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The cyclicity of period annuli of degenerate quadratic Hamiltonian systems with elliptic segment loops

Published online by Cambridge University Press:  07 May 2002

SHUI-NEE CHOW ,
CHENGZHI LI  and
YINGFEI YI
SHUI-NEE CHOW
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA and Department of Mathematics, National University of Singapore, Singapore 119260
CHENGZHI LI
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China
YINGFEI YI
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA and Department of Computational Science, National University of Singapore, Singapore 119260
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Abstract

We study the cyclicity of period annuli (or annulus) for general degenerate quadratic Hamiltonian systems with an elliptic segment or a saddle loop, under quadratic perturbations. By using geometrical arguments and studying the respective Abelian integral based on the Picard–Fuchs equation, it is shown that the cyclicity of period annuli (or annulus) for such systems equals two. This result, together with those of Gavrilov and Iliev (2000), Iliev (1996), Zhao et al (2000) and Zhao and Zhu (2001) gives a complete solution to the infinitesimal Hilbert 16th problem in the case of degenerate quadratic Hamiltonian systems under quadratic perturbations.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 22 , Issue 2 , April 2002 , pp. 349 - 374
Copyright
2002 Cambridge University Press

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