phase oscillators

The purpose of this study is to acquaint the reader with one of the effective approaches to describing processes in adaptive networks, built in the framework of the well-known Kuramoto model. Methods. The solution to this problem is based on the analysis of the results of works devoted to the study of the dynamics of oscillatory networks with adaptive couplings.

Great interest in the field of dynamic systems and nonlinear processes is caused by research in the field of energy networks. A power grid is a complex network of coupled oscillators that demonstrates collective behavior by synchronizing
network elements at the base frequency of a power grid.

Synchronization of two reactively coupled van der Pol oscillators with external force is investigated in this paper. We consider and compare quasi-periodic motion of oscillators with frequency-locked mode. The paper includes maps of Lyapunov’s exponents, twoparametric bifurcation diagrams and phase portraits. Possible types of motion in driven system are discussed.

Synchronization phenomena are studied in phase dynamics approximation in the periodically driven system of two coupled oscillators. The cases are discussed when the autonomous oscillators demonstrate phase locking or beats with incommensurate frequencies. Lyapunov charts are presented, the possible regimes of dynamics of the driven system are discussed. Different types of two-dimensional tori are revealed and classified.

The problem of the dynamics of three coupled self-oscillators and three coupled periodically driven self-oscillators is discussed, in the last case only one of the oscillators is directly exited by the external fore. The regions of complete synchronization, two-, threeand four-frequency tori and chaos are revealed. Three typical situations of synchronization of three self-oscillators by the external driving are found. First situation refers to the mode locking of autonomous oscillators.

We investigate dynamical states formed in a network of coupled phase oscillators in which strength of interactions between oscillators evolve dynamically depending on their relative phases. The feature of the system is co-evolution of coupling weights and states of elements. It is ascertained that depending on the parameters the network exhibit several types of behavior: globally synchronized state, two-cluster and multi-cluster states, various synchronized states with a fixed phase relationship between oscillators and desynchronized state.

The work is devoted to investigation of multistability of running waves in a ring of periodic oscillators with diffusive non-local couplings. It analyzes the influence of long-range couplings and their change with distance on the stability of spatially-periodic regimes with different wave numbers. The research are carried out by numerical (computer) experiments. The system under study is an ensemble of identical phase oscillators.

Aim. The aim of the study is to formulate an effective model of the power grid, to determine the stable modes of its operation, to identify differences in the considered modes and to test the stability of the system to changes in control parameters, initial conditions and to various types of external influence. Method. The effective model of the energy network, which consists of three coupled oscillators, is considered for different methods of setting the initial conditions and variation of the control parameters.