  In 1996, Edward Lorenz introduced a system of ordinary differential equations
that describes a single scalar quantity as it evolves on a circular array of
sites, undergoing forcing, dissipation, and rotation invariant advection.
Lorenz constructed the system as a test problem for numerical weather
prediction. Since then, the system has also found widespread use as a test case
in data assimilation. Mathematically, it belongs to a class of dynamical
systems with a single bifurcation parameter (rescaled forcing) that undergoes
multiple bifurcations and exhibits chaotic behavior for large forcing. In this
paper, the main characteristics of the advection term in the model are
identified and used to describe and classify a number of possible
generalizations of the system. A graphical method to study the bifurcation
behavior of constant solutions is introduced, and it is shown how to use the
rotation invariance to compute normal forms of the system analytically.
Problems with site-dependent forcing, dissipation, or advection are considered
and basic existence and stability results are proved for these extensions. We
address some related topics in the appendices, wherein the Lorenz '96 system in
Fourier space is considered, explicit solutions for some advection-only systems
are found, and it is demonstrated how to use advection-only systems to assess
numerical schemes.
