Observation of synchronization at kuramoto model complex network topolojies

Synchronization, which is a common phenomenon in many different field of science, nature and daily life, has great importance. Scientists have developed several mathematical methods in order to explain the formation of synchronization. The kuramoto model is the most preferred and frequently used model in studies of these methods. Kuramoto model is defined as a multiple system that occur from oscillators' phases connected with each other. In such a system, synchronization is shaped according to the size of connection strength between oscillators connected with each other. In this thesis, the dynamics of a system in the global network structure were primarily analyzed by using Kuramoto model equations. In order to get an idea about the synchronization status from system dynamics, the amplitude of order parameter and the phase of oscillators have been benefited. In this direction, it was observed that as the connection strength increases, as the synchronous behavior occurs and the phase of each oscillator is locked to the same value in the system. However, it has been found that the value of the connection strength should be greater than a certain value, as well as to be smaller than a specific value in order for synchronization to occur in a system. It was observed that the distribution of each oscillator is different and synchronization did not ocur when the connection strength between oscillators was smaller than critical threshold. With the detection of global network structure did not reflect the real systems, different network models have been proposed. In this context, in the scope of thesis, small-world and scale-free network models and in systems defined by using Kuramoto equations, synchronization were studied. It was determined that synchronization performance was dependent on the connection strength between oscillators (K) and network parameters (probabilities of the oscillators in the network setting up the connection (p) and average degree of connectivity (k)),in a system defined by using small-world network. An increase in any of the network parameters has shown another increase in synchronization of a system. In contrast, the connection strength must be at a certain range for increasing synchronization of the systems. A rapid decrease in synchronization was observed at connection strength values outside of this range. On the other hand, increasing number of oscillators in the system did not have a significant effect on the synchronization, and it was only effective in determination of lower critical threshold value of connection strength. Finally, in this thesis, the synchronization of a system that was created with scale-free network structure was investigated. With the increasing number of oscillators in the system, it was observed that synchronization occurred in a shorter period, and also determined clearly the value of the critical threshold. In addition, it was found that network parameter (m: The number of connection that is created by new added node) had no dominant effect on synchronization in systems defined by scale-free network as distinct from small-world network.