|
Shooting and Bifurcating with
Mathematica
Jay Wolkowisky
University of Colorado
Download talk material:
The first part of the presentation explains a version of the so-called "shooting
method" to solve two-point boundary-value problems for systems of
ordinary
differential equations that contain a parameter. This version of the
shooting method is
not a trial-and-error method, as some would use it. The application
here is to boundary-value problems that do not have a unique solution (sometimes called
nonlinear eigenvalue
problems). This is the setting that would arise when bifurcation
occurs.
The second part of the presentation
uses the above method to investigate the buckled states of a circular
plate loaded in the
plane of the plate and resting on an elastic foundation. This problem
arose as an
application to geophysics. Mathematically this problem leads to a
system of three coupled,
second-order ordinary differential equations that have a cubic
nonlinearity. Boundary
conditions are specified at the center and at the edge. There are two
parameters that
arise in this problem. One is the load parameter, which measures the
compressive load in
the plane of the plate, and the other is the parameter that measures the stiffness
of the foundation.
The shooting method lends itself very nicely in the buckled
states (bifurcating
branches) that bifurcate from the unbuckled state as the load and
stiffness parameters are
varied. The results that are obtained are very interesting and lead to
unexpected buckled
states which arise from the interaction of different bifurcating
branches (secondary
bifurcations). The animation capabilities of Mathematica are
used very heavily to
illustrate the results.
The third part of the presentation tries to explain the unusual
bifurcations that were
mentioned previously. This is done by looking at an algebraic problem
with a cubic
nonlinearity and two parameters, analogous to our buckled-plate
problem. This algebraic
problem is easy to solve and is analyzed using Mathematica's
graphic and animation
abilities. The graphical results of this simple algebraic problem are
surprisingly similar
to the bifurcation diagrams of the buckled-plate problems. Therefore,
we get an insight
into what is really causing the interesting phenomena of our much
more-complicated
problem.
| 
