Nonlinear Dynamics

, Volume 73, Issue 1–2, pp 665–670 | Cite as

Limit cycles and integrability in a class of system with a high-order critical point

Original Paper


In this paper, a class of polynomial differential system with high-order critical point are investigated. The system could be changed into a system with a 3-order nilpotent critical point. Finally, an example was given, with the help of computer algebra system MATHEMATICA, the first three quasi-Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there three small amplitude limit cycles created from the 3-order nilpotent critical point is also proved.


High-order critical point Nilpotent critical point Center Focus Bifurcation of limit cycle 



This research is partially supported by the National Nature Science Foundation of China (11201211, 61273012) and Nature Science Foundation of Shan-dong Province (ZR2012AL04).


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of ScienceLinyi UniversityLinyiP.R. China

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