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.
Some accreting neutron stars and young stars show unexplained episodic flares in the form of quasi-periodic oscillations or recurrent outbursts. In a series of two papers we present new work on an instability that can lead to episodic outbursts when
the accretion disc is truncated by the stars strong magnetic field close to the corotation radius (where the Keplerian frequency matches the stars rotational frequency). In this paper we outline the physics of the instability and use a simple parameterization of the disc-field interaction to explore the instability numerically, which we show can lead to repeated bursts of accretion as well as steady-state solutions, as first suggested by Sunyaev and Shakura. The cycle time of these bursts increases with decreasing accretion rate. These solutions show that the usually assumed `propeller state, in which mass is ejected from the system, does not need to occur even at very low accretion rates.
