Thesis advisor: Renato Mirollo / There are countless occurrences of oscillating systems in nature. Climate cycles and planetary orbits are a few that humans experience daily. Man has also incorporated, to his benefit, oscillation into his craft; the grandfather clock, for example, can keep track of time with astounding accuracy using the period of a long pendulum. Such systems can range in complexity in a number of ways. The governing equation for a given oscillator could be as simple as a sine curve, or its motion could appear so erratic that oscillatory motion is undetectable to viewers. The number of oscillators in a system can also vary, and oscillators can be coupled; that is, oscillators can be affected by the motion of neighboring oscillators. It is this last case we wish to study. We will briefly look at the case of finitely many oscillators and then move to analyzing a model consisting of infinitely many identical oscillators. Synchrony is the simplest collective behavior. We will study a more complicated pattern called splay states in which oscillators are equally staggered in phase, i.e. phase locked such that the system will return to this pattern if it is disturbed by an arbitrarily small amount. Mathematically, this requires us to find attracting fixed points in the system. We will approximate the local behavior of our model by linearizing the system near its fixed points. We will then apply our findings to a few specific cases of such models including: uniform density, linear distribution, alpha-function pulses, and integrate-and-fire. / Thesis (BS) — Boston College, 2013. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: College Honors Program. / Discipline: Mathematics.

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We address the question of performance of large-scale networks with distributed feedback control. We consider networked dynamical systems with single and double integrator dynamics, subject to distributed disturbances. We focus on two types of problems. First, we consider problems modeled over regular lattice structures. Here, we treat consensus and vehicular formation problems and evaluate performance in terms of measures of “global order”, which capture the notion of network coherence. Second, we consider electric power networks, which we treat as dynamical systems modeled over general graphs. Here, we evaluate performance in terms of the resistive power losses that are incurred in maintaining network synchrony. These losses are associated with transient power flows that are a consequence of “local disorder” caused by lack of synchrony. In both cases, we characterize fundamental limitations to performance as networks become large. Previous studies have shown that such limitations hold for coherence in networks with regular lattice structures. These imply that connections in 3 spatial dimensions are necessary to achieve full coherence, when the controller uses static feedback from relative measurements in a local neighborhood. We show that these limitations remain valid also with dynamic feedback, where each controller has an internal memory state. However, if the controller can access certain absolute state information, dynamic feedback can improve performance compared to static feedback, allowing also 1-dimensional formations to be fully coherent. For electric power networks, we show that the transient power losses grow unboundedly with network size. However, in contrast to previous results, performance does not improve with increased network connectivity. We also show that a certain type of distributed dynamic feedback controller can improve performance by reducing losses, but that their scaling with network size remains an important limitation. / <p>QC 20160504</p>