Dynamical analysis in scalar field cosmology
  We give a general method to find exact cosmological solutions for
scalar-field dark energy in the presence of perfect fluids. We use the
existence of invariant transformations for the Wheeler De Witt (WdW) equation.
We show that the existence of a point transformation under which the WdW
equation is invariant is equivalent to the existence of conservation laws for
the field equations, which indicates the existence of analytical solutions. We
extend previous work by providing exact solutions for the Hubble parameter and
the effective dark-energy equation of state parameter for cosmologies
containing a combination of perfect fluid and a scalar field whose
self-interaction potential is a power of hyperbolic functions. We find
solutions explicitly when the perfect fluid is radiation or cold dark matter
and determine the effects of non-zero spatial curvature. Using the Planck 2015
data, we determine the evolution of the effective equation of state of the dark
energy. Finally, we study the global dynamics using dimensionless variables. We
find that if the current cosmological model is Liouville integrable (admits
conservation laws) then there is a unique stable point which describes the
de-Sitter phase of the universe.
