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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 07 Feb 2024 03:52:14 +0000Topics in bifurcation theory
https://resolver.caltech.edu/CaltechTHESIS:03252013-104520935
Authors: {'items': [{'id': 'Decker-D-W', 'name': {'family': 'Decker', 'given': 'Dwight William'}, 'show_email': 'NO'}]}
Year: 1978
DOI: 10.7907/xv6r-jf75
<p>I. Existence and Structure of Bifurcation Branches</p>
<p>The problem of bifurcation is formulated as an operator equation
in a Banach space, depending on relevant control parameters, say of
the form G(u,λ) = 0. If dimN(G_u(u_O,λ_O)) = m the method of Lyapunov-Schmidt
reduces the problem to the solution of m algebraic equations.
The possible structure of these equations and the various types of
solution behaviour are discussed. The equations are normally derived
under the assumption that G^O_λεR(G^O_u). It is shown, however, that
if G^O_λεR(G^O_u) then bifurcation still may occur and the local structure
of such branches is determined. A new and compact proof of the
existence of multiple bifurcation is derived. The linearized
stability near simple bifurcation and "normal" limit points is then
indicated.</p>
<p>II. Constructive Techniques for the Generation of Solution Branches</p>
<p>A method is described in which the dependence of the solution
arc on a naturally occurring parameter is replaced by the dependence
on a form of pseudo-arclength. This results in continuation procedures
through regular and "normal" limit points. In the neighborhood
of bifurcation points, however, the associated linear operator
is nearly singular causing difficulty in the convergence of continuation
methods. A study of the approach to singularity of this
operator yields convergence proofs for an iterative method for determining
the solution arc in the neighborhood of a simple bifurcation
point. As a result of these considerations, a new constructive
proof of bifurcation is determined.</p>https://thesis.library.caltech.edu/id/eprint/7552