Photoactuated self-oscillating systems have received increasing attention due to their capability to convert light energy into sustained mechanical motion. Nevertheless, most reported systems exhibit smooth and continuous oscillations, with limited ability to accumulate elastic energy and release it abruptly for burst-like motion. This study proposes a bistable photoactuated liquid crystal elastomer (LCE)-magnet beam composed of a bilayer composite beam, a tip magnet, and two metal blocks. Through the coupling of photothermal contraction and magnetic adhesion, the proposed beam exhibits the autonomous self-flipping behavior without external control. A nonlinear mechanical model is developed to clarify the underlying mechanism, showing that the self-flipping motion results from the competition between the equivalent force induced by LCE contraction and the magnetic adhesion forces from the two metal blocks. Three motion states, namely static, single-flip, and continuous self-flipping states, are identified. Two critical contraction conditions for state transitions are derived analytically, providing criteria for the onset and persistence of self-flipping oscillation. Parametric analysis further shows that key system parameters govern the transition boundaries between motion states and affect the oscillation frequency. The results provide useful insight into the design of photoactuated smart beam structures with self-sustained burst-like periodic motion.