Alex Vladimirsky
Department of Mathematics, Cornell University.
| Invariant manifolds are fundamental geometric structures that partition phase spaces into sets of points with the same forward and backward limit sets. Their applications range from designing efficient controllers, to exploring the structure of weak shock waves for systems of conservation laws, to studying the global bifurcations. We show how the problem of constructing an invariant manifold (of co-dimension k) can be locally reduced to solving a (system of k) quasi-linear PDE(s), which can be efficiently solved using an Ordered Upwind Method (OUM). Such methods, originally introduced in a joint work with J.A. Sethian for static Hamilton-Jacobi PDEs, rely on careful use of the direction of information propagation to systematically advance the computed "boundary" and to de-couple the discretized system. We illustrate our approach by constructing invariant manifolds of hyperbolic saddle points for several "geometrically stiff" systems. This work is a joint project with J. Guckenheimer. |
Back to AML Seminar Schedule