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The ability to steer the state of a dynamical network towards a desired state within a time horizon is intrinsically dependent on the number of driven nodes considered, as well as the network’s topology. The trade-off between time-to-control and the minimum number of driven nodes is captured by the notion of the actuation spectrum (AS). We study the actuation spectra of a variety of artificial and real-world networked systems, modeled by fractional-order dynamics that are capable of capturing non-Markovian time properties with power-law dependencies. We find evidence that, in both types of networks, the actuation spectra are similar when the time-to-control is less or equal to about 1/5 of the size of the network. Nonetheless, for a time-to-control larger than the network size, the minimum number of driven nodes required to attain controllability in networks with fractional-order dynamics may still decrease in comparison with other networks with Markovian properties. These differences suggest that the minimum number of driven nodes can be used to determine the true dynamical nature of the network. Furthermore, such differences also suggest that new generative models are required to reproduce the actuation spectra of real fractional-order dynamical networks. 

Abstract Since its invention, optical frequency comb has revolutionized a broad range of subjects from metrology to spectroscopy. The recent development of microresonator-based frequency combs (microcombs) provides a unique pathway to create frequency comb systems on a chip. Indeed, microcomb-based spectroscopy, ranging, optical synthesizer, telecommunications and astronomical calibrations have been reported recently. Critical to many of the integrated comb systems is the broad coverage of comb spectra. Here, microcombs of more than two-octave span (450 nm to 2,008 nm) is demonstrated throughχ⁽²⁾andχ⁽³⁾nonlinearities in a deformed silica microcavity. The deformation lifts the circular symmetry and creates chaotic tunneling channels that enable broadband collection of intracavity emission with a single waveguide. Our demonstration introduces a new degree of freedom, cavity deformation, to the microcomb studies, and our microcomb spectral range is useful for applications in optical clock, astronomical calibration and biological imaging.