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We study the Landau model for uniaxial incommensurate-commensurate systems ofthe I class by keeping Umklapp terms of third and fourth order in the expansionof the free energy. It applies to systems in which the soft mode minimum liesbetween the corresponding commensurate wave numbers. The minimization of theLandau functional leads to the sine-Gordon equation with two nonlinear terms,equivalent to the equation of motion for the well-known classical mechanicalproblem of two mixing resonances. We calculate the average free energies forperiodic, quasiperiodic and chaotic solutions of this equation, and show thatin the regime of finite strengths of Umklapp terms only periodic solutions areabsolute minima of the free energy, so that the phase diagram contains onlycommensurate configurations. The phase transitions between neighboringconfigurations are of the first order, and the wave number of ordering goesthrough harmless staircase with a finite number of steps. These results are thebasis for the interpretation of phase diagrams for some materials from the Iclass of incommensurate-commensurate systems, in particular of those forA$_2$BX$_4$ and BCCD compounds. Also, we argue that chaotic barriers whichseparate metastable periodic solutions represent an intrinsic mechanism forobserved memory effects and thermal hystereses.Comment: 12 pages, 14 figures, LaTeX, to be published in Phys. Rev.

Landau model for uniaxial systems with complex order parameter