Order of Partial Synchronies derived from Network Structure

Dr. Hiroko Kamei (University of Dundee)

Thursday 17th January, 2013 14:00-15:00 326


In many areas of science, interacting dynamical systems can be represented as a network. We study how the network structure constrains the behaviour of the system, in particular all possible partial synchronies and their ordered structure. The coupled cell system formalism used can describe general interacting (coupled) individual systems (cells), focusing on the network structure and the types of interaction, while ignoring precise details such as function forms or parameter values. The resulting dynamical system can capture which individual systems would be expected to behave the same in model-independent manner, and as a result, it is suitable to analyse synchronous behaviour of a network. We represent the structure of a given network by a symbolic adjacency matrix, which encodes the different types of interaction, and show that adjacency matrix manipulation enables us to search all possible partial synchronies (stable or unstable) of a given network. Furthermore, these partial synchronies can be ordered and have a mathematical structure as a complete lattice. We demonstrate our computational method using networks from application problems and discuss the stability of possible partial synchronies which is a consequence of the detailed of the model system. For identically coupled identical systems, we show ordered structures have a direct link to the eigenvalues and eigenvectors of an adjacency matrix and can be used to analyse synchrony-breaking bifurcation of a given network.

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