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DY: Dynamik und Statistische Physik

DY 12: Statistical physics of complex networks

DY 12.1: Talk

Monday, March 24, 2003, 11:30–11:45, G\"OR/226

Synchronization of Pulse-Coupled Oscillators on Complex Networks — •Marc Timme, Fred Wolf, and Theo Geisel — Max-Planck-Institut für Strömungsforschung, 37073 Göttingen

Complex networks appear as a variety of natural and artificial systems. While recent studies have focused on their structure, the dynamics on such networks constitutes a challenging task of current and future research. Here we develop an exact stability analysis of synchronous states for arbitrarily connected networks of pulse-coupled oscillators [1]. Networks of regular, random, and more complex connectivities are considered. As opposed to conventional stability analysis, stability is not determined by a single stability matrix but by a multitude of operators. We solve this multi-operator problem analytically. Using the Gershgorin theorem, we find exact bounds on the eigenvalues of all operators. Due to the multi-operator problem for complex connectivities, eigenvalues and eigenvectors do not immediately imply any stability properties. Using concepts from graph theory, we completely analyze the stability and asymptotic stability of the synchronous state. For inhibitory interactions the synchronous state is stable, independent of the parameters and the network connectivity.
[1] M. Timme, F. Wolf, and T. Geisel; Phys. Rev. Lett. 89:258701 (2002).