Skip to main content

The geometry of the elementary catastrophes (1). the cuspoids

Part of the Lecture Notes in Mathematics book series (LNM,volume 373)

Abstract

The Elementary Catastrophes arise as stable singularities in a system of potentials parameterized by a manifold C (the Control Space) on a manifold X (the Behavior Space) and represented by a smooth map:

$$V:CxX \to R.$$

The present paper describes the geometry of these singularities for potentials of the type:

$$V = \frac{x}{{n + 2}}n + 2 + A\frac{x}{n}n + B\frac{x}{{n - 1}}n - 1 + ... + Rx.$$

and termed the Cuspoids.

Keywords

  • Control Space
  • Stable Singularity
  • Behavior Space
  • Cusp Catastrophe
  • Common Limb

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 work was begun when the authors were both at the Mathematics Institute, University of Warwick, Coventry CV4 7AL, Warwickshire, England.

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.

Bibliography

  1. Poston, T. & Woodcock, A. E. R., Zeeman's Catastrophe Machine, Proc. Camb. Phil. Soc., to appear.

    Google Scholar 

  2. Zeeman, E. C., Catastrophe Machines, Towards a Theoretical Biology IV, ed, C. H. Waddington.

    Google Scholar 

  3. Thom, R., Stabilité Structurelle et Morphogéneśe, Benjamin-Addison-Wesley. (1973).

    Google Scholar 

  4. Woodcock, A. E. R. & Poston, T., The Geometrical Properties of the Hyperbolic & Elliptic Umbilics, (this volume).

    Google Scholar 

  5. Woodcock, A. E. R. & Poston, T., The Geometrical Properties of the Parabolic Umbilic, (this volume).

    Google Scholar 

  6. Woodcock, A. E. R. & Poston, T., The Geometrical Properties of the Reduced Double Cusp, (this volume).

    Google Scholar 

  7. Godwin, N., Ph.D. Thesis, University of Warwick (1970).

    Google Scholar 

  8. Godwin, N., Thom's Parabolic Umbilic, I.H.E.S. No. 40, p.117–138, (1972).

    Google Scholar 

  9. Godwin, N., The Compactified Parabolic Umbilic, Proc. Lond. Math. Soc., to appear.

    Google Scholar 

  10. Woodcock, A. E. R., The Elementary Catastrophes, A Film (to appear).

    Google Scholar 

  11. Edwards, An Elementary Treatise on the Differential Calculus, Macmillan, London, 1892.

    Google Scholar 

  12. Thom, R., Sur La théorie des enveloppes, Journ. de Math t.XLI-Fasc. 2, 1962.

    Google Scholar 

  13. Godwin, N. & Poston, T., On the Maxwell set of catastrophes with finite noise, to appear.

    Google Scholar 

  14. Woodcock, A, E. R., Stereographic Reconstructions of the Catastrophe Mfds of the Cuspoid Catastrophes (this volume).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 1974 Springer-Verlag

About this chapter

Cite this chapter

Woodcock, A.E.R., Poston, T. (1974). The geometry of the elementary catastrophes (1). the cuspoids. In: A Geometrical Study of the Elementary Catastrophes. Lecture Notes in Mathematics, vol 373. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0068968

Download citation

  • DOI: https://doi.org/10.1007/BFb0068968

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06681-1

  • Online ISBN: 978-3-540-37941-6

  • eBook Packages: Springer Book Archive