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Topological bifurcation for the double cusp polynomial

Published online by Cambridge University Press:  24 October 2008

A. N. Godwin
A. N. Godwin
Affiliation:
Lanchester Polytechnic, Rugby
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Extract

In his work on elementary catastrophes Zeeman(1) has considered what he has named as the double cusp catastrophe. This catastrophe is defined by the unfolding of the two variable polynomial x4 + y4. Using Mather's results (2) on stability of singular germs of C∞ maps we can find an expression for the unfolding. The eight dimensional unfolding can then be considered as a polynomial in two variables with eight parameters.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 77 , Issue 2 , March 1975 , pp. 293 - 312
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

REFERENCES

(1)Zeeman, E. C.Applications of catastrophe theory. Proc. London. Math. Soc. (To appear.)Google Scholar
(2)Mather, J. N.Stability of C∞ mapping, IV.Google Scholar
(3)Kaplan, W.Regular curve-families filling the plane, I. Duke Math. J. 7 (1940), 154–185.CrossRefGoogle Scholar
(4)Kaplan, W.Regular curve-families filling the plane, II. Duke Math. J. 8 (1941), 11–46.CrossRefGoogle Scholar
(5)Kaplan, W.The structure of a curve-family on a surface in the neighbourhood of an isolated singularity. Am. J. Math. 64 (1942), 1–35.CrossRefGoogle Scholar
(6)Bendixson, I.Sur les courbes defines par des equation differentielles. Acta Mathematica 24 (1901), 1–88.CrossRefGoogle Scholar
(7)Kuo, T. C.A complete determination of C˚ – sufficiency in J' (2, 1). Invent. Math. 8 (1969), 226–235.CrossRefGoogle Scholar
(8)Hodge, W. V. D. and Pedoe, D.Methode of algebraic geometry, vol. I (Cambridge, 1947).Google Scholar
(9)Godwin, A. N. Elementary catastrophes Ph.D. Thesis Warwick University (1971).Google Scholar
(10)Kuo, T. C., On C˚ – sufficiency of jets of potential functions. Topology 8 (1969), 167–171.CrossRefGoogle Scholar
(11)Lu, Y. C.Sufficiency of jets in Jn (2, 1) via decomposition. Invent. Math. 10 (1970), 119–127.CrossRefGoogle Scholar
(12)Walker, R. J.Algebraic curves (Dover; New York 1950).Google Scholar
(13)Thom, R.Stabilité structurelle et morphogenèse, (Benjamin). (To appear.)CrossRefGoogle Scholar
(14)Sansone, G. and Conti, R.Nonlinear differential equations (Pergamon Press; New York, 1964).Google Scholar
(15)Nemytskii, V. V. and Stepanov, V. V.Qualitative theory of differential equations (Princeton University Press 1960).Google Scholar
(16)Lefschetz, S.Differential equations: geometric theory (Interscience; New York, 1959).Google Scholar