The phenomenon of synchronization in systems of coupled oscillators is a subject of intense study and increasing importance. It has been found as a crucial aspect in many fields including biological, technological and physical systems, such as the synchronization behavior of groups of cardiac pacemaker cells, coupled metronomes, large groups of fireflies, biochemical oscillators, oscillating neutrinos and neuronal synchronization. The Kuramoto model describes the synchronization of oscillators represented by a single phase variable$\phi$. The set of oscillators can be pictured as particles rotating on a unit circle, or 2D sphere. In the limit of infinite oscillators, the system can be described by a low dimensional set of equations that can be easily handled. The complexity reduction also allowed the analysis of the system under external forces and a complete bifurcation diagram was constructed. The 2D Kuramoto model was recently extended to any number of dimensions, depicting particles moving on the surface of high dimensional spheres. Inspired on this formulation and influenced by the works of Ott and Antonsen, we proposed an analogous mathematical formulation for the case of oscillators on a D-dimensional sphere.In order to study the synchronization in systems of oscillators coupled by a network it is also important to consider the symmetries of the system and how structural factors of the network and of the dynamics of the oscillators can affect the behavior of the system and contribute for the synchronization. Recent studies has shown that symmetry breaking can be an important factor for the synchronization of the oscillators and asymmetry can induce symmetry in a wide class of networks of oscillators. In this project, we aim to study the role of symmetry and structural factors of the network and of the oscillators in the phenomenon of synchronization.During the period of collaboration with professor Adilson E. Motter, the student intends to deepen her research in the area of oscillator synchronization connected by a network. This includes the study of synchronization phenomena in systems with external forces, synchronization in clusters and asymmetry-induced synchronization (AISync). This project intends to be the first to address AISync and cluster sync stabilized by asymmetry in this class of generalized Kuramoto oscillators. Our main objectives for this internship are: (i) interact with prof. Motter's research group; (ii) study if AISync behavior among systems with other symmetric network structures and oscillator dynamics; (iii) study is the existence and stability of a completely synchronous state; (iv) study the cluster synchronization in a network of identical oscillators; (v) search for a set of parameters that can be changed in order to optimize the synchronization of the clusters.
News published in Agência FAPESP Newsletter about the scholarship: