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Mathematical Theory of Bifurcation

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Bifurcation Phenomena in Mathematical Physics and Related Topics

Part of the book series: NATO Advanced Study Institutes Series ((ASIC,volume 54))

Abstract

One of the first studies of bifurcation is due to Euler (1744), who treated the buckling of a column subjected to axial compression (the so-called Elastica). He considered the following boundary-value problem

$$\left\{ {\begin{array}{*{20}{c}} {\theta '' + \lambda \sin \theta = o} \\ {\theta '\left( o \right) = \theta '(1) = o} \\ \end{array} } \right.$$
(1)

where θ is the angle between the tangent to the column and the real axis, λ is the thrust applied and 1 is the length of the column:

fig. 1

Obviously θ=0 is a solution of (0.l) for any real λ.This means that the column remains straight in equilibrium. But it is not difficult to see that as λ exceeds π2 a new family of solutions appears and the column buckles.

Notes by E. Buzano and C. Canuto

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  1. Mathematics Research Center, Madison, Wisconsin, USA

    Michael G. Crandall & Paul H. Rabinowitz

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  1. Départment de Mathematiques, Université de Paris-Nord, Villetaneuse, France

    Claude Bardos

  2. CEN, CEA-Saclay, Gif-sur-Yvette, France

    Daniel Bessis

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Crandall, M.G., Rabinowitz, P.H. (1980). Mathematical Theory of Bifurcation. In: Bardos, C., Bessis, D. (eds) Bifurcation Phenomena in Mathematical Physics and Related Topics. NATO Advanced Study Institutes Series, vol 54. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9004-3_1

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