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Abstract

The Nobel Prize winning confirmation in 1998 of the accelerated expansion of
our Universe put into sharp focus the need of a consistent theoretical model to
explain the origin of this acceleration. As a result over the past two decades
there has been a huge theoretical and observational effort into improving our
understanding of the Universe. The cosmological equations describing the
dynamics of a homogeneous and isotropic Universe are systems of ordinary
differential equations, and one of the most elegant ways these can be
investigated is by casting them into the form of dynamical systems. This allows
the use of powerful analytical and numerical methods to gain a quantitative
understanding of the cosmological dynamics derived by the models under study.
In this review we apply these techniques to cosmology. We begin with a brief
introduction to dynamical systems, fixed points, linear stability theory,
Lyapunov stability, centre manifold theory and more advanced topics relating to
the global structure of the solutions. Using this machinery we then analyse a
large number of cosmological models and show how the stability conditions allow
them to be tightly constrained and even ruled out on purely theoretical
grounds. We are also able to identify those models which deserve further in
depth investigation through comparison with observational data. This review is
a comprehensive and detailed study of dynamical systems applications to
cosmological models focusing on the late-time behaviour of our Universe, and in
particular on its accelerated expansion. In self contained sections we present
a large number of models ranging from canonical and non-canonical scalar
fields, interacting models and non-scalar field models through to modified
gravity scenarios. Selected models are discussed in details and interpreted in
the context of late-time cosmology.