On deducing the presence of catastrophes

Poston, Tim

Mathématiques et Sciences humaines, Tome 64 (1978), p. 71-99 / Harvested from Numdam

Détail

Publié le : 1978-01-01

MR 538884 | Zbl 0411.00016

Bibliographie

[1] R. Abraham & J.W. Robbin, Transversal Mappings and Flows, Benjamin, N.Y. 1967. | MR 240836 | Zbl 0171.44404

[2] V.I. Arnol'D, Wave front evolution and equivariant Morse Lemma, Comm. Pure Appl. Math. 29 (1976) 557-582. | MR 436200 | Zbl 0343.58003

[3] J. Chazarain, Solutions asymptotiques et caustiques, in Rencontre de Cargèse sur les singularités et leurs applications (ed. F. Pham) publ. Math. Dept. Univ. Nice 1975.

[4] R. Courant & H. Robbins, What Is Mathematics?, Oxford University Press 1941. | JFM 67.0001.05 | MR 5358 | Zbl 0060.12302

[5] N.F. Dixon, On the Psychology of Military Incompetence, Johnathan Cape, London 1976.

[6] A.S. Eddington, Gravitation and the Principle of Relativity, Royal Institution Discourse 1918.

[7] P.J. Holmes & D.A. Rand, The bifurcations of Duffing's equation: an application of catastrophe theory, J. Sound Vib.44 (1976) 237-253. | Zbl 0337.34049

[8] L. Van Hove, The occurrence of singularities in the elastic frequency distribution of a crystal, Phys. Rev. 89 (1953), 1189-1193. | MR 56512 | Zbl 0050.23605

[9] C.A. Isnard & E.C. Zeeman, Some models from catastrophe theory in the social sciences, in Use of Models in the Social Sciences (ed. L. Collins), Tavistock, London 1975.

[10] J.J. Kozak and C.J. Benham, Denaturation; an example of a catastrophe I, Proc. Nat. Acad. Sci. USA 71 (1974), 1977-1981. (The 'review article' [31] was written without awareness of parts II, III in J. Theor. Biol. 63 (1976) 125-149, 66 (1977) 679-693 which had already dealt with those objections in [31] not dependent on misrepresentation.) | Zbl 0289.92005

[11] H. Kurland & J. Robbin, Infinite codimension and Transversality, in Dynamical Systems - Warwick 1974 (ed. A. Manning), Lecture Notes in Math. 468, Springer, Berlin 1975, 135-150. | MR 649273 | Zbl 0317.58011

[12] L.D. Landau & E.M. Lifshitz, Statistical Physics, 2nd English edition (trans Peierls & Peierls), Pergamon, Oxford 1959. | MR 586944 | Zbl 0080.19702

[13] K. Lorenz, On Aggression, Bantam Books, N.Y. 1967.

[14] R.J. Magnus & T. Poston, On the full unfolding of the von Kármán equation at a double eigenvalue, Math. Report 109, Battelle, Geneva 1977.

[15] A. Du Plessis, Maps without certain singularities, to appear in Comm. Math. Helv. | MR 397779 | Zbl 0313.58010

[16] T. Poston, Au courant with differential equations, Manifold 18 (1976) 6-9.

[17] T. Poston, The elements of catastrophe theory or The honing of Occam's razor, to appear in Transformations: Mathematical Approaches to Culture Change (eds K.L. Cooke and A.C. Renfrew), Academic Press, N.Y. 1978.

[18] T. Poston & I.N. Stewart, Taylor Expansions and Catastrophes, Research Notes in Math. 7, Pitman, London & S. Francisco 1976. | MR 494231 | Zbl 0335.58008

[19] T. Poston & I.N. Stewart, Catastrophe Theory and its Applications, Pitman, London & S. Francisco 1978. | Zbl 0382.58006

[20] M. Spivak, Calculus on Manifolds, Benjamin, N.Y. 1965. | MR 209411 | Zbl 0141.05403

[21] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747-817. | MR 228014 | Zbl 0202.55202

[22] I.N. Stewart, The Seven Elementary Catastrophes, New Scientist 68 (1975) , 447-454.

[23] I.N. Stewart, Catastrophe Theory, Special Supplement to Britannica Book of the Year 1977.

[24] H.J. Sussmann, Some properties of vector fields that are not altered by small perturbations, J. Diff. Eq. 20 (1976) no 2, 292-315. | MR 394756 | Zbl 0346.49036

[25] H.J. Sussmann, Catastrophe theory - a preliminary critical study, PSA 1976 vol. 1 (eds. F. Suppe & P.D. Asquith), Phil. Sci. Assoc., East Lansing, Michigan 1976.

[26] H.J. Sussmann & R.S. Zahler, Catastrophe theory as applied to the social and biological Sciences : a critique, to appear in Synthèse. All quotations are from the widely circulated preprint, Rutgers Univ. Math. Dept., Feb. 1977. | MR 495176 | Zbl 0389.92002

[27] H.J. Sussmann & R.S. Zahler, Catastrophe theory: mathematics misused, The Sciences 17 no 6 (1977) 20-23.

[28] R. Thom, Answer to Christopher Zeeman's reply, in Dynamical Systems - Warwick 1974 (ed. A. Manning), Lecture Notes in Math. 468, Springer, Berlin 1975, 384-389. | MR 649921

[29] R. Thom, Structural stability catastrophe theory, and applied mathematics, SIAM Review 19, 189-201, 1977. | MR 451284 | Zbl 0447.58010

[30] A.E.R. Woodcock & T. Poston, A Geometrical Study of the Elementary Catastrophes, Lecture Notes in Math. 373, Springer, Berlin 1974. | Zbl 0279.58004

[31] R.S. Zahler & H.J. Sussmann, Claims and accomplishments of applied catastrophe theory, Nature 269 (1977), 759-763. Letters pointing out many errors, misquotations etc. are in ibid. 270 (1977), 381-384, 658, and a reply by Zahler with some new ones in 271 (1978), 401.

[32] E.C. Zeeman, Differential equations for the heartbeat and nervous impulse, in Towards a Theoretical Biology IV (ed. C.H. Waddington), Edinburgh Univ. Press, 1972, 8-67, Dynamical Systems (ed. M.M. Peixoto), Academic Press, N.Y. 1973, and [35] 81-140. | MR 342207

[33] E.C. Zeeman, Primary and secondary waves in developmental biology, AAAS 1974, Some Mathematical Questions in Biology VIII, Lectures on Mathematics in the Life Sciences 7, Amer. Math. Soc., Providence R.I. 1974, 69-161, and [35] 141-233. | MR 484531 | Zbl 0317.92002

[34] E.C. Zeeman, On the unstable behaviour of stock exchanges, J. Math. Econ. 1 (1974) 39-49, and [35], 361-371. | MR 418874 | Zbl 0297.90002

[35] E.C. Zeeman, Catastrophe Theory: Selected Papers (1972-1977), Addison-Wesley, N.Y. 1977. | MR 474383 | Zbl 0398.58012

[36] E.C. Zeeman, A boundary value problem involving cusps, to appear.

See also I.N. Stewart and T. Poston, The Catastrophe Theory Controversy, to appear in the New Scientist, and the September 1978 special 'Catastrophe Theory' issue of Behavioral Science, where further arguments for catastrophe modelling are variously developed.