Feedback methods used to control chaos as discussed in this paper
are most closely related to the algorithm first
proposed by Ott, Grebogi, and Yorke (OGY)[10].
The original OGY method required the measurement of two system variables
at the Poincaré surface (in a three dimensional system) to determine
the appropriate feedback for control.
However, many experimental situations (such as that described in this paper)
donot easily allow the measurement of more than one variable.
It was recently
found[3,11,12] that the
original OGY method must be modified
to apply to the situation where single time series data is used. In general,
this requires nontrivial modification of the algorithm and makes the prescribed
change in the control parameter on the nth cycle depend on the changes
that were made
on previous cycles. It was shown in reference [3] that in highly
dissipative systems this reduces to a simple recursive algorithm where the
change on the nth cycle depends only on the change made on the 
th
cycle as indicated in Eq. 1. Furthermore, it was shown that
the recursive term goes to zero (R = 0) if the attractor
(in the neighborhood of the fixed point at the Poincaré section)
does not shift in the direction normal to its plane in state space
when small changes are made in the control parameter.
Figure: Poincaré sections reconstructed using time delay
embedding of the measured time series of the anodic current.
The anodic current is minimum at times 
and
600 msec. The open
and closed circles are for values of the anodic potential that
differ by 4 mV.
Figure 6 shows the Poincaré section of the reconstructed
attractor in a delay
coordinate embedding for the electrochemical cell at parameter values
where we used RPF to control on a period-1 orbit. The time 
is
the time when the anodic current goes through the n-th minimum near the
period-1 orbit. Two sets of data are shown with the anodic
potential, V, held constant in each case. The open circles are for
V = 0.720 V and the closed circles for V = 0.724 V. This figure
showing experimental data should be compared with Fig. 1b of
reference [3]. While the observed experimental shift in the
Poincaré section is small, there is clear evidence that a shift is
present.