  We discuss visual observations spanning nearly 70 years of the nearby
semiregular variable R Doradus. Using wavelet analysis, we show that the star
switches back and forth between two pulsation modes having periods of 332 days
and about 175 days, the latter with much smaller amplitude. Comparison with
model calculations suggests that the two modes are the first and third radial
overtone, with the physical diameter of the star making fundamental mode
pulsation unlikely. The mode changes occur on a timescale of about 1000 d,
which is too rapid be related to a change in the overall thermal structure of
the star and may instead be related to weak chaos.
  The Hipparcos distance to R Dor is 62.4 +/- 2.8 pc which, taken with its
dominant 332-day period, places it exactly on the period-luminosity relation of
Miras in the Large Magellanic Cloud. Our results imply first overtone pulsation
for all Miras which fall on the P-L relation. We argue that semiregular
variables with long periods may largely be a subset of Miras and should be
included in studies of Mira behaviour. The semiregulars may contain the
immediate evolutionary Mira progenitors, or stars may alternate between periods
of semiregular and Mira behaviour.
