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Stability

  • Carl Ludwig Siegel
  • Jürgen K. Moser
Part of the Grundlehren der mathematischen Wissenschaften book series (CLASSICS, volume 187)

Abstract

We begin with the definition of stability and instability. Let ℜ be a topological space whose points we denote by þ, and let a be a certain point in ℜ. By a neighborhood here we will always mean a neighborhood of a in ℜ. Let þ1 = Sþ be a topological mapping of a neighborhood U1 onto a neighborhood B1 whereby a = Sa is mapped onto itself. The inverse mapping p-1 = S-1þ then carries B1 onto U1, and in general þn = Snþ (n = 0, ± 1, ± 2,…) is a topological mapping of a neighborhood Un onto a neighborhood Bn, having a as a fixed-point. For each point þ = þ0 in the intersection U1∩B1=M we construct the successive images þk+1 =Sþk (k = 0,1,…), as long as þk lies in U1 and similarly þ-k-1 =S-1þ-k as long as þ-k lies in B1. If the process terminates with a largest k+ 1 = n, then þ0,…, þn-1 all still lie in U1, but þn no longer does; similarly for the negative indices. In this way, to each þ in M there is associated a sequence of image points þk (k =…, — 1,0,1,…), which is finite, infinite on one side, or infinite on both sides.

Keywords

Normal Form Hamiltonian System Equilibrium Solution Formal Power Series Invariant Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Carl Ludwig Siegel
    • 1
  • Jürgen K. Moser
    • 2
    • 3
  1. 1.Mathematisches InstitutUniversität GöttingenDeutschland
  2. 2.ETH Zentrum, MathematikZürichSchweiz
  3. 3.Courant Institute of Mathematical SciencesNew YorkUSA

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