On dynamical systems approaches and methods in $f(R)$ cosmology
  We discuss dynamical systems approaches and methods applied to flat
Robertson-Walker models in $f(R)$-gravity. We argue that a complete description
of the solution space of a model requires a global state space analysis that
motivates globally covering state space adapted variables. This is shown
explicitly by an illustrative example, $f(R) = R + \alpha R^2$, $\alpha > 0$,
for which we introduce new regular dynamical systems on global compactly
extended state spaces for the Jordan and Einstein frames. This example also
allows us to illustrate several local and global dynamical systems techniques
involving, e.g., blow ups of nilpotent fixed points, center manifold analysis,
averaging, and use of monotone functions. As a result of applying dynamical
systems methods to globally state space adapted dynamical systems formulations,
we obtain pictures of the entire solution spaces in both the Jordan and the
Einstein frames. This shows, e.g., that due to the domain of the conformal
transformation between the Jordan and Einstein frames, not all the solutions in
the Jordan frame are completely contained in the Einstein frame. We also make
comparisons with previous dynamical systems approaches to $f(R)$ cosmology and
discuss their advantages and disadvantages.
