Abstract and Applied Analysis
Volume 2012 (2012), Article ID 720830, 28 pages
http://dx.doi.org/10.1155/2012/720830
Research Article

Limit Cycle Bifurcations from a Nilpotent Focus or Center of Planar Systems

Maoan Han1,2 and Valery G. Romanovski2,3

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Faculty of Natural Science and Mathematics, University of Maribor, 2000 Maribor, Slovenia
3Center for Applied Mathematics and Theoretical Physics, University of Maribor, 2000 Maribor, Slovenia

Received 11 August 2012; Accepted 28 October 2012

Academic Editor: Jaume Giné

Copyright © 2012 Maoan Han and Valery G. Romanovski. 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 analytic properties of the Poincaré return map and generalized focal values of analytic planar systems with a nilpotent focus or center. We use the focal values and the map to study the number of limit cycles of this kind of systems and obtain some new results on the lower and upper bounds of the maximal number of limit cycles bifurcating from the nilpotent focus or center. The main results generalize the classical Hopf bifurcation theory and establish the new bifurcation theory for the nilpotent case.