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

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Bifurcation and Nonlinear Eigenvalue Problems

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

Abstract

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.

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References

  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 

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C. Bardos J. M. Lasry M. Schatzman

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© 1980 Springer-Verlag

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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. https://doi.org/10.1007/BFb0090436

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  • DOI: https://doi.org/10.1007/BFb0090436

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  • Print ISBN: 978-3-540-09758-7

  • Online ISBN: 978-3-540-38637-7

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