Abstract
Bifurcation theories for the instability of slowly evolving systems have been developed in various disciplines, and a first step is here taken towards some desirable unification. A modern account of the authors' general branching theory for discrete systems is first presented, some new features being the introduction of principal imperfections and the delineation of the important semi-symmetric points of bifurcation. This theory, embedded in a perturbation approach ideal for quantitative analysis, is complementary to the far-reaching qualitative catastrophe theory of René Thom which offers a profound topological classification of instability phenomena. For this reason, we present here a detailed correlation of the two theories.
Also presented in the paper is a survey of some fields of application ranging from classical fields such as hydrodynamics, through thermodynamics, crystallography and cosmology, to the newer domains of biology and psychology.
Zusammenfassung
Verzweigungstheorien für die Instabilität sich allmählich entwickelnder Systeme wurden in verschiedenen Disziplinen entwickelt; es wird hier der erste Schritt zu einer erwünschten Vereinheitlichung getan. Einleitend wird ein moderner Abriss der allgemeinen Verzweigungstheorie diskreter Systeme, wie sie von den Verfassern entwickelt wurde, dargestellt. Einige neue Elemente sind die Einführung von ‘Hauptimperfektionen’ und die Beschreibung halbsymmetrischer Verzweigungspunkte. Diese Theorie, verbunden mit der für eine quantitative Analyse idealen Störungsrechnung, ergänzt die weitreichende qualitative Katastrophentheorie von René Thom, die eine tiefschürfende topologische Klassifizierung der Instabilitätserscheinungen bietet. Aus diesem Grunde wird hier die Wechselbeziehung der zwei Theorien eingehend dargestellt.
Anschliessend wird eine Uebersicht einiger Anwendungsgebiete geboten—vom klassischem Feld der Hydrodynamik über die Thermodynamik, Kristallographie und Kosmologie zu den neueren Bereichen der Biologie und Psychologie.
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References
H. Poincaré,Les méthodes nouvelles de la mécanique céléste, Vols 1–3, Gauthier-Villars, Paris, 1892–1899.
A.M. Liapunov,Sur les figures d'equilibre peu différentes des ellipsoides d'une masse liquide homogène douée d'un mouvement de rotation, Zap. Akad. Nauk. St. Petersburg1, 1, 1906.
E. Schmidt,Zur Theorie der linearen und nicht linearen Integralgleichungen, Math. Ann.65, 370, 1910.
T.H. Hildebrandt andL.M. Graves,Implicit Functions and their Differentials in General Analysis, Trans. Amer. Math. Soc.29, 127, 1927.
M.A. Krasnoselskii,Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1964.
M. Vainberg andP.G. Aizengendler,The Theory and Methods of Investigation of Branch Points of Solutions, Progress in Mathematics, Vol. 2, Plenum Press, New York, 1968.
I. Stakgold,Branching of Solutions of Nonlinear Equations, SIAM Rev.13, 289, 1971.
G.H. Pimbley,Eigenfunction Branches of Nonlinear Operators and their Bifurcations, Springer, New York, 1969.
J.B. Keller andS. Antman, ed.,Bifurcation Theory and Nonlinear Eigenvalue Problems, Benjamin, New York, 1969.
H. Leipholz, ed.,Instability of Continuous Systems, Springer, Berlin, 1971.
R. Thom,Stabilité structurelle et morphogénèse, Benjamin, Reading, 1972.
R. Thom,Topological models in biology, Topology,8, 313, 1969.
W.T. Koiter,On the Stability of Elastic Equilibrium, Dissertation, Delft, Holland, 1945 (English translation: NASA, Tech. Trans.F10, 833, 1967).
B. Budiansky, ‘Theory of Buckling and Post-buckling Behaviour of Elastic Structures’,Advances in Applied Mechanics, Vol. 14, Academic Press, New York, 1974.
J.W. Hutchinson, ‘Plastic Buckling’,Advances in Applied Mechanics Vol. 14, Academic Press, New York, 1974.
J.M.T. Thompson,A General Theory for the Equilibrium and Stability of Discrete Conservative Systems, Z. angew. Math. Phys.20 797, 1969.
