Effective analytical-numerical methods for the study of regular and chaotic oscillations in dynamical systems

Abstract

This dissertation examines the difficulties in analyzing the onset of oscillations in the process of loss of stability in various nonlinear dynamical systems. The study of the onset of oscillations originated with the discovery of periodic regimes in automatic control systems, as well as with the discovery of chaos associated with attempts to explain a laminar fluid flow becoming turbulent. One of the first methods revealing and analyzing stability of periodic oscillations applied to automatic control systems with one scalar nonlinearity was the Andronov pointmapping method, which is applicable only to piecewise linear systems of low order. Van der Pol, Krylov and Bogolyubov suggested the harmonic balance method, which is applicable to systems of arbitrary dimension with scalar nonlinearity of a general form. However, this method is approximate and may incorrectly predict the loss of stability and existence of oscillations. In this dissertation, for systems with one scalar nonlinearity, the discussion of the classical harmonic balance and the point-mapping methods has been carried out. Advantages and disadvantages of the locus of a perturbed relay system (LPRS) method, which is an extension of the harmonic balance method, were discussed and new examples demonstrating difficulties of studying scenarios of the loss of stability and onset of oscillations in relay systems were presented. None of the above mentioned methods are applicable when oscillations emerging in the system after the loss of stability demonstrate complex chaotic behavior. Such phenomenon was first noticed by famous scientist Lorenz in the study of turbulent convection of a fluid layer. One of the first explanations to the birth of such oscillations was given via a homoclinic bifurcation, in which a homoclinic oscillation appears in the phase space. In general, proving the existence of a homoclinic oscillation and giving a full description of the loss of stability and the onset of chaos via a homoclinic bifurcation remain open challenges. In this dissertation, for a class of Lorenz-like systems, the conditions of the existence of a homoclinic oscillation have been analytically obtained and a numerical investigation of several new homoclinic bifurcation scenarios have been carried out. For the Lorenz system, to visualize unstable periodic oscillations, which may appear during homoclinic bifurcations and are embedded in chaotic attractor, the Pyragas control algorithm has been implemented. Keywords: global stability, periodic and homoclinic oscillations, chaos