Collective Dynamics and Bifurcations in Symmetric Networks of Phase Oscillators. I
- PDF / 250,822 Bytes
- 28 Pages / 594 x 792 pts Page_size
- 107 Downloads / 188 Views
COLLECTIVE DYNAMICS AND BIFURCATIONS IN SYMMETRIC NETWORKS OF PHASE OSCILLATORS. I O. A. Burylko
UDC 517.91
The present paper is a brief survey of the history and development of the famous Kuramoto model of coupled phase oscillators. We consider several systems generalizing the classical Kuramoto model and given on symmetric oscillatory networks with different functions of interaction between the elements. We describe the collective dynamics and bifurcations of transitions between different modes of interacting elements, namely: partial and complete synchronization, global antiphase mode, slow switching, and chimera states. We show the relationship between the symmetries of networks and the existence of invariant manifolds of the system, cluster states, and more complicated collective behaviors. In part II, we plan to consider several models with nonglobal symmetric coupling.
1. Introduction The first experiments revealing the antiphase synchronization of two clock pendula were carried out by the famous physicist Ch. Huygens in the 17th century [1, 2]. After this, for more than 200 years, no significant achievements were attained in the fields of accumulation of experimental data and theoretical substantiation of the phenomenon of synchronization. As an exception, we can mention the effects of synchronization of the flares of fireflies and circadian rhythms of the plants discovered by biologists. Rapid intensification of scientific interest in the phenomenon of synchronization occurred at the end of the 19th century and the beginning of the 20th century, which was explained by the appearance of electrodynamics and successes in the investigation of the interaction of neurons. This can be explained by the self-oscillating nature of the collective behavior of interacting objects for which a theoretical comprehension had not been yet available at that time. The subsequent advances in the development of the theory of synchronization and collective dynamics were connected with the appearance of computers, cybernetics, radiophysics, theory of oscillations, theory of bifurcations, achievements in the investigation of human brain, creation of mathematical neural models (first of all, of the Hodgkin–Huxley model), diverse biological observations of the collective dynamics of living organisms, discovery of oscillatory chemical reactions, lasers, devices stimulating the operation of heart and brain, and computerized and artificial neural networks. Among the researchers who made significant contributions to the formation of the theory of synchronization, we can especially mention Rayleigh, van der Pol, Andronov, Witt, and Wiener. In the middle of the last century, the researchers understood that the collective dynamics of interacting objects is quite complicated and, thus, it is necessary to develop mathematical models capable of description of the common features of the phenomenon of synchronization even independently of the nature and complexity of the interacting objects. At the same time, these models should be sufficien
Data Loading...
