We demonstrate, with a class of exactly solvable Frenkel-Kontorova models, the emergence of the non-recurrent rotationally ordered con_gurations whenever certain resonance conditions are satis_ed among the openings in the hull function in both the commensurate and the incommensurate cases. If one insists on depicting them with hull functions, suitably de_ned extended numbers must be employed. The defect-mediated transition in the incommensurate case is also discussed.