THE INFLUENCE OF ANISOTROPIC INTERACTION OF THE INCOMENSURATE SUPERSTRUCTURE ON THE BEHAVIOR OF THE LYAPUNOV'S EXPONENTS

Abstract

Lyapunov’s exponents for a wide range of parameters were first calculated for the A₂BX₄ family of crystals, which have an incommensurate superstructure. Lyapunov's exponents determine the dynamics of the incommensurate superstructure. They were calculated by the implicit Adams-Multon and Runge-Kut method in the Python software environment using the jitcode library.

The Adams-Multon (BDF) method, in comparison with the Runge-Kut method (RK45), allows us to obtain a thin structure of Lyapunov's exponents.

The positive value of one Lyapunov’s exponent, and the negative value of the other three exponents, is characteristic of the incommensurate superstructure, which was established. Since the third exponent acquires a value that far exceeds the sum of all others, the incommensurate superstructure is characterized as a system having an attractor. For an incommensurate superstructure which is described by a two-component order parameter, the characteristic range of Lyapunov's exponents is characterized, which is characterized by constant positive values of the first Lyapunov’s exponent. The strongly degenerate abnormal behavior of Lyapunov's third and fourth exponents shows that the incommensurable superstructure is inherent in both hyperchaos and the establishment of a quasi-stable state with the appearance of long-periodic commensurate phases.

A incommensurate phase in sinusoidal mode can be considered as a phase characterized by a strange attractor with one positive Lyapunov’s exponent (+, -, -, -). The soliton mode of the incommensurate superstructure is characterized by the appearance of long-periodic commensurate phases, and the chaotic phase in the transition between two commensurate long-periodic phases. So there is a situation called hyperсhаos: (+, +, -, -), when there are two positive exponents.

The Fourier obtained spectra of the incommensurate superstructure indicate that the ground state of the system is characterized by a boundary cycle. The Fourier spectrum of the boundary cycle is discrete with distinct peaks at frequencies corresponding to the fundamental harmonics of the cycle. The distribution of the spectral density of the chaotic attractor, which arises at T = 1.0 and K = 2.0 in comparison with the limit cycle, is continuous, but all peaks are preserved in it, which, conventionally speaking, is a "memory" of the harmonics of the missing boundary cycle. They are clearly distinguished in a continuous Fourier spectrum. The abnormal spatial behavior of the amplitude and phase of the order parameter is characterized by a chaotic attractor.

Key words: incommensurate superstructures, phase portraits, Lyapunov’s exponents.