  We present a new analysis of the long-period variables in the Large
Magellanic Cloud (LMC) from the MACHO Variable Star Catalog. Three-quarters of
our sample of evolved, variable stars have periodic light curves. We
characterize the stars in our sample using the multiple periods found in their
frequency spectra. Additionally, we use single-epoch Two Micron All Sky Survey
measurements to construct the average infrared light curves for different
groups of these stars. Comparison with evolutionary models shows that stars on
the red giant branch (RGB) or the early asymptotic giant branch (AGB) often
show non-periodic variability, but begin to pulsate with periods on the two
shortest period-luminosity sequences (3 & 4) when they brighten to K_s ~ 13.
The stars on the thermally pulsing AGB are more likely to pulsate with longer
periods that lie on the next two P-L sequences (1 & 2), including the sequence
associated with the Miras in the LMC. The Petersen diagram and its variants
show that multi-periodic stars on each pair of these sequences (3 & 4, and 1 &
2) typically pulsate with periods associated only with that pair. The periods
in these multi-periodic stars become longer and stronger as the star evolves.
We further constrain the mechanism behind the long secondary periods (LSPs)
seen in half of our sample, and find that there is a close match between the
luminosity functions of the LSP stars and all of the stars in our sample, and
that these star's pulsation amplitudes are relatively wavelength independent.
Although this is characteristic of stellar multiplicity, the large number of
these variables is problematic for that explanation.
