| Foliations and invariant manifolds are fundamental tools to describe the qualitative behavior of the local flows in the study of the dynamics near an equilibrium. In a foliation of phase space through each (base) point there is a pair of manifolds. One, which we refer to as the leaf in the stable foliation, intersects the inertial manifold at the tracking initial condition of the base point. In this thesis, we study the accurate computation of foliations, inertial manifolds, and tracking initial conditions for dynamical systems near a hyperbolic fixed point. Several algorithms are presented. They are variations of a contraction mapping method in [32] to compute inertial manifolds, which itself represents a particular leaf in the unstable foliation. Such a mapping is combined with one for the leaf in the stable foliation to compute the tracking initial condition for a given solution. In particular, a unified approach has been developed and implemented in a public domain software package, FOLI8PAK, available at [8]. In addition to being universal, this approach provided greater accuracy. The algorithms are demonstrated on the Kuramoto-Sivashinsky equation. The other part of the thesis is to apply Newton-like methods to compute foliations. Newton's method, when convergent, is more efficient than the successive iteration method. Since the framework of our problem is in a Banach space, a number of classic convergence results for the Newton-like methods have been generalized to this infinite dimensional case. |
