Landau model for uniaxial systems with complex order parameter

We study the Landau model for uniaxial incommensurate-commensurate systems of class I by keeping umklapp terms of third and fourth order in the expansion of the free energy. It applies to systems in which the soft-mode minimum lies between the corresponding commensurate wave numbers. The minimization of the Landau functional leads to the sine-Gordon equation with two nonlinear terms, equivalent to the equation of motion for the well-known classical mechanical problem of two mixing resonances. We calculate the average free energies for periodic, quasiperiodic, and chaotic solutions of this equation, and show that in the regime of finite strengths of umklapp terms only periodic solutions are absolute minima of the free energy, so that the phase diagram contains only commensurate configurations. The phase transitions between neighboring configurations are of the first order, and the wave number of ordering goes through a harmless staircase with a finite number of steps. These results are the basis for the interpretation of phase diagrams for some materials from class I of incommensurate-commensurate systems, in particular of those for $A_2{\mathrm{BX}}_4$ and betaine-calciumchloride-dihydrate compounds. Also, we argue that chaotic barriers which separate metastable periodic solutions represent an intrinsic mechanism for observed memory effects and thermal hystereses.