THE FEATURES OF BEHAVIOR OF THE LYAPUNOV'S EXPONENTS FROM THE SYMMETRY OF THERMODYNAMIC POTENTIAL DESCRIBED BY LIFSHITZ INVARIANT

Abstract

The influence of the thermodynamic potential (n) symmetry on the behavior of the Lyapunov exponents is investigated. The calculation of the system of second-order differential equations was performed by a multi-step variable-order BDF method. The Lyapunov coefficients were calculated using Python software using Skipy, JiTCODE libraries.

It is established that for values of n ≥ 5, the dynamics of the incommensurate superstructure is determined by the spatial change in the amplitude of the order parameter (the first integral of the motion of the incommensurate superstructure). For n < 5, the dynamics of a incommensurate superstructure is determined by the spatial change of the phase of the order parameter and its perturbation. It is shown that the dynamics of the superstructure is not affected by the ferroelectric or ferroelastic nature of the incommensurate superstructure.

In order to obtain a detailed picture of the dynamics of the incommensurate superstructure, phase portraits were investigated in the coordinates R, R`, and φ at different values of the anisotropic interaction of the superstructure and the initial phase values of the order parameter φ₀. It is established that the dynamics of this system is determined by the cascades of transition to chaos and areas of self-organization of the superstructure. It is shown that at the final stage of the transition to chaos, the system is characterized by the emergence of a chaotic attractor, which has a more complex phase portrait structure.

The projection of such an attractor consists of two symmetrical parts with respect to the axis of the abscissa. The movement of a typical trajectory of a chaotic attractor can be divided into two phases. In the first phase of the trajectory, chaotic movements are made along the turns of the upper (lower) part of the attractor from time to time, passing to the boundary of the localization region where the stability of the upper (lower) symmetric boundary cycles is lost. At an unpredictable time, the trajectory moves from the upper (lower) part of the attractor to its upper (lower) part. This process is repeated infinitely many times. Thus, the transition to chaos has features characteristic of the Feigenbaum scenario (infinite cascade of bifurcations of doubling cycles), and for mixing (unpredictable mixing between the upper and lower parts of the emerging chaotic attractor). The self-organization process can be associated with the localization of a wave vector of incommensurate superstructure at a commensurate higher order value.

It is shown that the distortion of the order parameter phase at the edge of the crystal can be related to boundary conditions in the first approximation. According to the authors, such cascading transitions to chaos are related to the modes of existence of a incommensurate superstructure. The emergence of chaos in the third stage is caused by the transition of a incommensurate superstructure to its stochastic mode of existence. In a stochastic mode of existence of a incommensurate superstructure, a chaotic, possibly incommensurate phase arises, which is a prototype of this chaotic state. It should be noted that the initial conditions (boundary conditions) stabilize the system, thereby leading to the removal of the degeneracy of the system, and the transition of a incommensurate superstructure to a state characterized by the existence of long-period proportional phases.

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