  Geometric confinements are frequently encountered in soft matter systems and
in particular significantly alter the dynamics of swimming microorganisms in
viscous media. Surface-related effects on the motility of microswimmers can
lead to important consequences in a large number of biological systems, such as
biofilm formation, bacterial adhesion and microbial activity. On the basis of
low-Reynolds-number hydrodynamics, we explore the state diagram of a
three-sphere microswimmer under channel confinement in a slit geometry and
fully characterize the swimming behavior and trajectories for neutral swimmers,
puller- and pusher-type swimmers. While pushers always end up trapped at the
channel walls, neutral swimmers and pullers may further perform a gliding
motion and maintain a stable navigation along the channel. We find that the
resulting dynamical system exhibits a supercritical pitchfork bifurcation in
which swimming in the mid-plane becomes unstable beyond a transition channel
height while two new stable limit cycles or fixed points that are symmetrically
disposed with respect to the channel mid-height emerge. Additionally, we show
that an accurate description of the averaged swimming velocity and rotation
rate in a channel can be captured analytically using the method of hydrodynamic
images, provided that the swimmer size is much smaller than the channel height.
