Center-focus and Hopf bifurcation for a class of quartic Kukles-like systems
© Wu and Xu; licensee Springer. 2014
Received: 6 August 2014
Accepted: 1 September 2014
Published: 24 September 2014
In the present article, we solve the center-focus problem for a class of quartic Kukles-like systems with third-order nilpotent singularities and prove the existence of five limit cycles in the neighborhood of the origin.
One of the most classical problems in the qualitative theory of planar analytic differential systems is to characterize the local phase portrait at an isolated singular point. This problem has been solved except if the singular point is a center or a focus. The problem in distinguishing between a center and a focus is called the center-focus problem. Once we have made a distinction between a center and a focus, another problem is to find the number of limit cycles bifurcated from the focus.
where and are real analytic functions without constant and linear terms, defined in a neighborhood of the origin. Takens  proved that the Lyapunov system can be formally transformed into a generalized Liénard system. Moussu  found the normal form for analytic nilpotent centers. Stróżyna and Żoła̧dek  studied orbital normal forms for analytic planar vector fields with nilpotent singularity. Álvarez and Gasull  proved that the generalized Liénard system could be simplified even more by a reparameterization of the time. At the same time, Giacomini et al. [5, 6] showed that the analytic nilpotent systems with a center can be expressed as limit of non-degenerate systems with a center. The conditions of center and isochronous center at the origin for a class of non-analytic quintic systems are studied in .
Remember that the so-called stability problem and the center problem can be solved for smooth non-degenerate critical points via the Lyapunov constants. On the other hand the same problem but for nilpotent singular point is far to be solved in general, due to the invalidation of classical methods. In 2011, Liu and Li  dealt with the integral factor method for solving the above mentioned problems of third-order nilpotent singularities. This method is based into a different way of computing the so-called quasi-Lyapunov constants. Using the integral factor method,  investigated center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of septic polynomial differential systems.
The rest of paper is organized as follows. In Section 2, we give some preliminary knowledge presented in , which is helpful throughout the paper. In Section 3, we compute the first several quasi-Lyapunov constants at the third-order nilpotent singular point of system (1.2) and provide sufficient and necessary conditions in order that system (1.2) have a center in a neighborhood of the origin. We end this paper in Section 4 by applying Theorem 2.1 to generate limit cycles for system (1.2).
2 Some preliminary results
Before state our results we need to introduce some well-known definitions, lemmas, and theorems.
where n is a positive integer.
we say that the origin is a high-order singular point of system (2.1) with the multiplicity m.
By using the results in , we attain the following conclusion.
Lemma 2.1 The origin of system (2.1) is a three-order singular point which is a saddle point or a center, if and only if , .
Otherwise, by letting , and making the transformation , , we obtain the mentioned result.
We have the following result.
where is a polynomial of , , () with rational coefficients.
It is differential from the center-focus problem for the elementary critical points, we know from Lemma 2.2 that when for the first non-zero , k is an even integer.
For any positive integer m, is called the m th focal value of system (2.5) in the origin.
If , then the origin of system (2.5) is called one-order weakened focus. In addition, if there is an integer , such that , but , then the origin is called a m-order weakened focus of system (2.5).
If for all positive integer m, we have , then the origin of system (2.5) is called a center.
We say that is equivalent to , denoted by .
If and for all positive integers m, , we say that the function sequences and are equivalent, denoted by .
We know from Lemma 2.2 and Definition 2.2 that for the sequence , , we have , .
For an integer k, letting be the k-order focal value of the origin of system (2.24).
then there exist two positive numbers and , such that for , , in a neighborhood of the origin, system (2.24) has at most m limit cycles which enclose the origin (an elementary node) . In addition, under the above conditions, there exist , , such that when , , there exist exactly m limit cycles of (2.24) in a small neighborhood of the origin.
We give the following key results, which define the quasi-Lyapunov constants and provide a way of computing them.
We see from (2.27) and (2.30) that when (2.8) holds, .
Definition 2.4 For system (2.5), is called the m th quasi-Lyapunov constant of the origin.
where for all k, is a k-homogeneous polynomial in x, y and .
It is easy to see that (2.36) is linear with respect to the function M, so that we can easily find the following recursive formulas for the calculation of and .
