The dayside magnetosphere contains a region where the field strength has a local maximum. This region, located just inside the magnetopause around the equatorial plane and between the cusps, has a width of 2--3 Re. When a drift shell with a sufficiently small mirror field intersects this region, it will bifurcate into two branches near local noon, each branch going across one cusp and joining together at the symmetrical local time. The particle then drifts around the Earth over a single branch until it comes back to local noon. In the neighborhood of the bifurcation points, the bounce period tends to infinity, and thus the adiabaticity of the bounce motion is broken there, but not elsewhere. This breaking causes a small but finite jump Delta I in the second invariant. Repeated crossings lead to a random walk in second invariant space, and thus to radial diffusion. We use theory and simulations to determine the magnitude of DeltaI. Our study is limited to static magnetic fields, but it can be extended to general fields. Our results indicate that DeltaI is sensitively dependent on bounce phase at bifurcation, and it can grow significantly for some initial conditions. When the initial second invariant I0 is much larger than the mirror gyroradius rhom, we use separatrix crossing theory. The average of DeltaI over bounce phases is zero, and the rms DeltaI is of the order of rho m. When I0 is comparable to rhom , the equation of bounce motion is approximated as the second Painleve equation, whose asymptotic solutions are used to determine Delta I. In this limit, the rms DeltaI is still O (rhom); however, the average is nonzero, in the form exp(- I0/rhom). Drift-shell bifurcation leads to significant radial diffusion. For MeV electrons, the diffusion coefficient can be several R e per day. Also, because of bifurcation, some quasitrapped particles can remain in the magnetosphere for a finite number of drifts before they leave permanently. Such behavior leads to metastable particles, a new kind of trapping. These results can be useful for radiation-belt modeling efforts.