In　this　work,　we　derive　exact　solutions　of　a　dynamical　equation,　which　can　represent　all　two-level　Hermitian　systems　driven　by　periodic　N-step　driving　fields.　For　different　physical　parameters,　this　dynamical　equation　displays　various　phenomena　for　periodic　N-step　driven　systems.　The　time-dependent　transition　probability　can　be　expressed　by　a　general　formula　that　consists　of　cosine　functions　with　discrete　frequencies,　and,　remarkably,　this　formula　is　suitable　for　arbitrary　parameter　regimes.　Moreover,　only　a　few　cosine　functions　(i.e.,　one　to　three　main　frequencies)　are　sufficient　to　describe　the　actual　dynamics　of　the　periodic　N-step　driven　system.　Furthermore,　we　find　that　a　beating　in　the　transition　probability　emerges　when　two　(or　three)　main　frequencies　are　similar.　Some　applications　are　also　demonstrated　in　quantum　state　manipulations　by　periodic　N-step　driving　fields.　©　2021　American　Physical　Society.