Stuart Hasting, Dept. of Mathematics, University of Pittsburgh
" Layers, chaos, and global attractors for some one-dimensional reaction diffusion problems"
We consider equations of the form with Neumann boundary conditions. The key early reference is by Angenent, Mallet-Paret and Peletier (1987), who found all stable equilibrium solutions for a class of function which are cubic in , with for large . In joint work with S.B. Ai we study the ``Duffing'' case , and find a large class of unstable equilibria, including solutions with multiple internal layers. We also study the ode satisfied by the steady states as a nonlinear oscillator, and obtain some chaos results which do not treat as a small parameter.
In the case of the Neumann problem for small we are able to obtain the Morse index of the new solutions. Further, we consider the bifurcation of solutions as a function of , and we anticipate that near the initial bifurcation point it will be possible to apply techniques of Fiedler and Rochas to determine the global attractor.
TUESDAY, October 31, 2000