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Bifurcation and hysteresis varieties for the thermal-chainbranching model with a negative modal parameter

  • Ian N. Stewart (a1)
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The theory of unfoldings of singularities, or ‘elementary catastrophe theory’ (Thom(10), Poston and Stewart(9), Golubitsky and Guillemin(3), Gibson(2)) has been generalized by Golubitsky and Schaeffer(5,6), providing a powerful method for analysing imperfect bifurcation. One recent application by Golubitsky, Keyfitz, and Schaeffer(4) concerns the ‘explosion peninsula’ in chemical reactions such as that between hydrogen and oxygen. They show that the ‘thermal-chainbranching model’ of Gray and Yang(7,8) is capable of reproducing the necessary qualitative features, a result for which only heuristic arguments had previously been available.

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(1)Callahan, J.Special bifurcations of the double cusp (preprint), (University of Warwick, 1978).
(2)Gibson, C. G.Singular points of Smooth Mappings (Research Notes in Mathematics 25, Pitman, London and Boston 1979).
(3)Golubitsky, M. and Guillemin, V.Stable Mappings and their Singularities (Springer-Verlag, Berlin, Heidelberg, New York, 1973).
(4)Golubitsky, M., Keyfitz, B. and Schaeffer, D. A Singularity Theory analysis of a Thermal-chainbranching model for the explosion peninsula (in the Press).
(5)Golubitsky, M. and Schaeffer, D.A theory for imperfect bifurcation via singularity theory. Commun. Pure Appl. Math. 32 (1979), 2198.
(6)Golubitsky, M. and Schaeffer, D.Imperfect bifurcation in the presence of symmetry. Commun. Math. Phys. 67 (1979), 205232.
(7)Gray, B. F.Theory of branching reactions with chain interaction. Trans. Faraday Soc. 66 (1970), 11181126.
(8)Gray, B. F. and Yang, C. H.On the unification of the thermal and chain theories of explosion limits. J. Phys. Chem. 69 (1965) 2747.
(9)Poston, T. and Stewart, I. N.Catastrophe Theory and its Applications (Pitman, London and Boston, 1978).
(10)Thom, R.Structural Stability and Morphogenesis (translation, Fowler, D. H.), (Benjamin, Reading Mass. 1975).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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