** Authors: **

J.B. van den Berg

J. D. Mireles James

** Abstract:**
The present work deals with numerical methods for computing slow stable
invariant manifolds as well as their invariant stable and unstable normal
bundles. The slow manifolds studied here are sub-manifolds of the stable
manifold of a hyperbolic equilibrium point. Our approach is based on studying
certain partial differential equations equations whose solutions parameterize
the invariant manifolds/bundles. Formal solutions of the partial differential
equations are obtained via power series arguments, and truncating the formal
series provides an explicit polynomial representation for the desired chart
maps. The coefficients of the formal series are given by recursion relations
which are amenable to computer calculations. The parameterizations conjugate
the dynamics on the invariant manifolds and bundles to a prescribed linear
dynamical systems. Hence in addition to providing accurate representation
of the invariant manifolds and bundles our methods describe the dynamics on
these objects as well. Example computations are given for vector fields which
arise as Galerkin projections of a partial differential equation. As an
application we illustrate the use of the parameterized slow manifolds and
their linear bundles in the computation of heteroclinic orbits.

Discrete and Continuous Dynamical Systems, Vol. 36, No. 9, pp. 4637--4664 (2016).