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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The stability of the equilibrium
of reversible systems


Author: Bin Liu
Journal: Trans. Amer. Math. Soc. 351 (1999), 515-531
MSC (1991): Primary 58F13
MathSciNet review: 1422614
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Abstract: In this paper, we consider the system

\begin{displaymath}\dot x=a(t)y^{2m+1}+f_1(x,y,t),\quad\dot y=-b(t)x^{2n+1}+f_2(x,y,t),\end{displaymath}

where $m,n\in \mathbf Z_+$, $m+n\ge 1$, $a(t)$ and $b(t)$ are continuous, even and 1-periodic in the time variable $t$; $f_1$ and $f_2$ are real analytic in a neighbourhood of the origin $(0,0)$ of $(x,y)$-plane and continuous 1-periodic in $t$. We also assume that the above system is reversible with respect to the involution $G\colon(x,y)\mapsto(-x,y)$. A sufficient and necessary condition for the stability in the Liapunov sense of the equilibrium $(x,y)=(0,0)$ is given.


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Additional Information

Bin Liu
Affiliation: Department of Mathematics, Peking University, Beijing 100871, China
Email: bliu@pku.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-99-01965-0
PII: S 0002-9947(99)01965-0
Keywords: Reversible systems, stability, invariant curves
Received by editor(s): March 17, 1995
Received by editor(s) in revised form: December 4, 1995
Additional Notes: Research was supported by the NNSF of China
Article copyright: © Copyright 1999 American Mathematical Society