  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.
