Symmetric attractors in systems of coupled logistic maps
Arne Jakobsen
Department of Mathematical Sciences
The Norwegian Institute of Technology
N-7034 Trondheim, Norway
In this paper we consider how the presence of 
-symmetry in a
coupled map lattice (CML) of N cells imposes restrictions to the
dynamics of the system. Bifurcations of stable states are explained by
use of equivariant bifurcation theory. We list all possible symmetry
types of period two points of the system and illustrate them with
numerical examples. We also consider arbitrary types of chaotic
attractors of such a system. Numerical results from simulations are given
and the structure of the attractor is explained in terms of admissible
and strongly admissible subgroups for such a system.
Lattice cell 
has local dynamics given by the one
dimensional logistic map 
. We
shall only consider nearest neighbor diffusive coupling. If we denote by
the state vector 
at time 
, the dymanics of
cell i at time 
is given by
Furthermore, we assume periodic boundary condition, that is, 
and 
. From this it follows that the map 
is -equivariant,
that is, 
for all
, where is the dihedral group.
We say that the 
-limit set 
of x under 
is
an attractor for if A is stable.
Now, assume that 
is a finite group acting
linearly on 
. The symmetry group of A is defined by
.
An important result is that the attractor of the CML-dynamical system under consideration can possess a well defined global symmetry eve if the attractor is chaotic. Example of this phenomenon is given for N=3.
A result by Melbourne, Dellnitz and Golubitsky gives that the symmetry
groups of chaotic attractors are subject to restrictions. The admissible
subgroups depend crucially on the geometry of those reflection
hyperplanes in that correspond to reflections that are in
but not in 
. This enables us to explain some of the
symmetric chaotic attractors observed in simulations. For N=3, there
only exists attractors with symmetry conjugate to 
and
. In the case we can also conclude the
attractor is strongly admissible, that is, admissible and connected.