Classical adiabatic invariants in actual adiabatic processes possess intrinsic dynamical fluctuations. The magnitude of such intrinsic fluctuations is often thought to be negligible. This widely believed physical picture is contested here. For adiabatic following of a moving stable fixed-point solution facing a pitchfork bifurcation, we show that intrinsic dynamical fluctuations in an adiabatic process can assist in a deterministic and robust selection between two symmetry-connected fixed-point solutions, irrespective of the rate of change of adiabatic parameters. Using a classical model Hamiltonian also relevant to a two-mode quantum system, we further demonstrate the formation of an adiabatic hysteresis loop in purely Hamiltonian mechanics and the generation of a Berry phase via changing one single-valued parameter only.