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Shooting and Bifurcating with Mathematica

Jay Wolkowisky
University of Colorado

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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.