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Bifurcation theory of nonlinear boundary value problems


Langford, William Finlay (1971) Bifurcation theory of nonlinear boundary value problems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BP03-ZM29.


The theory of bifurcation of solutions to two-point boundary value problems is developed for a system of nonlinear first order ordinary differential equations in which the bifurcation parameter is allowed to appear nonlinearly. An iteration method is used to establish necessary and sufficient conditions for bifurcation and to construct a unique bifurcated branch in a neighborhood of a bifurcation point which is a simple eigenvalue of the linearized problem. The problem of bifurcation at a degenerate eigenvalue of the linearized problem is reduced to that of solving a system of algebraic equations. Cases with no bifurcation and with multiple bifurcation at a degenerate eigenvalue are considered.

The iteration method employed is shown to generate approximate solutions which contain those obtained by formal perturbation theory. Thus the formal perturbation solutions are rigorously justified. A theory of continuation of a solution branch out of the neighborhood of its bifurcation point is presented. Several generalizations and extensions of the theory to other types of problems, such as systems of partial differential equations, are described.

The theory is applied to the problem of the axisymmetric buckling of thin spherical shells. Results are obtained which confirm recent numerical computations.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Applied Mathematics
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Keller, Herbert Bishop
Thesis Committee:
  • Unknown, Unknown
Defense Date:9 July 1970
Record Number:CaltechTHESIS:04082013-100223262
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7584
Deposited By: Benjamin Perez
Deposited On:08 Apr 2013 18:21
Last Modified:09 Nov 2022 19:20

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