About this Journal Submit a Manuscript Table of Contents
Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 403836, 6 pages
http://dx.doi.org/10.1155/2013/403836
Research Article

Bifurcation of Limit Cycles and Center Conditions for Two Families of Kukles-Like Systems with Nilpotent Singularities

1School of Mathematics and Computer, Hunan Normal University, Changsha, Hunan 410083, China
2School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, Henan 471023, China

Received 26 May 2013; Revised 26 August 2013; Accepted 28 August 2013

Academic Editor: Mark McKibben

Copyright © 2013 Peiluan Li 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 solve theoretically the center problem and the cyclicity of the Hopf bifurcation for two families of Kukles-like systems with their origins being nilpotent and monodromic isolated singular points.

1. Introduction

One of the main open problems in the qualitative theory of planar analytic differential systems is the case when the singular point is monodromic (the orbits move around the singular point). A monodromic point of an analytic system is either a center (i.e., a singular point with a punctures neighborhood filled with periodic orbits) or a focus (i.e., a singular point with a neighborhood where all the orbits are spirals which arrive at the equilibrium point in forward or backward time). The problem of distinguishing when a monodromic singular point is either a center or a focus is called the center problem.

The so-called center problem for planar vector fields has been intensively and extensively studied over the last century and is also closely related to Hilbert’s 16th problem. The singular point can be elementary or not in the sense whether the corresponding Jacobian matrix has zero eigenvalues. If the eigenvalues of the quoted matrix are imaginary with real part null, the singular point may be a focus or a center, which is known as the celebrated Poincaré-Lyapunov center problem and has been theoretically solved by Poincaré [1] and Liapunov [2]. If the matrix of the linear part at the singular point has its two eigenvalues equal to zero, but it is not identically null, by Andreev [3] we know what is the behavior of the solutions in a neighborhood of the singular point, except if it is a center of a focus (nilpotent center problem).

Yet for the bifurcation of limit cycles and center problem of nilpotent singular points in a planar vector field, its intrinsic dynamics is still far away from understanding due to the complexity and technical difficulties. Therefore, it is natural to restrict our study to nilpotent singularities. An analytic system of differential equations in the plane having an isolated nilpotent singularity, in some suitable coordinates, can be written as with and real analytic functions without constant nor linear terms defined in a certain neighborhood of the singularity.

The study of nilpotent singularities and their unfolding for vector fields is important not only for mathematical interest but also for practical reasons. To solve the finite cyclicity problem in the second part of Hilbert’s 16th problem, Dumortier et al. [4, 5] presented an impressive list of 121 graphics that occur in quadratic systems, among which many are nilpotent cases, studied in [611]. On the other hand, the center problem for nilpotent singularities is another interesting issue and has been partially solved; see [3, 1219], and so forth.

Liu and Li [20] originated a method in studying the center problem of third-order nilpotent singularities by integral factor method, where a new definition of the focal values, quasi-Lyapunov constants, is provided. At the same time, the equivalence of quasi-Lyapunov constants and focal values is proved. A linear recursive formula for computing quasi-Lyapunov constants is also derived. Afterward, they proved that if the third-order nilpotent origin is a -order weakened focus, then, by a small perturbation for the unperturbed system, there exist limit cycles in a neighborhood of the origin. Meanwhile, the origin becomes an elementary critical point and two complex singular points.

In the present paper, we are interested in the study of two families of Kukles-like systems with third-order nilpotent part of the form and, respectively, nonlinear part given by quartic and quintic homogeneous polynomial with an isolated singular point at the origin; that is,

This paper will be organized as follows. In Section 2 we summarize some general definitions and results about third-order nilpotent singularities. In Section 3 we present the conditions for third-order nilpotent singularities to be a center and give the proof of the cyclicity for system (2). In the last section, we present the parallel results for system (3).

2. The Formula to Compute the Quasi-Lyapunov Constants

The origin is a third-order monodromic critical point if and only if the system is of the following form:

The quasi-Lyapunov constant is defined and a way of computing them is provided in [20].

Theorem 1. For any positive integer and a given number sequence one can construct successively the terms with the coefficients satisfying of the formal series such that where for all , is a -homogeneous polynomial of and .

