Skip to main content

Dynamic Pade' approximant and behavior singularities in nonlinear physico-chemical systems

  • Conference paper
  • First Online:
Bifurcation and Nonlinear Eigenvalue Problems

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 782))


Spatio-temporal phenomena in nonlinear systems have been found to be of great variety including periodic and chaotic structures.1 We shall find here that the method of Pade' approximants may be extended to describe some of these phenomena including the chemical center wave. Catastrophe (or more generally singularity) theory is also shown here to be of great utility in obtaining classification theorems for systems with multiple space or time scales. The idea of symmetry broken singularities is introduced. Finally, unlike in the case of ordinary differential systems, it is shown that in some cases the phenomena must be understood in terms of the geometry of function space via "behavior functionals". These ideas shall be introduced through a discussion of various physical problems including crystal growth and reaction diffusion systems.

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

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.95
Price excludes VAT (USA)
  • 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

Institutional subscriptions


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. P.Glansdorff and I. Prigogine, Thermodynamic Theory of Structure Stability, and Fluctuations (Wiley, N.Y., 1971).

    Google Scholar 

  2. H.Eyring ed., Periodicies in Chemistry and Biology, Theoret. Chem. 4, (Academic Press, 1978).

    Google Scholar 

  3. Adv. in Chem. Phys. XXXVIII ed. I. Prigogine and S.A. Rice (Interscience, John Wiley and Sons, 1978).

    Google Scholar 

  4. Y. Bottinga, A. Kudo and P. Weill, Amer. Mineral., 51, 792 (1966).

    Google Scholar 

  5. J. Chadam, D. Feinn, S. Hasse and P. Ortoleva, "Chemical Kinetic Theory of Periodic Zoning in Plageoclose Feldspars" (submitted for publication).

    Google Scholar 

  6. R. Thom, Stability, Structure and Morphogenesis (Benjamin, N.Y., 1972); A.E. Woodcock and T. Poston A Geometric Study of the Elementary Catastrophes, Lecture Notes in Mathematics, No. 373 (Springer, Berlin, 1974).

    Google Scholar 

  7. D. Feinn and P. Ortoleva, J. Chem. Phys. 67, 5 (1977).

    Article  MathSciNet  Google Scholar 

  8. M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities (Springer-Verlag, N.Y., 1973).

    Book  MATH  Google Scholar 

  9. S. Schmidt and P. Ortoleva, "Symmetry Breaking Singularities in Reaction-Diffusion Systems" (in preparation).

    Google Scholar 

  10. A.T. Winfree, Sci. Amer. 230, 82 (1974).

    Article  Google Scholar 

  11. See Refs. 3 and 4 in citation 10 below.

    Google Scholar 

  12. P. Ortoleva, J. Chem. Phys. 69, 300 (1978).

    Article  ADS  MathSciNet  Google Scholar 

  13. J. Dreitlein and M.-L. Smoes, J. Theor. Biol. 46, 559 (1974)

    Article  Google Scholar 

  14. N. Kopell and L. Howard, Stud. in Appl. Math, 52, 291 (1973)

    Article  MathSciNet  Google Scholar 

  15. P. Ortoleva and J. Ross, J. Chem. Phys. 60, 5090 (1974).

    Article  ADS  Google Scholar 

  16. N. Minorsky, Nonlinear Oscillations, (Kreiger, Huntington, N.Y., 1974).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Editor information

C. Bardos J. M. Lasry M. Schatzman

Rights and permissions

Reprints and permissions

Copyright information

© 1980 Springer-Verlag

About this paper

Cite this paper

Ortoleva, P. (1980). Dynamic Pade' approximant and behavior singularities in nonlinear physico-chemical systems. In: Bardos, C., Lasry, J.M., Schatzman, M. (eds) Bifurcation and Nonlinear Eigenvalue Problems. Lecture Notes in Mathematics, vol 782. Springer, Berlin, Heidelberg.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09758-7

  • Online ISBN: 978-3-540-38637-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics