  We examine a model system where attractors may consist of a heteroclinic
cycle between chaotic sets; this `cycling chaos' manifests itself as
trajectories that spend increasingly long periods lingering near chaotic
invariant sets interspersed with short transitions between neighbourhoods of
these sets. This behaviour can be robust (i.e., structurally stable) for
systems with symmetries and provides robust examples of non-ergodic attractors
in such systems; we examine bifurcations of this state.
  We discuss a scenario where an attracting cycling chaotic state is created at
a blowout bifurcation of a chaotic attractor in an invariant subspace. This is
a novel scenario for the blowout bifurcation and causes us to introduce the
idea of set supercriticality to recognise such bifurcations. The robust cycling
chaotic state can be followed to a point where it loses stability at a
resonance bifurcation and creates a series of large period attractors.
  The model we consider is a 9th order truncated ODE model of three-dimensional
incompressible convection in a plane layer of conducting fluid subjected to a
vertical magnetic field and a vertical temperature gradient. Symmetries of the
model lead to the existence of invariant subspaces for the dynamics; in
particular there are invariant subspaces that correspond to regimes of
two-dimensional flows. Stable two-dimensional chaotic flow can go unstable to
three-dimensional flow via the cross-roll instability. We show how the
bifurcations mentioned above can be located by examination of various
transverse Liapunov exponents. We also consider a reduction of the ODE to a map
and demonstrate that the same behaviour can be found in the corresponding map.
