Local bifurcation of limit cycles and center problem for a class of quintic nilpotent systems
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Local bifurcation of limit cycles and center problem for a class of quintic nilpotent systems Yusen Wu1*, Luju Liu1 and Feng Li2 * Correspondence: [email protected] 1 School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, Henan, P. R. China Full list of author information is available at the end of the article
Abstract For a class of fifth degree nilpotent system, the shortened expressions of the first eight quasi-Lyapunov constants are presented. It is shown that the origin is a center if and only if the first eight quasi-Lyapunov constants are zeros. Under a small perturbation, the conclusion that eight limit cycles can be created from the eightorder weakened focus is vigorously proved. It is different from the usual Hopf bifurcation of limit cycles created from an elementary critical point. Mathematical Subject Classification: 34C07; 37G10. Keywords: quintic nilpotent system, quasi-Lyapunov constant, bifurcation of limit cycles, center problem
1 Introduction and statement of the main results Two main open problems in the qualitative theory of planar analytic differential systems are characterizing the local phase portrait at an isolated critical point and the determination and distribution of limit cycles. Recall that a critical point is said to be of focus-center type if it is either a focus or a center. In what follows, this problem is called the focus-center problem or the monodromy problem, which is usually done by the blow-up procedure. Of course, if the linear part of the critical point is non-degenerate (i.e., its determinant does not vanish) the characterization is well known. The problem has also been solved when the linear part is degenerate but not identically null, see [1-3]. On the other hand, once we know that a critical point is of focus-center type, one comes across another classical problem, usually called the center problem or the stability problem, that is of distinguishing a center from a focus. The Poincaré-Lyapunov theory was developed to solve this problem in the case where the critical point is non-degenerate, see [4,5]. From a theoretical viewpoint, the study of this problem for a concrete family of differential equations goes through the calculation of the so-called Lyapunov constants, which gives the necessary conditions for center, see [6,7]. To completely solve the stability problem of polynomial systems of a fixed degree, although the Hilbert basis theorem asserts that the number of needed Lyapunov constants is finite, which is the number is still open. Probably the most studied degenerated critical points are the nilpotent critical points. For these points, zero is a double eigenvalue of the differential matrix, but it is not identically zero. Nevertheless, given an analytic system with a nilpotent monodromic critical point it is not an easy task to know if it is a center or a focus. Analytic systems having a nilpotent critical point at the origin were studied by Andreev [1] in order to obtain their © 2012
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