  In this work we explore the dynamical system phase space of
Einstein-Gauss-Bonnet theory in the cosmological minisuperspace. This approach
binds the main features of the theory through a system of autonomous
differential equations, in the context of a flat
Friedmann-Lema\^{i}tre-Robertson-Walker spacetime. We analyze the critical
points that feature in this system to assess their stability criteria. The
phase space of this form of scalar-tensor gravity is very rich due to the
fourth-order contributions of the Gauss-Bonnet invariant together with the
second order contribution of the scalar field together with their coupling
dynamics. We find additional critical points as compared with previous works in
the literature which may be important for understanding the larger evolution of
standard background cosmology within this class of gravitational models.
