Abstract
This chapter deals with completely deterministic processes. The question of stability of motion plays a central role. When certain parameters change, stable motion may become unstable and completely new types of motion (or structures) appear. Though many of the concepts are derived from mechanics, they apply to many disciplines.
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References
N. N. Bogoliubov, Y. A. Mitropolsky: Asymptotic Methods in the Theory of Nonlinear Oscillations (Hindustan Publ. Corp., Delhi 1961)
N. Minorski: Nonlinear Oscillations (Van Nostrand, Toronto 1962)
A. Andronov, A. Vitt, S. E. Khaikin: Theory of Oscillators (Pergamon Press, London-Paris 1966)
D. H. Sattinger In Lecture Notes in Mathematics, Vol. 309: Topics in Stability and Bifurcation Theory, ed. by A. Dold, B. Eckmann (Springer, Berlin-Heidelberg-New York 1973)
M. W. Hirsch, S. Smale: Differential Equations, Dynamical Systems, and Linear Algebra (Academic Press, New York-London 1974)
V. V. Nemytskii, V. V. Stepanov: Qualitative Theory of Deferential Equations (Princeton Univ. Press, Princeton, N.J. 1960)
H. Poincaré: Oeuvres, Vol. 1 (Gauthiers-Villars, Paris 1928)
H. Poincaré: Sur l’equilibre d’une masse fluide animée d’un mouvement de rotation. Acta Math. 7 (1885)
H. Poincaré: Figures d’equilibre d’une masse fluide (Paris 1903)
H. Poincaré: Sur le problème de trois corps et les équations de la dynamique. Acta Math. 13 (1890)
H. Poincaré: Les méthodes nouvelles de la méchanique céleste (Gauthier-Villars, Paris 1892–1899)
Stability
J. La Salle, S. Lefshetz: Stability by Ljapunov’s Direct Method with Applications (Academic Press, New York-London 1961)
W. Hahn: Stability of Motion. In Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 138 (Springer, Berlin-Heidelberg-New York 1967)
D. D. Joseph: Stability of Fluid Motions I and II, Springer Tracts in Natural Philosophy, Vols. 27, 28 (Springer, Berlin-Heidelberg-New York 1976)
Exercises 5.3: F. Schlögl: Z. Phys. 243, 303 (1973)
Examples and Exercises on Bifurcation and Stability
A. Lotka: Proc. Nat. Acad. Sci. (Wash.) 6, 410 (1920)
V. Volterra: Leçons sur la théorie mathematiques de la lutte pour la vie (Paris 1931)
N. S. Goel, S. C. Maitra, E. W. Montroll: Rev. Mod. Phys. 43, 231 (1971)
B. van der Pol: Phil. Mag. 43, 6, 700 (1922);
B. van der Pol: Phil. Mag. 2, 7, 978 (1926);
B. van der Pol: Phil. Mag. 3, 7, 65 (1927)
H. T. Davis: Introduction to Nonlinear Differential and Integral Equations (Dover Publ. Inc., New York 1962)
G. Ioss, D. D. Joseph: Elementary Stability and Bifurcation Theory (Springer, Berlin, Heidelberg, New York 1980)
Classification of Static Instabilities, or an Elementary Approach to Thorn’s Theory of Catastrophes
R. Thorn: Structural Stability and Morphogenesis (W. A. Benjamin, Reading, Mass. 1975) Thorn’s book requires a good deal of mathematical background. Our “pedestrian’s” approach provides a simple access to Thorn’s classification of catastrophes. Our interpretation of how to apply these results to natural sciences, for instance biology is, however, entirely different from Thorn’s.
T. Poston, I. Steward: Catastrophe Theory and its Applications (Pitman, London 1978)
E. C. Zeeman: Catastrophe Theory (Addison-Wesley, New York 1977)
P. T. Sounders: An Introduction to Catastrophe Theory (Cambridge University Press, Cambridge 1980)
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© 1983 Springer-Verlag Berlin Heidelberg
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Haken, H. (1983). Necessity. In: Synergetics. Springer Series in Synergetics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-88338-5_5
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