  We consider the motion of a harmonically trapped overdamped particle, which
is submitted to a self-phoretic force, that is proportional to the gradient of
a diffusive field for which the particle itself is the source. In agreement
with existing results for free particles or particles in a bounded domain, we
find that the system exhibits a transition between an immobile phase, where the
particle stays at the center of the trap, and an oscillatory state. We perform
an exact analysis giving access to the bifurcation threshold, as well as the
frequency of oscillations and their amplitude near the threshold. Our analysis
also characterizes the shape of two-dimensional oscillations, that take place
along a circle or a straight line. Our results are confirmed by numerical
simulations.
