  We investigate the evaporation of a two-dimensional droplet on a solid
surface. The solid is flat but with smooth chemical variations that lead to a
space-dependent local contact angle. We perform a detailed bifurcation analysis
of the equilibrium properties of the droplet as its size is changed, observing
the emergence of a hierarchy of bifurcations that strongly depends on the
particular underlying chemical pattern. Symmetric and periodic patterns lead to
a sequence of pitchfork and saddle-node bifurcations that make stable solutions
to become saddle nodes. Under dynamic conditions, this change instability
suggests that any perturbation in the system can make the droplet to shift
laterally while relaxing to the nearest stable point, as is confirmed by
numerical computations of the Cahn-Hilliard and Navier-Stokes system of
equations. We also consider patterns with an amplitude gradient that creates a
set of disconnected stable branches in the solution space, leading to a
continuous change of the droplet's location upon evaporation.
