  In this paper, we study the phase space of cosmological models in the context
of Einstein-Gauss-Bonnet theory. More specifically, we consider a generalized
dynamical system that encapsulates the main features of the theory and for the
cases that this is rendered autonomous, we analyze its equilibrium points and
stable and unstable manifolds corresponding to several distinct cosmological
evolutions. As we demonstrate, the phase space is quite rich and contains
invariant structures, which dictate the conditions under which the theory may
be valid and viable in describing the evolution Universe during different
phases. It is proved that a stable equilibrium point and two invariant
manifolds leading to the fixed point, have both physical meaning and restrict
the physical aspects of such a rich in structure modified theory of gravity.
More important we prove the existence of a heteroclinic orbit which drives the
evolution of the system to a stable fixed point. However, while on the fixed
point the Friedman constraint corresponding to a flat Universe is satisfied,
the points belonging to the heteroclinic orbit do not satisfy the Friedman
constraint. We discuss the origin of this intriguing issue in some detail.
