Abstract: In this paper we study high order expansions of chart maps for local finite dimensional unstable
manifolds of hyperbolic equilibrium solutions of scalar parabolic partial differential equations. Our approach is
based on studying an infinitesimal invariance equation for the chart map that recovers the dynamics on the manifold
in terms of a simple conjugacy. We develop formal series solutions for the invariance equation and efficient
numerical methods for computing the series coefficients to any desired finite order. We show, under mild
non-resonance conditions, that the formal series expansion converges in a small enough neighborhood of the
equilibrium. An a-posteriori computer assisted argument proves convergence in larger neighborhoods. We implement
the method for a spatially inhomogeneous Fisher’s equation and numerically compute and validate high order
expansions of some local unstable manifolds for morse index one and two. We also provide a computer assisted
existence proof of a saddle-to-sink heteroclinic connecting orbit.
Authors:
Christian Reinhardt and J.D. Mireles James
Preprint:
(Submitted).
Computer Assisted Proof Codes:
(codes require the IntLab package)