Computational Study Of Dynamical Behaviours In Mobile Sensor Networks And The Human Endocrine System

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2021-08

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Abstract

This research develops and leverages ideas in applied and computational mathematics to solve interdisciplinary research problems. This thesis includes two projects. The first uses ideas from algebraic topology and computational geometry to solve pursuit evasion problem in mobile sensor networks. The other project leverages ideas from ergodic theory and dynamical systems to under-stand a novel phenomenon we call delay-induced uncertainty. Delay-induced uncertainty refers to a specific type of sustained temporal chaos that arises in physiological systems such as the human glucose-insulin system. Suppose one wants to monitor a domain with sensors each sensing small ball-shaped regions,but the domain is hazardous enough that one cannot control the placement of the sensors. Aprohibitively large number of randomly placed sensors would be required to obtain static coverage.Instead, one can use fewer sensors by providing only mobile coverage, a generalization of the static setup wherein every possible intruder is detected by the moving sensors in a bounded amount of time. Here, we use topology to implement algorithms certifying mobile coverage that use only local data to solve the global problem. Our algorithms do not require knowledge of the sensors’ locations. We experimentally study the statistics of mobile coverage in two dynamical scenarios.We allow the sensors to move independently (billiard dynamics and Brownian motion), or to locally coordinate their dynamics (collective animal motion models). Our detailed simulations show, for example, that collective motion enhances performance: The expected time until the mobile sensor network achieves mobile coverage is lower for the D’Orsogna collective motion model than for the billiard motion model. Further, we show that even when the probability of static coverage is low, all possible evaders can nevertheless be detected relatively quickly by mobile sensors. Our numerical experiments also show mobile sensor initialization affects intruder time detection. Finally, we show the intruder detection time is slower in circular geometry compared to square geometry of the same area for uniform random initialization of mobile sensors over the whole domain. Medicine rests on the fundamental assumption that medical intervention is reliably predictable. However, we have evidence that this prediction reliability can fail for physiological systems with meaningful delay. One such system is the glucose-insulin system. We show that when the glucose-insulin system is subjected to forcing drives used in the ICU setting, chaos that is both sustained in time and observable can result. This phenomenon is referred to as delay-induced uncertainty (DIU). The mathematical theory behind DIU uses ideas from the theory of nonuniformly hyperbolic dynamics in general and the theory of rank one maps in particular. We use a finite-dimensional stationary model known as the Ultradian model to demonstrate the presence of DIU in the human endocrine system. The top Lyapunov exponent is used as a tool to diagnose the presence of chaos in the glucose-insulin system. In our experiments, we show that DIU emerges for both insulin and glucose forcing drives. These experiments show that DIU emerges robustly in the Ultradian model.

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Dynamical systems, computational geometry, mathematical biology.

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