Zinchenko A. Computer modeling of deterministic chaos in processes with quadratic nonlinearity

The thesis is devoted to solving the actual scientific and technical objectives of the study of deterministic chaos in complex non-linear systems. Existing research methodology of deterministic chaos in complex non-linear systems were analyzed and improved. The specific numerical methods (20 methods) were assembled together, and this helped to solve the problems of direct (research modes of behavior depending on the parameters) and inverse (reconstruction of dynamic system parameter identification) research of non-linear dynamics based on existing and new methods of research. The numerical method for parametric identification of non-linear systems with chaos is proposed on the basis of chaotic synchronization and adaptive control in the observation of a single scalar implementation or implementation of a one-dimensional function of the phase coordinates. The method is based on the accuracy of the numerical solution of the system and improves the accuracy of the estimation of unknown parameters. A method for estimating the "window" of reconstruction is proposed based on minimizing relative error of calculations of correlation integrals taking into account the time sequence of observations that are correlated. This method improves calcula-tion of correlation integrals in the method of correlation dimension of Brock's residual test, BDS-test, a method of calculating the Kolmogorov entropy, as well as the Grassberger-Procaccia algorithm. The theorems for a class of nonlinear systems of the type YU.-SH. Chen are formulated and proven: the existence of a global exponential attractor of the system, pe-riodic solutions, the presence of bifurcations of the Poincare-Andronov-Hopf theorem as well as the management of attractors-for example control of deterministic chaos, which puts the system in a chaotic regime to regular regime was found, and control system for managing the general form in which it is fully synchronized with the driv-ing system was found. In addition, the estimated probability region of the attractor of the system that is limited by this system, is estimated.