  We investigate complex synchronization patterns such as cluster
synchronization and partial amplitude death in networks of coupled
Stuart-Landau oscillators with fractal connectivities. The study of fractal or
self-similar topology is motivated by the network of neurons in the brain. This
fractal property is well represented in hierarchical networks, for which we
present three different models. In addition, we introduce an analytical
eigensolution method and provide a comprehensive picture of the interplay of
network topology and the corresponding network dynamics, thus allowing us to
predict the dynamics of arbitrarily large hierarchical networks simply by
analyzing small network motifs. We also show that oscillation death can be
induced in these networks, even if the coupling is symmetric, contrary to
previous understanding of oscillation death. Our results show that there is a
direct correlation between topology and dynamics: Hierarchical networks exhibit
the corresponding hierarchical dynamics. This helps bridging the gap between
mesoscale motifs and macroscopic networks.

Adaptation and Self-Organizing Systems;Pattern Formation and Solitons