  We show that discs accreting onto the magnetosphere of a rotating star can
end up in a 'trapped' state, in which the inner edge of the disc stays near the
corotation radius, even at low and varying accretion rates. The accretion in
these trapped states can be steady or cyclic; we explore these states over wide
range of parameter space. We find two distinct regions of instability, one
related to the buildup and release of mass in the disk outside corotation, the
other to mass storage within the transition region near corotation. With a set
of calculations over long time scales we show how trapped states evolve from
both nonaccreting and fully accreting initial conditions, and also calculate
the effects of cyclic accretion on the spin evolution of the star. Observations
of cycles such as found here would provide important clues on the physics of
magnetospheric accretion. Recent observations of cyclic and other unusual
variability in T Tauri stars (EXors) and X-ray binaries are discussed in this
context.
