Skip to main content

Part of the book series: Modern Birkhäuser Classics ((MBC,volume 164))

  • 421 Accesses

Abstract

As an introduction to the theory of bifurcations, in this chapter we want to consider individual vector fields, i.e., families of vector fields with a 0-dimensional parameter space. We will present two fundamentals tools: the desingularization and the asymptotic expansion of the return map along a limit periodic set. In the particular case of an individual vector field these techniques give the desired final result: the desingularization theorem says that any algebraically isolated singular point may be reduced to a finite number of elementary singularities by a finite sequence of blow-ups. If X is an analytic vector field on S 2, then the return map of any elementary graphic has an isolated fixed point. As a consequence, in this special case there is no accumulation of limit cycles in the phase space. In other words, the cyclicity of each limit periodic set is less than one and any analytic vector field on the sphere has only a finite number of limit cycles.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
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.

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer Basel AG

About this chapter

Cite this chapter

Roussarie, R. (1998). The 0-Parameter Case. In: Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem. Modern Birkhäuser Classics, vol 164. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8798-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8798-4_3

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9778-5

  • Online ISBN: 978-3-0348-8798-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics