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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 | References | Similar Articles | Additional Information

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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  • 1. Lamberto Cesari, Asymptotic behavior and stability problems in ordinary differential equations, 3rd ed., Springer-Verlag, New York-Heidelberg, 1971. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 16. MR 0350089
  • 2. R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist-theorem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 1, 79–95. MR 937537
  • 3. Bin Liu, The stability of the equilibrium of a conservative system, J. Math. Anal. Appl. 202 (1996), no. 1, 133–149. MR 1402592,
  • 4. Wilhelm Magnus and Stanley Winkler, Hill’s equation, Dover Publications, Inc., New York, 1979. Corrected reprint of the 1966 edition. MR 559928
  • 5. J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962 (1962), 1–20. MR 0147741
  • 6. Jürgen Moser, Stable and random motions in dynamical systems, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. With special emphasis on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J; Annals of Mathematics Studies, No. 77. MR 0442980
  • 7. Rafael Ortega, The stability of the equilibrium of a nonlinear Hill’s equation, SIAM J. Math. Anal. 25 (1994), no. 5, 1393–1401. MR 1289144,
  • 8. M. B. Sevryuk, Reversible systems, Lecture Notes in Mathematics, vol. 1211, Springer-Verlag, Berlin, 1986. MR 871875
  • 9. Carl Ludwig Siegel and Jürgen K. Moser, Lectures on celestial mechanics, Springer-Verlag, New York-Heidelberg, 1971. Translation by Charles I. Kalme; Die Grundlehren der mathematischen Wissenschaften, Band 187. MR 0502448

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Additional Information

Bin Liu
Affiliation: Department of Mathematics, Peking University, Beijing 100871, China

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