Using the same systems of transformed equations, we compute 140 years worth of time series for 500 realizations. As before, we ignore the first 40 years worth of transient solution, and the data are used to create a histogram representing the probability density p SI of the S and I values. Figure 6͑a͒ shows the histogram associated with the complete stochastic system of transformed equations, while Fig.  6͑b͒ shows the histogram associated with the reduced system of equations found using the normal form coordinate transform. The color-bar values in Figs. 6͑a͒ and 6͑b͒ have been normalized by 10 −3 .