Skip to main content

Instability of dynamical systems with several degrees of freedom

  • Chapter
Collected Works

Part of the book series: Vladimir I. Arnold - Collected Works ((ARNOLD,volume 1))

Abstract

1. Recent progress in perturbation enables us to find many conditionally periodic motions in every nonlinear dynamical system which is close to an integrable system (see [1,2]. The stability of all the motions of the system follows from these results only when the dimension of the phase space is ≤ 4. The purpose of the present note is to give an example (3) of a system with a 5-dimensional phase space which satisfies all the conditions of [1,2] but is nonstable. The secular changes l 2 in the system (3) have the velocity exp (-1/√ε) and consequently cannot be dealt with by any approximation of the classical theory of perturbations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 98 (1954), 527. MR 16, 924.

    MATH  MathSciNet  Google Scholar 

  2. V. I. Arnol'd, Uspehi Mat. Nauk 18 (1963), no. 5(113), 13, 91.

    Google Scholar 

  3. K. A. Sitnikov, Dokl. Akad. Nauk SSSR 133 (1960), 303 = Soviet Physics Dok!. 5 (1961), 647. MR 23 #B435.

    MathSciNet  Google Scholar 

  4. An. M. Leontovic, Dokl. Akad. Nauk SSSR 145 (1962), 523 = Soviet Math. Dokl. 3 (1962), 1049. MR 25 #2287.

    MathSciNet  Google Scholar 

  5. H. Poincare; Les methodes nouvelles de la mecanique celeste, 3, Paris, 1899.

    Google Scholar 

  6. V. K. Mel'nikov, Trudy Moskov. Mat. Oble.- 12 (1963), 3 = Trans. Moscow Math. Soc. 12 (1963) (to appear). MR 27 #5981.

    Google Scholar 

Download references

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

(2009). Instability of dynamical systems with several degrees of freedom. In: Givental, A., et al. Collected Works. Vladimir I. Arnold - Collected Works, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01742-1_26

Download citation

Publish with us

Policies and ethics