FEATURES OF THE BEHAVIOR OF THE LYAPUNOV’S EXPONENTS UNDER CONDITIONS OF THE TETRAEDRIC GROUPS CORRELATED MOVEMENT SPATIAL AREAS EXISTENCE

Abstract

The dynamics of an incommensurate superstructure at the moment of it appear is investigated in the work. The spectra of the magnitude of the Lyapunov’s exponents and the maps of the dynamic modes of the incommensurate phase are analyzed for the 0.0 ÷ 1.0 interval of the T and K parameters. The calculation of Lyapunov’s exponents is performed in the Python environment using the libraries Skipy, JiTCODE.

It is established that a chaotic state arises in this system at small values of the T and K (0.0015 ÷ 0.03) parameters, which is caused by the interaction between different spatial regions of the correlated motion of tetrahedral groups. The appearance of a chaotic state (at the origin of a incommensurate superstructure) and its existence at small values of the parameter T are evidenced by both the spectra of the Lyapunov’s exponents λ₃ and λ₄ and the maps of dynamic regimes. As the parameter T increases, the chaotic state disappears.

Thus, a system characterized by two differential equations to describe a incommensurate superstructure in the presence of the two-component order parameter is characterized by the complex dynamics in the process of its origin. An important role in the dynamics of the origin of the incommensurate superstructure is played by a quantity that describes the degree of long-range interaction. At small values of its magnitude, the spatial regions of the correlated motion of tetrahedral groups do not interact with each other. The anharmonic oscillations of the main lattice and the sublattice (which describes the modulation superstructure) cause an increase in the long-range interaction, which causes the expansion of the spatial region of the correlated motion of tetrahedral groups, and their interaction. The interaction of the spatial regions of the correlated motion of tetrahedral groups leads to a violation of the modulation picture of the motion of tetrahedral groups, thereby causing the appearance of a chaotic state of the superstructure. Studies of Lyapunov’s exponents λ₃ and λ₄, which describe the phase behavior of the order parameter and the phase velocity of the superstructure, respectively, indicate the appearance of the chaotic state and its dependence on the initial conditions.

The considered dissipative system is characterized by energy loss in the process of its development, so the initial region in phase space decreases, and, finally, trajectories from all initial conditions taken in a certain region (gravity basin) converge to some attractor installed in phase space. Typically, a dissipative dynamic system has only one or more such attractors at certain parameter values. It is known that the number of attractors increases with decreasing dissipation, but there are few chaotic between them. This phenomenon is strongly associated with the crises that the attractor experiences when changing parameters, and causes a collision with the boundaries of the pool when changing the nonlinearity parameter. As a result, the chaotic attractor disappears immediately after its appearance from the cascade of doubling the period. Changes also occur with “gravity” pools and pool boundaries. The former become quite small for most coexisting attractors.

The obtained maps of dynamic modes show that the underdeveloped chaotic state of the incommensurate superstructure that arises in the process of its formation exists at small values of the parameter T (the magnitude of the long-range interaction) and disappears with its increase.

Key words: the incommensurate superstructure, the phase portrait, the dynamic modes, the spectra of Lyapunov’s exponents.