Figure 1: A coupled map chain.
A coupled map chain, illustrated in figure 1, consists of N cells,
labelled 
. The state of
the i-th cell at (discrete) timestep t is denoted 
.
A timestep is defined such that at its completion, the state of all N cells
has been updated exactly once.
The i-th cell implements a nonlinear function 
,
where 
is a weighted sum of 
, i.e. the state of this cell
at the previous timestep, and the state of one other cell, say the j-th, at
a former timestep, say 
. There are various possible values for j
and 
. Obviously, if the CMC is to be deterministic, 
, otherwise
future states would be needed to calculate the current state. In this paper
we consider one possibility, that the state of the i-th cell depends only on
its own state and the state of the 
-th cell, both at the previous timestep.
This is in contrast to the CMC described in [3] in which the
i-th cell is also coupled to the 
-th cell. In our case,
In this case, 
if 
for all i
and t. Throughout the paper we choose 
with
, so that
and hence 
and t.
This case is referred to as simultaneous updating, since all cells could in
principle be updated at the same time, given suitable buffering
on their outputs. (Buffering is required to transfer the new value
of the state variable to the output only at the end of the timestep.)
Each cell is described by two parameters
(nonlinearity parameter 
and a weight 
) and one state
variable, 
.
