Topics in Local Bifurcation Theory.
BROWN UNIV PROVIDENCE R I LEFSCHETZ CENTER FOR DYNAMICAL SYSTEMS
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Suppose Lambda, X, Z are Banach spaces, M Lambda x X yields Z is a mapping continuous together with derivatives up through some order r. A Bifurcation surface for the equation 1 Mlambda,x 0 is a surface in parameter space Lambda for which the number of solutions x of 1 changes as lambda crosses this surface. Under certain generic hypotheses on M, it is shown that one can systematically determine the bifurcation surfaces by elementary scaling techniques and the implicit function theorem. This document gives a summary of these results for the case of bifurcation near an isolated solution or families of solutions of the equation Mlambda sub 0, x 0. The results have applications of the buckling theory of plates and shells under the effect of external forces, imperfections, curvature and variations in shape. The results on bifurcation near families has applications in nonlinear oscillations and the theory of homoclinic orbits.
- Theoretical Mathematics