–335, 1993.

P. de Maesschalck and F. Dumortier, “The period function of classical Liénard equations,” Journal of Differential Equations, vol. 233, no. 2, pp. 380–403, 2007.

J. Chavarriga and M. Sabatini, “A survey of isochronous centers,” Qualitative Theory of Dynamical Systems, vol. 1, no. 1, pp. 1–70, 1999.

W. S. Loud, “Behavior of the period of solutions of certain plane autonomous systems near centers,” Contributions to Differential Equations, vol. 3, pp. 21–36, 1964.

I. I. Pleshkan, “On isochronicity conditions of the system of two differential equations,” Journal of Differential Equations, vol. 4, pp. 1991–1993, 1968.

C. Chicone and M. Jacobs, “Bifurcation of critical periods for plane vector fields,” Transactions of the American Mathematical Society, vol. 312, no. 2, pp. 433–486, 1989.

A. Cima, A. Gasull, and P. R. da Silva, “On the number of critical periods for planar polynomial systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 7, pp. 1889–1903, 2008.

A. Gasull and J. Yu, “On the critical periods of perturbed isochronous centers,” Journal of Differential Equations, vol. 244, no. 3, pp. 696–715, 2008.

A. Gasull and Y. Zhao, “Bifurcation of critical periods from the rigid quadratic isochronous vector field,” Bulletin des Sciences Mathématiques, vol. 132, no. 4, pp. 292–312, 2008.

X. Chen, V. G. Romanovski, and W. Zhang, “Critical periods of perturbations of reversible rigidly isochronous centers,” Journal of Differential Equations, vol. 251, no. 6, pp. 1505–1525, 2011.

J.-P. Françoise, “The successive derivatives of the period function of a plane vector field,” Journal of Differential Equations, vol. 146, no. 2, pp. 320–335, 1998.

E. Freire, A. Gasull, and A. Guillamon, “Period function for perturbed isochronous centres,” Qualitative Theory of Dynamical Systems, vol. 3, no. 1, pp. 275–284, 2002.