Skip to main content

Stability

  • Chapter
  • 1343 Accesses

Part of the Grundlehren der mathematischen Wissenschaften book series (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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-642-87284-6_3
  • Chapter length: 101 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   54.99
Price excludes VAT (USA)
  • ISBN: 978-3-642-87284-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   69.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Siegel, C.L., Moser, J.K. (1995). Stability. In: Lectures on Celestial Mechanics. Grundlehren der mathematischen Wissenschaften, vol 187. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-87284-6_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-87284-6_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58656-2

  • Online ISBN: 978-3-642-87284-6

  • eBook Packages: Springer Book Archive