(The assessment consists of 4 homework assignments, one for each part of the course.
Physical examples will also be discussed. The appearance of integrability in the Kuramoto-Sakaguchi model and a few Model and Kuramoto models on weighted networks. WeĪddress the original model with the mean-field coupling and discuss some of its generalisations, such as the Kuramoto-Sakaguchi Martynchuk, we discuss the phenomenon of synchronization in the context of the Kuramoto model. Phase transition, with respect to the connectivity probability parameter, of the size of the largest connected component of this Valesin, the topic is the Erdös-Renyi random graph model; the main result that is stated and proved is the Camlibel, the notion of system controllability is discussed with emphasis on systems defined on graphs,ĭiffusively coupled leader/follower systems, and concepts such as controllable space, leader selection, and targeted controllability. Multidimensional Fourier transform, and cryptography (using encryption schemes based on quasicrystals). Interactions, paving here a way to Mendeleev's periodic table of elements, to singularities in bifurcation or catastrophe theory, to Systems give the main classification of objects' Complexity in Mathematics: from Platonic solids and Lie algebras in fundamental Part I of the course, by A.V.Kiselev, is about root systems, i.e. This is an interdisciplinary course that presents different aspects of the theme Complexity and Networks from both the pure andĪpplied mathematics points of view. characterise the transition to synchronization in the Kuramoto model using the coupling strength and order parameter; compute the critical coupling and order parameter for the Cauchy distribution; model (numerically) the emergence of synchronization; perform the reduction of the Kuramoto-Sakaguchi model of identical oscillators and show its integrability for special parameter values bound controllable space by using graph partitions, verify whether a given set of nodes is a zero forcing set and apply the theory to specific classes of graphs in order to select leaders rendering the system controllable Ĥ. bound controllable space by using graph partitions, verify whether a given set of nodes is a zero forcing set and apply the theory to specific classes of graphs in order to select leaders rendering the system controllable ģ. find bases in root systems and act by their automorphism groups; express the classification of semisimple Lie algebras in terms of Dynkin diagrams; find Weyl bases in these algebras Ģ. At the end of the course, the student is able to:ġ.