The thresholding behaviour described above occurs for a narrow range
of parameters. We now describe a more prevalent mode of operation of the CML,
which is best described as smoothing, or nonlinear low pass filtering.
Such behaviour has been reported elsewhere; for instance, in [4],
it is shown numerically that every sixth iterate of a logistic map with delay
lies on a smooth curve. Figure 5 illustrates a typical case for a
CMC, in which N=12, 
,
, 
and 
for i > 1. Again, these parameters
are such that 
generates a chaotic sequence, whereas 
would generate a period-2 sequence in the absence of self-coupling.
Qualitatively the same smoothing effect takes place if the input is pseudo-random
numbers instead of a chaotic sequence.
Figure 5: Illustration of nonlinear low-pass filtering by a CML.
The input time series is shown at the back right of the plot. Moving from
back right to front left corresponds to travelling along the CML; from back
left to front right corresponds to moving forward in time. The z-axis is the
state of the cells as a function of distance and time, measured every other
timestep.
The simplest digital low pass filter consists of a running average. In a CMC each cell adds a weighted sum of all previous states because of the local feedback. This structure is similar to an infinite impulse response linear digital filter so it is not surprising that we see smoothing.
Smoothing behaviour is also only observed
if the states of the cells are measured every other timestep. It would be
desirable therefore to find an expression that gives the value of
as a function of the states of the i-th and possibly other
cells, two timesteps ago. In this way, we eliminate the need for states
of cells at the intermediate timesteps, allowing us to observe the smoothing
behaviour directly. It is possible to find such an expression; combining
equations 1 and 2 gives
where the function 
depends on the parameters 
and 
. Using
this equation again for 
and 
gives
The function 
depends on the parameters 
, 
, 
and
. More importantly though, it depends only on the state variables of
, 
and 
at timestep t-2, i.e. two timesteps ago.
The function 
is quite messy in the general case, but for
and 
with
where the superscript t-2 is understood. The function G has the four fixed points
The stability of these fixed points is currently being investigated.
