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Dynamical systems approaches and methods in cosmology

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I will with simple examples from spatially homogeneous and isotropic cosmology illustrate the importance of respecting the global features of a state space for a given model when reformulating field equations to useful dynamical systems. In particular I will use examples from f(R) gravity and GR with a minimally coupled scalar field. In this context I will also illustrate how various dynamical systems methods, such as, e.g., monotonic functions, center manifold techniques, averaging methods, can yield a global understanding of the solution spaces as well as approximations, complementing, e.g., the slow-roll approximation.