The rate of dynamical chaos during propagation of the positive Lyapunov exponents region under nonlocalilty conditions

Abstract

Background. When considering the problems of nonequilibrium dynamics of chaotic systems, it is of interest to study processes at short observation times, including using nonlocal models, and to study the motion of the order-chaos boundary. Materials and methods. We compare the solutions of the classical heat equation with the solutions for two nonlocal heat transfer models. The telegraph equation and the random walk model were considered. The system responses to perturbations in the form of the Dirac delta function and the Heaviside step function were investigated. The dynamics of the system is investigated when one part of the system initially behaves in a regular manner and the other in a chaotic one. The propagation of the chaos is considered as the motion of a region with the largest Lyapunov exponent greater than zero. Results. The time dependencies of the chaos propagation parameters were calculated for the classical and non-local models of nonstationary heat transfer. It is shown that for an initial perturbation of the delta function type, the deviations of the nonlocal models from the classical solution decrease with time as ln t / t and faster, and for the boundary condition of the Heaviside step function type the relative deviations from the classical solution decrease with time as 1/ t . Conclusions. The significant differences in the behavior of the systems under consideration are observed at times t < 102 in the selected system of units, which corresponds, for example, in Lennard- Jones systems t < 10−9 sec.

Keywords

• dynamical chaos
• propagation
• velocity
• lyapunov exponent
• heat conduction
• non-local models