Clearly, the recursive formulas presented by Theorem 2.4 is linear with respect to all . Accordingly, it is convenient to realize the computations of quasi-Lyapunov constants by using a computer algebraic system like Mathematica.
3 Quasi-Lyapunov constants and center conditions
It is easy to see that the origin of system (1.2) is a third-order nilpotent singular point which is a center or a focus. Now we start the preparation of computing the quasi-Lyapunov constants at the origin of system (1.2).
One of our main results is the following.
- (I)By letting(3.4)
where in the expression of , we have already let , .
the vector field of system (3.6) is symmetric with respect to axis x, so the origin is a center.
the vector field of system (3.7) is symmetric with respect to axis y, so the origin is also a center. □
4 Limit cycle bifurcation
Next we will prove that the third-order nilpotent singular is at most a weak focus of order five, moreover, based on this conclusion, the perturbed system of (1.2) can produce five limit cycles enclosing an elementary node at the origin.
, easily yields.
The above considerations imply the following main result.
Theorem 4.2 If the origin of system (1.2) is a five-order weak focus, for , making a small perturbation to the coefficients of system (1.2), then, for system (4.2), in a small neighborhood of the origin, there exist exactly five small amplitude limit cycles enclosing the origin , which is an elementary node.
where , are arbitrary non-zero real constants, s (≥3) is a natural number.
Then, for , system (4.2) has five limit cycles in a small neighborhood of the origin, where , .
The authors are grateful to both reviewers for their helpful suggestions and comments. This work is supported in part by the National Nature Science Foundation of China (11101126 and 11261010) and Scientific Research Foundation for Doctoral Scholars of HAUST (09001524).
- Takens F: Singularities of vector fields. Publ. Math. IHÉS 1974, 43: 47-100. 10.1007/BF02684366MathSciNetView ArticleMATHGoogle Scholar
- Moussu R: Symétrie et forme normale des centres et foyers dégénérés. Ergod. Theory Dyn. Syst. 1982, 2: 241-251.MathSciNetView ArticleMATHGoogle Scholar
- Stróżyna E, Żołądek H: The analytic and formal normal form for the nilpotent singularity. J. Differ. Equ. 2002, 179: 479-537. 10.1006/jdeq.2001.4043View ArticleMathSciNetMATHGoogle Scholar
- Álvarez MJ, Gasull A: Generating limit cycles from a nilpotent critical point via normal forms. J. Math. Anal. Appl. 2006, 318: 271-287. 10.1016/j.jmaa.2005.05.064MathSciNetView ArticleMATHGoogle Scholar
- Giacomini H, Giné J, Llibre J: The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems. J. Differ. Equ. 2006, 227(2):406-426. 10.1016/j.jde.2006.03.012View ArticleMathSciNetMATHGoogle Scholar
- Giacomini H, Giné J, Llibre J: Corrigendum to: ‘The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems’ [J. Differential Equations 227 (2006), no. 2, 406-426]. J. Differ. Equ. 2007, 232(2):702. 10.1016/j.jde.2006.10.004View ArticleGoogle Scholar
- Li F, Qiu J, Li J: Bifurcation of limit cycles, classification of centers and isochronicity for a class of non-analytic quintic systems. Nonlinear Dyn. 2014, 76(1):183-197. 10.1007/s11071-013-1120-4MathSciNetView ArticleMATHGoogle Scholar
- Liu Y, Li J: On third-order nilpotent critical points: integral factor method. Int. J. Bifurc. Chaos 2011, 21: 1293-1309. 10.1142/S0218127411029161View ArticleMathSciNetMATHGoogle Scholar
- Li F, Liu Y, Li H: Center conditions and bifurcation of limit cycles at three-order nilpotent critical point in a septic Lyapunov system. Math. Comput. Simul. 2011, 81(12):2595-2607. 10.1016/j.matcom.2011.05.001View ArticleMathSciNetMATHGoogle Scholar
- Amelkin VV, Lukashevich NA, Sadovskii AN: Nonlinear Oscillations in the Second Order Systems. BGU Publ., Minsk; 1982. (in Russian)Google Scholar
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