Quantum states which evolve cyclically in their projective Hilbert space give rise to a geometric (or Aharonov-Anandan) phase. An aspect of primary interest is stable cyclic behaviour as realized under a periodic Hamiltonian. The problem has been handled by use of time-dependent transformations treated along the lines of Floquet's theory as well as in terms of exponential operators with a goal to examine the variety of initial states exhibiting cyclic behaviour. A particular case of special cyclic initial states is described which is shown to be important for nuclear magnetic resonance experiments aimed at the study of the effects of the geometric phase. An example of arbitrary spin j in a precessing magnetic field and spin j=1 subject to both axially symmetric quadrupolar interaction and a precessing magnetic field are presented. The invariant (Kobe's) geometric phase is calculated for special cyclic states.