Synchronization of oscillators, a phenomenon found in a wide variety of natural and engineered systems, is typically understood through a reduction to a first-order phase model with simplified dynamics. Here, by exploiting the precision and flexibility of nanoelectromechanical systems, we examined the dynamics of a ring of quasi-sinusoidal oscillators at and beyond first order. Beyond first order, we found exotic states of synchronization with highly complex dynamics, including weak chimeras, decoupled states, traveling waves, and inhomogeneous synchronized states. Through theory and experiment, we show that these exotic states rely on complex interactions emerging out of networks with simple linear nearest-neighbor coupling. This work provides insight into the dynamical richness of complex systems with weak nonlinearities and local interactions.

We consider the properties of protoplanetary discs that are undergoing inside-out clearing by photoevaporation. In particular, we aim to characterise the conditions under which a protoplanetary disc may undergo 'thermal sweeping', a rapid ( 10 4 years) disc destruction mechanism proposed to occur when a clearing disc reaches sufficiently low surface density at its inner edge and where the disc is unstable to runaway penetration by the X-rays. We use a large suite of 1D radiation-hydrodynamic simulations to probe the observable parameter space, which is unfeasible in higher dimensions. These models allow us to determine the surface density at which thermal sweeping will take over the disc's evolution and to evaluate this critical surface density as a function of X-ray luminosity, stellar mass and inner hole radius. We find that this critical surface density scales linearly with X-ray luminosity, increases with inner hole radius and decreases with stellar mass and we develop an analytic model that reproduces these results.This surface density criterion is then used to determine the evolutionary state of protoplanetary discs at the point that they become unstable to destruction by thermal sweeping. We find that transition discs created by photoevaporation will undergo thermal sweeping when their inner holes reach 20 − 40 AU, implying that transition discs with large holes and no accretion (which were previously a predicted outcome of the later stages of all flavours of photoevaporation model) will not form. Thermal sweeping thus avoids the production of large numbers of large, non-accreting holes (which are not observed) and implies that the majority of holes created by photoevaporation should still be accreting.We emphasise that the surface density criteria that we have developed apply to all situations where the disc develops an inner hole that is optically thin to X-rays. It thus applies not only to the case of holes originally created by photoevaporation but also to holes formed, for example, by the tidal influence of planets.

In work [1], a surface embedded in flat R 3 is associated to any three hermitian matrices. We study this emergent surface when the matrices are large, by constructing coherent states corresponding to points in the emergent geometry. We find the original matrices determine not only shape of the emergent surface, but also a unique Poisson structure. We prove that commutators of matrix operators correspond to Poisson brackets. Through our construction, we can realize arbitrary noncommutative membranes: for example, we examine a round sphere with a non-spherically symmetric Poisson structure. We also give a natural construction for a noncommutative torus embedded in R 3 . Finally, we make remarks about area and find matrix equations for minimal area surfaces.

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Signed networks have been a topic of recent interest in the network control community as they allow studying antagonistic interactions in multi-agent systems. Although dynamical characteristics of signed networks have been wellstudied, notions such as controllability and stabilizability for signed networks for protocols such as consensus are missing in the literature. Classically, graph automorphisms with respect to the input nodes have been used to characterize uncontrollability of consensus networks. In this paper, we show that in addition to the graph symmetry, the topological property of structural balance facilitates the derivation of analogous sufficient conditions for uncontrollability for signed networks. In particular, we provide an analysis which shows that a gauge transformation induced by structural balance allows symmetry arguments to hold for signed consensus networks. Lastly, we use fractional automorphisms to extend our observations to output controllability and stabilizability of signed networks.

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In this paper we consider the H2-norm of networked systems with multi-time scale consensus dynamics. We develop a general framework for such systems that allows for edge weighting, independent agent-based time scales, as well as measurement and process noise. From this general system description, we highlight an interesting case where the influences of the weighting and scaling can be separated in the design problem. We then consider the design of the time scale parameters for minimizing the H2-norm for the purpose of network resilience.