Hierarchical globally coupled systems

We have constructed prototype models of globally coupled systems on lattices with space-time hierarchy. In our models fully chaotic dynamical elements at a certain level in the hierarchy are coupled to the levels immediately above and below in the hierarchy through their mean fields. We report the wealth of spatiotemporal phenomena our models yield and give detailed bifurcation diagrams in coupling parameter, ε, space $(0\mathrm{}<~\epsilon <~1).$ We find that over a large range of strong coupling $(0.5<\epsilon <~1)$ our models quickly evolve into a spatiotemporal fixed point. We analyze the stability of this phase. For moderate coupling $0.15<\mathrm{}\epsilon <0.5,$ we have spatial inhomogeneity and temporal regularity, marked by the presence of, either exact $2^k,$ $k=1,2\mathrm{}\dots$ cycles, or noisy bands (characterized by δ spikes on a noisy background in the power spectra). An interesting feature of this phase is the presence of several coexisting attractors, with fractal basin boundaries. For small coupling $(\epsilon \sim 0.1)$ we show that our system, while temporally chaotic and spatially nonhomogeneous, develops certain broad periodicities in their mean field, especially at finer scales. In addition, if one looks at the fluctuation of the mean field at various levels in the hierarchy in this phase, one finds that the mean square deviation does not decrease as $1/N$ with $N,$ where $N$ is the number of elements at a particular level. Instead it displays marked nonstatistical behavior with the deviations saturating (or even increasing) for high $N.\mathrm{}$