  We study a generalized P\'{o}lya urn model with two types of ball. If the
drawn ball is red, it is replaced together with a black ball, but if the drawn
ball is black it is replaced and a red ball is thrown out of the urn. When only
black balls remain, the roles of the colors are swapped and the process
restarts. We prove that the resulting Markov chain is transient but that if we
throw out a ball every time the colors swap, the process is recurrent. We show
that the embedded process obtained by observing the number of balls in the urn
at the swapping times has a scaling limit that is essentially the square of a
Bessel diffusion. We consider an oriented percolation model naturally
associated with the urn process, and obtain detailed information about its
structure, showing that the open subgraph is an infinite tree with a single
end. We also study a natural continuous-time embedding of the urn process that
demonstrates the relation to the simple harmonic oscillator; in this setting,
our transience result addresses an open problem in the recurrence theory of
two-dimensional linear birth and death processes due to Kesten and Hutton. We
obtain results on the area swept out by the process. We make use of connections
between the urn process and birth--death processes, a uniform renewal process,
the Eulerian numbers, and Lamperti's problem on processes with asymptotically
small drifts; we prove some new results on some of these classical objects that
may be of independent interest. For instance, we give sharp new asymptotics for
the first two moments of the counting function of the uniform renewal process.
Finally, we discuss some related models of independent interest, including a
"Poisson earthquakes" Markov chain on the homeomorphisms of the plane.