It is easy to see that (7) is linear with respect to the function , so that we can easily find in [20] the following recursive formulae for the calculation of and .

Theorem 2. For , in (6) and (7), can be uniquely determined by the recursive formula For , can be uniquely determined by the recursive formula where Notice that in (10), one sets

We see from Theorem 1 that if the origin of system (4) is -class or -class, then, by choosing , such that we can obtain a solution group of of (12); thus, we have

Clearly, the recursive formulas presented by Theorem 2 are linear with respect to all . Therefore, it is convenient to realize the computations of quasi-Lyapunov constants by using computer algebraic system like MATHEMATICA.

3. Center Conditions and Limit Cycles of System (2)

By Theorems 1 and 2, we directly have the following.

Lemma 3. Assume that is a natural number. One can derive a power series (6) for system (2) under which (7) is satisfied, where In addition, for any natural numbers , is given by the following recursive formula: and, for any natural number , is given by the following recursive formula:

The straightforward computing according to Lemma 3 with MATHEMATICA gives

To make , we will consider two different cases: and .

Case 1 (). Taking expression (13) into account, the first four quasi-Lyapunov constants of system (2) are given by

Now we can prove the following result.

Theorem 4. In this case, the origin of system (2) is a center if and only if the first four quasi-Lyapunov constants vanish; that is, one of the following two series is satisfied:

Proof. It is direct to obtain conditions (19) and (20) by imposing , .
On the other hand, when condition (19) holds, system (2) goes over to whose vector field is symmetric with respect to the -axis. When condition (20) holds, system (2) goes over to whose vector field is symmetric with respect to the -axis. This ends the proof.

Case 2 (). Taking expression (13) into account, the first five quasi-Lyapunov constants of system (2) are given by

For the same reason as before, we obtain the following.

Theorem 5. In this case, the origin of system (2) is a center if and only if the first five quasi-Lyapunov constants vanish; that is, the following series is satisfied:

Proof. It is direct to obtain conditions (24) by imposing , .
On the other hand, when condition (24) holds, system (2) goes over to whose vector field is symmetric with respect to the -axis. Then, we have proved the theorem.

It is not hard to see that condition (20) is a particular case of condition (24); in summary, the following result characterizes the center of system (2).

Theorem 6. System (2) has a center at the origin if and only if one of the following two sets of conditions holds:
,
.

Remark 7. Indeed, in the light of the theory of [21], the origin is not only a center but also an analytic center under condition (I).

Referring to the expressions given in (18) and (23), we have the following.

Lemma 8. The origin of system (2) is a fine focus of maximum order five. Further, it is of order five if and only if

Consider the perturbed system of (2)

Theorem 9. Under a small perturbation of system (2), for a small parameter , in a neighborhood of the origin of system (27), there exist exactly 5 small amplitude limit cycles enclosing the elementary node .

The next example shows this fact.

Example 10. Take Straightforward computations by using expression (23) give the first 5 quasi-Lyapunov constants of the origin of system (27): Thereby, for , system (27) has 5 limit cycles in a small neighborhood of the origin, where ,  .

4. Center Conditions and Limit Cycles of System (3)

Similar to Lemma 3, we have the following.

Lemma 11. Assume that is a natural number. One can derive a power series (6) for system (3) under which (7) is satisfied, where In addition, for any natural numbers , is given by the following recursive formula: and, for any natural number , is given by the following recursive formula:

Taking expression (13) into account, the first four quasi-Lyapunov constants of system (3) are given by

Hence, we arrive at the following.

Theorem 12. System (3) has a center at the origin if and only if the following set of conditions holds:
.

Proof. It is direct to obtain condition (III) by imposing , .
On the other hand, when condition (III) is satisfied, system (3) reduces to whose vector field is symmetric with respect to the origin. The claim follows.

Remark 13. Indeed, in the light of the theory of [21], the origin is not only a center but also an analytic center under condition (III).

We obtain the following by expression (33).

Lemma 14. The origin of system (3) is a fine focus of maximum order four. Further, it is of order four if and only if

Consider the perturbed system of (3):

Theorem 15. Under a small perturbation of system (3), for a small parameter , in a neighborhood of the origin of system (36), there exist exactly 4 small amplitude limit cycles enclosing the elementary node .

