@article{BlasiusToenjes2009,
  author    = {Blasius, Bernd and Toenjes, Ralf},
  title     = {Zipf's Law in the popularity distribution of chess openings},
  issn      = {0031-9007},
  doi       = {10.1103/Physrevlett.103.218701},
  year      = {2009},
  abstract  = {We perform a quantitative analysis of extensive chess databases and show that the frequencies of opening moves are distributed according to a power law with an exponent that increases linearly with the game depth, whereas the pooled distribution of all opening weights follows Zipf's law with universal exponent. We propose a simple stochastic process that is able to capture the observed playing statistics and show that the Zipf law arises from the self-similar nature of the game tree of chess. Thus, in the case of hierarchical fragmentation the scaling is truly universal and independent of a particular generating mechanism. Our findings are of relevance in general processes with composite decisions.},
  language  = {en}
}
@article{ToenjesBlasius2009,
  author    = {Toenjes, Ralf and Blasius, Bernd},
  title     = {Perturbation analysis of complete synchronization in networks of phase oscillators},
  issn      = {1539-3755},
  doi       = {10.1103/Physreve.80.026202},
  year      = {2009},
  abstract  = {The behavior of weakly coupled self-sustained oscillators can often be well described by phase equations. Here we use the paradigm of Kuramoto phase oscillators which are coupled in a network to calculate first- and second-order corrections to the frequency of the fully synchronized state for nonidentical oscillators. The topology of the underlying coupling network is reflected in the eigenvalues and eigenvectors of the network Laplacian which influence the synchronization frequency in a particular way. They characterize the importance of nodes in a network and the relations between them. Expected values for the synchronization frequency are obtained for oscillators with quenched random frequencies on a class of scale-free random networks and for a Erdoumls-Reacutenyi random network. We briefly discuss an application of the perturbation theory in the second order to network structural analysis.},
  language  = {en}
}
@article{ToenjesBlasius2009,
  author    = {Toenjes, Ralf and Blasius, Bernd},
  title     = {Perturbation analysis of the Kuramoto phase-diffusion equation subject to quenched frequency disorder},
  issn      = {1539-3755},
  doi       = {10.1103/Physreve.79.016112},
  year      = {2009},
  abstract  = {The Kuramoto phase-diffusion equation is a nonlinear partial differential equation which describes the spatiotemporal evolution of a phase variable in an oscillatory reaction-diffusion system. Synchronization manifests itself in a stationary phase gradient where all phases throughout a system evolve with the same velocity, the synchronization frequency. The formation of concentric waves can be explained by local impurities of higher frequency which can entrain their surroundings. Concentric waves in synchronization also occur in heterogeneous systems, where the local frequencies are distributed randomly. We present a perturbation analysis of the synchronization frequency where the perturbation is given by the heterogeneity of natural frequencies in the system. The nonlinearity in the form of dispersion leads to an overall acceleration of the oscillation for which the expected value can be calculated from the second-order perturbation terms. We apply the theory to simple topologies, like a line or sphere, and deduce the dependence of the synchronization frequency on the size and the dimension of the oscillatory medium. We show that our theory can be extended to include rotating waves in a medium with periodic boundary conditions. By changing a system parameter, the synchronized state may become quasidegenerate. We demonstrate how perturbation theory fails at such a critical point.},
  language  = {en}
}