THE BEHAVIOR OF THE LYAPUNOV'S EXPONENTS FOR THE INCOMMENSURATE STRUCTURE WITH THE PARAMETER OF THE SPONTANEOUS DEFORMATION AS THE ORDER PARAMETER

Abstract

The Lyapunov’s exponents in a wide range of parameters that determine the dynamics of an incommensurate superstructure for ferroelectric crystals [N(CH₃)₄]₂CuCl₄, which have an incommensurate superstructure were calculated. The calculation of the Lyapunov’s exponents was performed by the Adams-Multon method in the Python software environment using the JiTCODE library.

It is established that the positive value of one Lyapunov exponent, and the negative value of the other three exponents for incommensurate superstructure, is characterized. The incommensurable superstructure is characterized by a strange attractor with a boundary cycle, since the third indicator acquires a value that far exceeds the sum of all others. The Lyapunov’s exponent spectrum, with a constant positive value of the first Lyapunov’s exponent for an incommensurate superstructure described by a two-component order parameter, is characterized. The strongly degenerate abnormal behavior of Lyapunov's third and fourth exponents shows that the incommensurable superstructure is characterized by hyperchaos, which, according to the authors, describes the appearance of a chaotic phase, and the establishment of a quasi-stable state with the appearance of long-periodic proportional phases.

The ground state of the system is characterized by a boundary cycle, as evidenced by the Fourier spectra of an incommensurate superstructure. Fourier spectrum of the boundary cycle is discrete with distinct peaks at frequencies that correspond to the fundamental harmonics of the cycle. In contrast to the boundary cycle, the distribution of the spectral density of the chaotic attractor, which occurs at T = 1.0 and K = 2.0, is continuous, but it preserves all peaks, which, conventionally speaking, are a "memory" of the harmonics of the missing boundary cycle. They are clearly distinguished in a continuous Fourier spectrum. Chaotic attractor is characterized by anomalous spatial behavior of the amplitude and phase of the order parameter. The study of phase-frequency characteristics of spatial oscillations of the amplitude of the order parameter were carried out. Gradual disappearance of the periodic structure as the magnitude K (0 < K <1.3) increases, the phase-frequency characteristic becomes chaotic in the condition of transition to chaos (at K ≥ 1.5), and its autocorrelation is a kind of "white noise".

Keywords: incommensurate superstructure, phase portrait, Lyapunov’s exponent.