Physica D
**122** (1998) 134-154.
## Cycling chaos: its creation, persistence and loss of stability in a
model of nonlinear magnetoconvection

Peter Ashwin(1)
and
A.M.Rucklidge(2).

(1) Department of Mathematical and Computing Sciences,

University of Surrey, Guildford GU2 5XH, UK

(2) Department of Applied Mathematics and Theoretical Physics,

University of Cambridge, Cambridge, CB3 9EW, UK

**Abstract.**
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.
Such behaviour is robust to perturbations that preserve the symmetry of the
system; 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
differs from the standard scenario for the blowout bifurcation in that in our
case, the blowout is neither subcritical nor supercritical. 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 ordinary differential equation
(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, with variation
in the vertical but only one of the two horizontal directions. 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. This allows us to describe and
predict a number of observed transitions in these models. The dynamics we
describe is new but nonetheless robust, and so should occur in
other applications.

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