Defended Dissertations in 2014

Summary
Oscillatory systems are common in nature and man-made equipments. The behavior of such systems is usually described by nonlinear differential equations. The oscillations in the conservative systems with infinite number of close periodic solutions are well investigated in theoretical physics, since this problem arises when considering the motion of planets. Whereas the oscillators containing limit cycles when in the neighborhood of a periodic solution there are no other such solutions, have been considered only for the last century. Such oscillators can be encountered e.g. in electronics, robotics, lasers, chemical reactions, biological systems and economical models. A great interest about limit cycle oscillators has been boosted in the last two decades with the development of neuroscience. It considers the mechanism of suppression and activation of the neuron, and control of synchronization and desynchronization in the neural networks. In nonlinear dynamics one may obtain the analytical results by means of linearization of underlying equations around the specific solution. In the case of the limit cycle, such linearization is described by the Floquet theory, which was formulated at the end of the 19th century. However, the main achievement has been made by I. Malkin in the middle of the 20th century, when the phase reduction method was formulated. Later, it was rediscovered by A. Winfree, who investigated the biological rhythms. The method is based on the idea that one is dealing only with a scalar phase instead of all the system variables. Such an approach enables to obtain analytical results for a weakly perturbed limit cycle oscillator or for several coupled oscillators with a weak interaction. In biological systems, in lasers and electronics, one often has to deal with the delay effect, when the system dynamics depends not only on the current value of dynamic variables but also on their delayed values. Such systems are described by delayed differential equations which are infinite-dimensional, in contrast to ordinary differential equations. Therefore, it is convenient to perform a phase reduction of these systems since an infinite-dimensional system is reduced to a system with a single dimension. In this work we will pay attention to this issue. viii Introduction The dynamical systems with an irregular behavior, called as chaotic systems, are hard to predict but relatively easy to control. One of the recent problems in the control theory is stabilization of the unstable periodic orbits embedded in a chaotic attractor. For solving this problem, K. Pyragas proposed in 1992 a control method using the delayed feedback. This method showed itself as very attractive in various experimental situations. Here the controlled orbits are described by equations with delay, and thus the phase reduction method such systems is also relevant since it may show whether the orbit is stabilizible, and reveal the other properties of the orbit. Another significant problem in nonlinear dynamics is the treatment of systems under a high frequency external force. The frequency is high in comparison with the intrinsic system frequencies. The vibrational mechanics is a field of science devoted to mechanical systems subject to high frequency perturbations. The high frequency of the external force can drastically change the system’s behavior. For example, the classical problem of the vibrational mechanics is the stabilization of the upside-down position of a rigid pendulum by vibrating its pivot up and down with a quite small amplitude. The stabilization can be achieved if the amplitude and the frequency satisfy some conditions. Another example – a sand spilled on an inclined plane, which is moved by high frequency force. The sand can start climb up if the angle of the plane, the frequency and the amplitude satisfy some conditions. These crazy experimental results can be explained by the mathematical analysis of the system’s equations. Here the phase reduction is not the appropriate tool for the theoretical analysis, since oscillator changes it’s behavior drastically. Therefore, here it is more useful to apply the averaging method which is based on the eliminating of the fast oscillating terms in order to get the equations determining the system behavior averaged over the period of high frequency oscillations. For patients with Parkinson’s disease, when the illness can’t be removed by drugs, there is applied a surgical procedure called deep brain stimulation. The electrode is implanted directly into the brain and the high frequency electrical current stimulates some parts of the brain. From the experiments we know that this leads to the positive results – involuntary movements are decreased or they disappear absolutely. But still it is not clear what happens with the synchronized neurons, which are responsible for the unwanted movement, under the high frequency electrical current. Therefore here we need to use the averaging method in order to analyze the system’s dynamical equations.