  A dynamical systems approach to competition of Saffman-Taylor fingers in a
channel is developed. This is based on the global study of the phase space
structure of the low-dimensional ODE's defined by the classes of exact
solutions of the problem without surface tension. Some simple examples are
studied in detail, and general proofs concerning properties of fixed points and
existence of finite-time singularities for broad classes of solutions are
given. The existence of a continuum of multifinger fixed points and its
dynamical implications are discussed. The main conclusion is that exact
zero-surface tension solutions taken in a global sense as families of
trajectories in phase space spanning a sufficiently large set of initial
conditions, are unphysical because the multifinger fixed points are
nonhyperbolic, and an unfolding of them does not exist within the same class of
solutions. Hyperbolicity (saddle-point structure) of the multifinger fixed
points is argued to be essential to the physically correct qualitative
description of finger competition. The restoring of hyperbolicity by surface
tension is discussed as the key point for a generic Dynamical Solvability
Scenario which is proposed for a general context of interfacial pattern
selection.
