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Zeeman's Catastrophe Machine

Published online by Cambridge University Press:  24 October 2008

T. Poston  and
A. E. R. Woodcock
T. Poston
Affiliation:
Mathematical Institute, The University of Warwick
A. E. R. Woodcock
Affiliation:
Mathematical Institute, The University of Warwick
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Extract

René Thom(5) has shown that if the state of a system is determined by the local minimization of a potential – that is, if the system is so highly dissipative that transients can safely be ignored–then, though a smooth change in the potential function may give rise to a discontinuous change of state, the ways in which this can happen are quite limited. Infact, if we have at most a four-parameter family of potentials, discontinuities of this kind can occur in only seven ways up to local diffeotype, if they are to be structurally stable. (This latter condition is the requirement that it be im-possible to alter the discontinuity type by an arbitrarily small perturbation of the family of potentials: roughly it requires that the behaviour of the system, considered as a function of possible families ofpotentials, be ‘continuous’ at the family concerned. It bears the same relationship as does continuity to computability: a computer given approximately correct data for which to compute the value of a function or the behaviour of a system will give an approximately correct answer if the function is continuous, the system structurally stable. If not, only analytic methods will serve, and the physical significance of the result will be dubious. Fortunately, in the dimensions with which we are concerned, structural stability is an open dense property in the space of possible families of potentials; thus unstable systems can be ignored for almost all purposes, just as we ignore the ‘possibility’ of balancing a pin on its point.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 2 , September 1973 , pp. 217 - 226
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

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