Complexity as a measure of paths to regularity

Abstract

In this thesis, a way to quantify the synchronization of a system is introduced. It is made from a codification of the paths towards synchronization for synchronizing flows defined over a network. The collection of paths toward synchronization defines a combinatorial structure, called the transition diagram, the main object of study. The cardinality of this collection defines a measure of complexity which depends on the dimension of the system. The transition diagram corresponding to the Laplacian flow over the complete graph K_N and the complete bipartite graph K_{N,N} is described, through a coding: the feasible states by increasing functions, and the transitions between them by consecutive functions that follow certain rules. These results are applied to the Kuramoto flow (over the same graph) when a neighborhood close to the diagonal is considered. Furthermore, it generalizes to flows that are monotonic (that is, its coordinates and the differences of the coordinates maintain the order). It is presented as well some numerical and analytical results concerning the Laplacian and Kuramoto flows over the cycle graph C_N, and the ring lattice family C(N,k). In this case there are a different perspective, due to their no-monotonic behavior.