Integrability and limit cycles in cubic Kukles systems with a nilpotent singular point

  • Feng LiEmail author
  • Shimin Li
Original Paper


In this paper, integrability problem and bifurcation of limit cycles for cubic Kukles systems which are assumed to have a nilpotent origin are investigated. A complete classification is given on the integrability conditions and proven to have a total of 7 cases. Bifurcation of limit cycles is also discussed; six or seven limit cycles can be obtained by two different perturbation methods. Integrability problem and bifurcation of limit cycles for the cubic Kukles systems with a nilpotent origin have been completely solved.


Kukles system Center–focus problem Nilpotent singular point Normal form Integrability 



We thank the anonymous reviewer for his/her valuable and detailed comments which have greatly improved our paper. This research was partially supported by the National Natural Science Foundation of China (Nos. 11601212, 2017A030313010,11501055), Natural Science Foundation of Shandong Province (No. ZR2018MA002) and Science and Technology Program of Guangzhou (No. 201707010426).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Linyi UniversityLinyiChina
  2. 2.Guangdong University of Finance and EconomicsGuangzhouChina

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