Example 16. Take Straightforward computations by using expression (33) give the first 4 quasi-Lyapunov constants of the origin of system (36): Thereby, for , system (36) has 4 limit cycles in a small neighborhood of the origin, where ,  .

Acknowledgments

This work is supported in part by the National Nature Science Foundation of China (11101126 and 11271110) and Scientific Research Start-up Fund for Doctoral Scholars of HAUST (09001524).

References

  1. H. Poincaré, “Mémoire sur les courbes définies par les équations différentielles,” Journal de Mathématiques, vol. 37, pp. 375–422, 1881.
  2. M. A. Liapunov, Problème Général De La Stabilité Du Mouvement, vol. 17 of Annals of Mathematics Studies, 1947.
  3. A. F. Andreev, “Investigation of the behaviour of the integral curves of a system of two differential equations in the neighbourhood of a singular point,” American Mathematical Society Translations, vol. 8, pp. 183–207, 1958. View at MathSciNet
  4. F. Dumortier, R. Roussarie, and C. Rousseau, “Hilbert's 16th problem for quadratic vector fields,” Journal of Differential Equations, vol. 110, no. 1, pp. 86–133, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  5. F. Dumortier, R. Roussarie, and C. Rousseau, “Elementary graphics of cyclicity 1 and 2,” Nonlinearity, vol. 7, no. 3, pp. 1001–1043, 1994. View at MathSciNet
  6. F. Dumortier, R. Roussarie, and J. Sotomayor, “Bifurcations of cuspidal loops,” Nonlinearity, vol. 10, no. 6, pp. 1369–1408, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  7. Yu. S. Il'yashenko, “Finiteness theorems for limit cycles,” Russian Mathematical Surveys, vol. 40, pp. 143–200, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  8. M. El Morsalani and A. Mourtada, “Degenerate and non-trivial hyperbolic 2-polycycles: appearance of two independant Écalle-Roussarie compensators and Khovanskiĭ's theory,” Nonlinearity, vol. 7, no. 6, pp. 1593–1604, 1994. View at MathSciNet
  9. R. Roussarie, “On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields,” Boletim da Sociedade Brasileira de Matemática, vol. 17, no. 2, pp. 67–101, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  10. R. Roussarie, “Cyclicité finie des lacets et des points cuspidaux,” Nonlinearity, vol. 2, no. 1, pp. 73–117, 1989. View at MathSciNet
  11. H. Zhu and C. Rousseau, “Finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type,” Journal of Differential Equations, vol. 178, no. 2, pp. 325–436, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  12. M. J. Álvarez and A. Gasull, “Monodromy and stability for nilpotent critical points,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 15, no. 4, pp. 1253–1265, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  13. M. J. Álvarez and A. Gasull, “Generating limit cycles from a nilpotent critical point via normal forms,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 271–287, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  14. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maĭer, Qualitative Theory of Second-Order Dynamic Systems, Wiley, New York, NY, USA, 1973. View at MathSciNet
  15. J. Chavarriga, H. Giacomin, J. Giné, and J. Llibre, “Local analytic integrability for nilpotent centers,” Ergodic Theory and Dynamical Systems, vol. 23, no. 2, pp. 417–428, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  16. W. W. Farr, C. Li, I. S. Labouriau, and W. F. Langford, “Degenerate Hopf bifurcation formulas and Hilbert's 16th problem,” SIAM Journal on Mathematical Analysis, vol. 20, no. 1, pp. 13–30, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  17. V. Mañosa, “On the center problem for degenerate singular points of planar vector fields,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 4, pp. 687–707, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  18. R. Moussu, “Symétrie et forme normale des centres et foyers dégénérés,” Ergodic Theory and Dynamical Systems, vol. 2, no. 2, pp. 241–251, 1982. View at MathSciNet
  19. F. Takens, “Singularities of vector fields,” Institut des Hautes Études Scientifiques, no. 43, pp. 47–100, 1974. View at MathSciNet
  20. Y. Liu and J. Li, “On third-order nilpotent critical points: integral factor method,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 21, no. 5, pp. 1293–1309, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  21. T. Liu, L. Wu, and F. Li, “Analytic center of nilpotent critical points,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 22, Article ID 1250198, 8 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet