Imperfection Theory and Isolated Solutions Which Perturb Bifurcation
Isolated solutions are probably very common in dynamical problems. One way to treat them is as a perturbation of problems which do bifurcate. This method of studying isolated solutions which are close to bifurcating solutions is known as imperfection theory. Some of the basic ideas involved in imperfection theory can be understood by comparing the bending of an initially straight column with an initially imperfect, say bent, column (see Figure III. 1). The first column will remain straight under increasing end loadings P until a critical load P c is reached. The column then undergoes supercritical, one-sided, double-point bifurcation (Euler buckling). In this perfect (plane) problem there is no way to decide if the column will buckle to the left or to the right. The situation is different for the initially bent column. The sidewise deflection starts as soon as the bent column is loaded and it deflects in the direction x < 0 of the initial bending. If the initial bending is small the deflection will resemble that of the perfect column. There will be a small, nonzero deflection with increasing load until a neighborhood of P c is reached; then the deflection will increase rapidly with increasing load. When P is large it will be possible to push the deflected bent column into a stable “abnormal” position (x > 0) opposite to the direction of initial bending.
KeywordsImplicit Function Theorem Double Point Steady Solution Find Solution Hyperbolic Paraboloid
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