M.J. Sewell,On the Branching of Equilibrium Paths, Proc. Roy. Soc. Ser.A,315, 499, 1970.
J.M.T. Thompson andG.W. Hunt,A General Theory of Elastic Stability, Wiley, London, 1973.
E.C. Zeeman, ‘A Catastrophe Machine,’ inTowards a Theoretical Biology, Vol. 4, (ed. C. H. Waddington) Edinburgh University Press, Edinburgh 1972.
J.J. Stoker,Nonlinear Vibrations in Mechanical and Electrical Systems, Wiley, New York, 1966.
S. Chandrasekhar,Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Oxford, 1968.
P. Glansdorff andI. Prigogine,Thermodynamic Theory of Structure Stability and Fluctuations, Wiley, London 1971.
M. Born andK. Huang,Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford, 1954.
J.M.T. Thompson andP.A. Shorrock,Bifurcational Instability of an Atomic Lattice, J. Mech. Phys. Solids23, 21, 1975.
R.A. Lyttleton,The Stability of Rotating Liquid Masses, Cambridge University Press, Cambridge, 1953.
P. Ledoux, ‘Stellar Stability,’ inHandbuch der Physik, Vol. LI (ed. S. Flügge), Springer, Berlin, 1958.
G.I. Taylor,Disintegration of Water Drops in an Electric Field, Proc. Roy. Soc. Ser.A,280, 383, 1964.
E.C. Zeeman,On the Unstable Behaviour of Stock Exchanges, J. Math. Economics (to appear).
C.A. Isnard andE.C. Zeeman, ‘Some Models from Catastrophe Theory in the Social Sciences,’ inUse of Models in the Social Sciences (ed. L. Collins), Tavistock, London, 1974.
D'Arcy W. Thompson,On Growth and Form, Cambridge University Press, Cambridge, 1971.
C.H. Waddington ed.,Towards a Theoretical Biology, Vols. 1–4, Edinburgh University Press, Edinburgh, 1968–1972.
E.C. Zeeman, ‘The Geometry of Catastrophe,’The Times Literary Supplement, December 10, 1971.
G.W. Hunt,Imperfection-sensitivity of Compound Branching, to be published.
K. Huseyin,Nonlinear Theory of Elastic Stability, Noordhoff, Leyden, 1974.
J. Roorda,Stability of Structures with Small Imperfections, J. Engng. Mech. Div. Am. Soc. civ. Engrs.91, 87, 1965.
D.H. Fowler, ‘The Riemann-Hugoniot Catastrophe and Van der Waals Equation,’ inTowards a Theoretical Biology, Vol. 4, (ed. C. H. Waddington) Edinburgh University Press, Edinburgh 1972.
J.P. Keener andH.B. Keller,Perturbed Bifurcation Theory, Arch. Rat. Mech. Anal.50, 159, 1973.
A.H. Chilver,Coupled Modes of Elastic Buckling, J. Mech. Phys. Solids15, 15, 1967.
W.J. Supple, ed.,Structural Instability, IPC Science and Technology Press, Guildford, 1973.
D. Chillingworth,Elementary Catastrophe Theory, IMA Bulletin,11, 155, 1975.
G.W. Hughes The Coupled Buckling of Some Stiffened Panels in Compression, Ph.D. Thesis, University of Birmingham, 1974.
V. Tvergaard,Imperfection-sensitivity of a Wide Integrally Stiffened Panel under Compression, Int. J. Solids Struct.9, 177, 1973.
K.C. Johns,Imperfection Sensitivity of Coincident Buckling Systems, Int. J. Nonlin. Mech.9, 1, 1974.
D. Ho,Buckling Load of Nonlinear Systems with Multiple Eigenvalues, Int. J. Solids Struct.10, 1315, 1974.
J.M.T. Thompson,Experiments in Catastrophe, Nature254, 392, 1975.
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Thompson, J.M.T., Hunt, G.W. Towards a unified bifurcation theory. Journal of Applied Mathematics and Physics (ZAMP) 26, 581–603 (1975). https://doi.org/10.1007/BF01594031
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DOI: https://doi.org/10.1007/BF01594031