The goal of this research project is to develop computational methods for investigating the dynamical behavior of chaotic attractors resulting from systems of differential equations of moderately high dimension. The emphasis is on systems of equations arising from inertial manifold reductions of PDE's governing fluid flows at and somewhat beyond the onset of instabilities and turbulence. The PI's interest is in characterizing the low dimensional dynamics of systems which potentially have 10-100 degrees of freedom. The principal components of the proposed work include: the development of accurate numerical methods for the precise calculation of stable and unstable manifolds of attractors in more than three dimensions; use of real-time color graphics for improved visualization of the geometric structure of saddle orbits and the associated manifolds; and the use of vectorized algorithms to exploit the power of new computer hardware. The algorithms and resulting software packages are intended to assist applied mathematicians, engineers, physicists, and students investigating the dynamical behavior of systems of nonlinear ordinary and partial differential equations.
