Scalar Field Cosmology -- Geometry of Dynamics
  We study the Scalar Field Cosmology (SFC) using the geometric language of the
phase space. We define and study an ensemble of dynamical systems as a Banach
space with a Sobolev metric. The metric in the ensemble is used to measure a
distance between different models. We point out the advantages of visualisation
of dynamics in the phase space. It is investigated the genericity of some class
of models in the context of fine tuning of the form of the potential function
in the ensemble of SFC.
  We also study the symmetries of dynamical systems of SFC by searching for
their exact solutions. In this context we stressed the importance of scaling
solutions. It is demonstrated that scaling solutions in the phase space are
represented by unstable separatrices of the saddle points. Only critical point
itself located on two dimensional stable submanifold can be identified as
scaling solution. We have also found a class of potentials of the scalar fields
forced by the symmetry of differential equation describing the evolution of the
universe. A class of potentials forced by scaling (homology) symmetries was
given. We point out the role of the notion of a structural stability in the
context of the problem of indetermination of the potential form of the Scalar
Field Cosmology. We characterise also the class of potentials which reproduces
the \Lambda CDM model, which is known to be structurally stable. We show that
the structural stability issue can be effectively used is selection of the
scalar field potential function. This enables us to characterise a structurally
stable and therefore a generic class of SFC models. We have found a nonempty
and dense subset of structurally stable models. We show that these models
possess symmetry of homology.
