### Abstract:

In this work the performance of ensembles generated by commonly used methods in a nonlinear system with
multiple attractors is examined. The model used here is a spectral truncation of a barotropic quasigeostrophic
channel model. The system studied here has 44 state variables, great enough to exhibit the problems associated
with high state dimension, but small enough so that experiments with very large ensembles are practical, and
relevant probability density functions (PDFs) can be evaluated explicitly. The attracting sets include two stable
limit cycles.
To begin, the basins of attraction of two known stable limit cycles are characterized. Large ensembles are
then used to calculate the evolution of initially Gaussian PDFs with a range of initial covariances. If the initial
covariances are small, the PDF remains essentially unimodal, and the probability that a point drawn from the
initial PDF lies in a different basin of attraction from the mean of that PDF is small. If the initial covariances
are so large that there is significant probability that a given point in the initial ensemble does not lie in the same
basin of attraction as the mean, the initial Gaussian PDF will evolve into a bimodal PDF. In this case, graphical
representation of the PDF appears to split into two distinct regions of relatively high probability.
The ability of smaller ensembles drawn from spaces spanned by singular vectors and by bred vectors to
capture this splitting behavior is then investigated, with the objective here being to see how well they capture
multimodality in a highly nonlinear system. The performance of similarly small random ensembles drawn without
dynamical constraints is also evaluated.
In this application, small ensembles chosen from subspaces of singular vectors performed well, their weakest
performance being for an ensemble with relatively large initial variance for which the Gaussian character of
the initial PDF remained intact. This was the best case for the bred vectors because of their tendency to align
tangent to the attractor, but the bred vectors were at a disadvantage in detection of the tendency of an initially
Gaussian PDF to evolve into a bimodal one, as were the unconstrained ensembles.