Hidden and self-excited attractors in radiophysical and biophysical models

Abstract

One of the central tasks of investigation of dynamical systems is the problem of analysis of the steady (limiting) behavior of the system after the completion of transient processes, i.e., the problem of localization and analysis of attractors (bounded sets of states of the system to which the system tends after transient processes from close initial states). Transition of the system with initial conditions from the vicinity of stationary state to an attractor corresponds to the case of a self-excited attractor. However, there exist attractors of another type: hidden attractors are attractors with the basin of attraction which does not have intersection with a small neighborhoods of any equilibrium points. Classification "hidden vs self-excited" attractors was introduced by Leonov and Kuznetsov. Discovery of the hidden chaotic attractor has shown the need for further study of the scenarios concerned with the appearance and properties of hidden attractors, since the appearance of such attractors in the system can lead to a qualitative change in the dynamics of the system. In the present work two directions have been chosen, for which the possibility of the appearance of hidden attractors can be critical: radiophysics and biophysics. The features of radiophysical generators which can be used for systems of secure communication based on the dynamical chaos are considered in detail. Using the Chua circuit as an example, we investigate the problem of synchronization between two coupled generators in case when the observed regimes are represented by hidden and self-excited attractors. This example shows that in case of hidden attractors under certain initial conditions desynchronization of the coupled subsystems is possible, and the system of secure communication becomes inoperative. Alternative new radiophysical generators with self-excited attractors are also proposed. In such generators, the dynamical chaos is stable to the variation of parameters, initial conditions. In the context of the biophysics problems, a simplified model describing the dynamics of beta-cells based on the Hodgkin-Huxley formalism is presented. It has a typical for such systems bursting attractor which became hidden. This model can be used for the description of various pathological states of cells formation.