Peculiarities of the Dynamics of Incommensurate Superstructures in Crystals of the [N(CH3)4]2MeCl4 Family with Factor n = 3

Abstract

This study investigates the dynamics of incommensurate superstructures in crystals of the (N(CH₃)₄)₂MeCl₄ family with a unit-cell multiplication factor n = 3 using numerical simulations within the Landau–Ginzburg framework. The incommensurate phase is described by a two-component order parameter with modulation induced by the Lifshitz invariant and analyzed using Fourier spectra, Lyapunov exponents, dynamic-regime maps, and modulation wave-vector evolution. The results reveal the existence of two structurally distinct incommensurate states within a single incommensurate phase. The first regime corresponds to sinusoidal modulation with narrow periodicity windows and localized chaotic behavior, whereas the second regime exhibits solitonic and stochastic dynamics characterized by higher harmonics, broad periodicity distributions, and hyperchaotic states. The transition between these regimes occurs near the control parameter K ≈ 0.4, where additional modulation harmonics emerge. For K > 1.2, the system enters a hyperchaotic regime with two positive Lyapunov exponents. The role of surface energy is shown to be essential for phase stability. Its increase modifies the effective anisotropic interaction, shifts the phase-transition temperatures, narrows the stability interval of the incommensurate phase, and induces transformations between distinct incommensurate structures through an intermediate commensurate phase.  The physical  significance  of  the case n = 3 is connected with the tripling of the unit cell and the emergence of threefold anisotropy in the thermodynamic potential, which determines the sequence of phase transitions and lock-in behavior. The obtained results provide a consistent physical interpretation of modulation dynamics in low-symmetry ferroelastic crystals and establish a framework for the analysis of complex incommensurate phases.