Abstract: It generalizes the Frenkel-Kontorova model to the Frenkel-Kontorova-Devonshire model where the interactions are the triple-well potential. By use of the effective potential method, numerical solutions of eigenvalue problem are used to work out the exact phase diagrams of W a triple-well potential and V a piecewise parabolic potential. According to the winding number ω and the rotation number Ω, the periodicity of the phase diagram is analyzed and some complex but regular phase structures are found.

Key words: CI transitions, effective potential, rotation number, winding number, Farey tree