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Stability and folds

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Abstract

It is known that when one branch of a simple fold in a bifurcation diagram represents (linearly) stable solutions, the other branch represents unstable solutions. The theory developed here can predict instability of some branches close to folds, without knowledge of stability of the adjacent branch, provided that the underlying problem has a variational structure. First, one particular bifurcation diagram is identified as playing a special role, the relevant diagram being specified by the choice of functional plotted as ordinate. The results are then stated in terms of the shape of the solution branch in this distinguished bifurcation diagram. In many problems arising in elasticity the preferred bifurcation diagram is the loaddisplacement graph. The theory is particularly useful in applications where a solution branch has a succession of folds.

The theory is illustrated with applications to simple models of thermal selfignition and of a chemical reactor, both of which systems are of Émden-Fowler type. An analysis concerning an elastic rod is also presented.

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Communicated by H. Weinberger

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Maddocks, J.H. Stability and folds. Arch. Rational Mech. Anal. 99, 301–328 (1987). https://doi.org/10.1007/BF00282049

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Keywords

  • Neural Network
  • Complex System
  • Simple Model
  • Nonlinear Dynamics
  • Variational Structure