Skip to main content

Generation of limit cycles from separatrix polygons in the phase plane

Contributed Papers

Part of the Lecture Notes in Mathematics book series (LNM,volume 810)

Abstract

Two-dimensional autonomous systems may have solution curves which form separatrix polygons in the phase plane. These are polygons the corner points of which are saddle points and the sides of which are separatrices connecting these saddle points. They are structurally unstable, and in this paper we will study the change in the phase portrait due to arbitrary small changes in the right hand sides of these systems. In particular, attention will be given to the generation of limit cycles from these polygons. The number of limit cycles generated by a separatrix polygon is seen to be related to the eigenvalues of the locally linearized system in the saddle points. For separatrix polygons with two saddle points, criteria, involving these eigenvalues are given when exactly one or exactly two limit cycles can be generated. For separatrix polygons with three or more saddle points similar criteria are given to ensure the generation of at least one, two, or more (till n for a n sided polygon) limit cycles.

Keywords

  • Saddle Point
  • Phase Portrait
  • Limit Continuum
  • Stable Limit Cycle
  • Closed Path

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 (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andronov, A.A., Gordon, J.J., Leontovich, E.A. and Maier, A.G.; Qualitative Theory of Second-Order Dynamic Systems, Israel Program for Scientific Translation, Jerusalem, 1973.

    Google Scholar 

  2. Andronov, A.A., Gordon, J.J., Leontovich, E.A. and Maier, A.G.; Theory of Bifurcations of Dynamic Systems on a Plane, Israel Program for Scientific Translation, Jerusalem, 1971.

    Google Scholar 

  3. Comtet, L.; Advanced Combinatorics; the art of finite and infinite expansions; revised and enlarged edition, Reidel, Dordrecht, 1974.

    MATH  Google Scholar 

  4. Dulac, H.; Sur les cycles limites, Bull. Soc. Math. de France, Vol. 51, pp. 45–188, 1923.

    MathSciNet  MATH  Google Scholar 

  5. Jablonski; Théorie des permutations et des arrangements completes, Journal de Liouville, 8, pp. 331–349, 1892.

    MATH  Google Scholar 

  6. Leontovich, E.A.; On the generation of limit cycles from separatrices, Dokl. Akad. Nank. U.S.S.R., Vol. 78, no. 4, pp. 641–644, 1951.

    MathSciNet  Google Scholar 

  7. Reyn, J.W.; A stability criterion for separatrix polygons in the phase plane, Nieuw Archief voor Wiskunde (3), XXVII, pp. 238–254, 1979.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1980 Springer-Verlag

About this paper

Cite this paper

Reyn, J.W. (1980). Generation of limit cycles from separatrix polygons in the phase plane. In: Martini, R. (eds) Geometrical Approaches to Differential Equations. Lecture Notes in Mathematics, vol 810. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089983

Download citation

  • DOI: https://doi.org/10.1007/BFb0089983

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10018-8

  • Online ISBN: 978-3-540-38166-2

  • eBook Packages: Springer Book Archive