  It has long been suggested that the mid-latitude atmospheric circulation
possesses what has come to be known as `weather regimes', loosely categorised
as regions of phase space with above-average density and/or extended
persistence. Their existence and behaviour has been extensively studied in
meteorology and climate science, due to their potential for drastically
simplifying the complex and chaotic mid-latitude dynamics. Several well-known,
simple non-linear dynamical systems have been used as toy-models of the
atmosphere in order to understand and exemplify such regime behaviour.
Nevertheless, no agreed-upon and clear-cut definition of a `regime' exists in
the literature. We argue here for an approach which equates the existence of
regimes in a dynamical system with the existence of non-trivial topological
structure of the system's attractor. We show using persistent homology, an
algorithmic tool in topological data analysis, that this approach is
computationally tractable, practically informative, and identifies the relevant
regime structure across a range of examples.
