Chaos **14** (2004) 571-582.
doi:10.1063/1.1769111
## Cycling chaotic attractors in two models for dynamics with invariant
subspaces

Peter Ashwin(1)
A.M.Rucklidge(2)
Rob Sturman(2).

(1) School of Mathematical Sciences, Laver Building,

University of Exeter, Exeter EX4 4QE, UK

(2) Department of Applied Mathematics,

University of Leeds, Leeds, LS2 9JT, UK

**Abstract.**
Nonergodic attractors can robustly appear in symmetric systems as structurally
stable cycles between saddle-type invariant sets. These saddles may be chaotic
giving rise to 'cycling chaos'. The robustness of such attractors appears by
virtue of the fact that the connections are robust within some invariant
subspace. We consider two previously studied examples and examine these in
detail for a number of effects: (i) presence of internal symmetries within the
chaotic saddles, (ii) phase-resetting, where only a limited set of connecting
trajectories between saddles are possible and (iii) multistability of periodic
orbits near bifurcation to cycling attractors.
The first model consists of three cyclically coupled Lorenz equations and was
investigated first by Dellnitz et al. (1995). We show that one can find a
'false phase-resetting' effect here due to the presence of a skew product
structure for the dynamics in an invariant subspace; we verify this by
considering a more general bi-directional coupling. The presence of internal
symmetries of the chaotic saddles means that the set of connections can never
be clean in this system, that is, there will always be transversely repelling
orbits within the saddles that are transversely attracting on average.
Nonetheless we argue that 'anomalous connections' are rare.

The second model we consider is an approximate return mapping near the stable
manifold of a saddle in a cycling attractor from a magnetoconvection problem
previously investigated by two of the authors. Near resonance, we show that the
model genuinely is phase-resetting, and there are indeed stable periodic orbits
of arbitrarily long period close to resonance, as previously conjectured. We
examine the set of nearby periodic orbits in both parameter and phase space and
show that their structure appears to be much more complicated than previously
suspected. In particular, the basins of attraction of the periodic orbits
appear to be pseudo-riddled in the terminology of Lai (2001).

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