文章基本信息

标题：Bifurcation Theory and the Type Numbers of Marston Morse
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作者：Melvyn S. Berger
期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：1972
卷号：69
期号：7
页码：1737-1738
DOI：10.1073/pnas.69.7.1737
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：Let H be a real Hilbert space and f(x,{lambda}) be a C2 operator mapping a small neighborhood U of (x0,{lambda}0) {varepsilon} (H x R1) into itself. We investigate the solutions of the equation f(x,{lambda}) = 0 near a solution (x0,{lambda}0), assuming that f(x,{lambda}) is a gradient mapping and 0 < dim Ker fx(x0,{lambda}0) < {infty}. In particular, we show that the type numbers of Marston Morse for an isolated critical point can be used to prove the existence of a point of bifurcation at (x0,{lambda}0). An application of this result is given to the discovery of periodic motions near a stationary point for a large class of nonlinear Hamiltonian systems in "resonant" cases.