| Abstract [eng] |
Nonlinear limit cycle oscillators are common in nature and man-made equipments, for example, they occur in electronics, robotics, lasers, chemical reactions, biological systems and economical models. Such oscillators demonstrate periodic behavior with fixed frequency and amplitude independently of the system’s initial conditions. The goal of the doctoral thesis is the development and application of phase reduction and averaging methods to analyze particular nonlinear problems in self-oscillatory systems. The phase reduction method allows us to reduce the dynamic of a weakly perturbed limit cycle oscillator to a single scalar equation that defines the dynamics of the phase. This method is usually applied to the systems described by ordinary differential equations. Here this method is extended for the systems with time delay. The phase reduction method is applied to analyze the delayed feedback control (DFC) algorithm. Such an approach allows us to obtain analytical results for slightly mismatched DFC scheme and to stabilize unstable periodic orbits with topological restriction. The averaging method is applied to self-oscillatory systems driven by high-frequency periodic force. The method allows to derive the equations for the slow motion, averaged over high-frequency oscillations. Using this method the mechanism of suppression of sustained neuronal spiking under high frequency electrical stimulation is investigated. |
