Frequently, in nonlinear differential equations arising from physically interesting models, there is an "obvious" near-equilibrium oscillatory solution. However, occasionally there can be evidence, both physical and mathematical, that other large amplitude solutions may exist for the same forcing terms. The main object of my research is to investigate when these occur, how they can be proven to exist, and how they can be numerically calculated. The main difference between studying nonlinear equations, as opposed to linear equations, is that linear equations typically have an analytically attainable solution which is unique. In the nonlinear case there is often an obvious solution which can be approximated by linearization. Finding this solution and its properties have been the preoccupation and achievement of the twentieth century. However, this solution to the linearized equation is only the tip of the iceberg of all possible solutions. How can the nonlinear analyst formulate a strategy for exploring the infinite-dimensional space of other possibilities? How does one know if there are other solutions out there? How does one locate solutions that one may know are out there? As of now there are few global strategies for approximating multiple solutions. Continuation, steepest descent, and mountain-pass variational algorithms come to mind, but these are only just the beginning. Because of the availability of computational experimentation, we are witnessing a new golden age in mechanics. This age began with the numerical experiments of Fermi, Pasta and Ulam in the fifties, Lorenz and Kruskal in the sixties, and gathered momentum with work on the forced pendulum in the seventies. My current research focuses on understanding the solution space associated with nonlinear mechanical models. Currently that focus has been centered on a nonlinear spring equation with very low frequency forcing. See the following relevant publications.
Mathematics
When a Mechanical Model Goes Nonlinear: Unexpected Responses to Low-Periodic Shaking (with P. J. McKenna), Faculty Publications (2005)
This paper had its origin in a curious discovery by the first author in research...
Multiple Periodic Solutions for a Nonlinear Suspension Bridge Equation (with P. J. McKenna), Faculty Publications (1999)
We investigate nonlinear oscillations in a fourth-order partial differential equation which models a suspension bridge....