Cyclic Pursuit: Symmetry, Reduction and Nonlinear Dynamics

In this dissertation, we explore the use of pursuit interactions as a building block for collective behavior, primarily in the context of constant bearing (CB) cyclic pursuit. Pursuit phenomena are observed throughout the natural environment and also play an important role in technological contexts, such as missile-aircraft encounters and interactions between unmanned vehicles. While pursuit is typically regarded as adversarial, we demonstrate that pursuit interactions within a cyclic pursuit framework give rise to seemingly coordinated group maneuvers. We model a system of agents (e.g. birds, vehicles) as particles tracing out curves in the plane, and illustrate reduction to the shape space of relative positions and velocities. Introducing the CB pursuit strategy and associated pursuit law, we consider the case for which agent i pursues agent i+1 (modulo n) with the CB pursuit law. After deriving closed-loop cyclic pursuit dynamics, we demonstrate asymptotic convergence to an invariant submanifold (corresponding to each agent attaining the CB pursuit strategy), and proceed by analysis of the reduced dynamics restricted to the submanifold. For the general setting, we derive existence conditions for relative equilibria (circling and rectilinear) as well as for system trajectories which preserve the shape of the collective (up to similarity), which we refer to as pure shape equilibria. For two illustrative low-dimensional cases, we provide a more comprehensive analysis, deriving explicit trajectory solutions for the two-particle "mutual pursuit" case, and detailing the stability properties of three-particle relative equilibria and pure shape equilibria. For the three-particle case, we show that a particular choice of CB pursuit parameters gives rise to remarkable almost-periodic trajectories in the physical space. We also extend our study to consider CB pursuit in three dimensions, deriving a feedback law for executing the CB pursuit strategy, and providing a detailed analysis of the two-particle mutual pursuit case. We complete the work by considering evasive strategies to counter the motion camouflage (MC) pursuit law. After demonstrating that a stochastically steering evader is unable to thwart the MC pursuit strategy, we propose a (deterministic) feedback law for the evader and demonstrate the existence of circling equilibria for the closed-loop pursuer-evader dynamics.

Increasing Stability of Lithium-Ion Batteries with Ordered Graphene Silicon Negative Electrodes

Currently, lackluster battery capability is restricting the widespread integration of Smart Grids, limiting the long-term feasibility of alternative, green energy conversion technologies. Silicon nanoparticles have great conductivity for applications in rechargeable batteries, but have degradation issues due to changes in volume during lithiation/delithiation cycles. To combat this, we use electrochemical deposition to uniformly space silicon particles on graphene sheets to create a more stable structure. We found the process of electrochemical deposition degraded the graphene binding in the electrode material, severely reducing charge capacity. But, the usage of mechanically mixing silicon particles with grapheme yielded batteries better than those that are commercially available.

The Effect of Diet-Induced Obesity and Subsequent Weight Loss on Body Composition, Glucose Clearance, Metabolite Profile and Liver Amp-Activated Protein Kinase in Mice

Obesity, currently an epidemic, is a difficult disease to combat because it is marked by both a change in body weight and an underlying dysregulation in metabolism, making consistent weight loss challenging. We sought to elucidate this metabolic dysregulation resulting from diet-induced obesity (DIO) that persists through subsequent weight loss. We hypothesized that weight gain imparts a change in “metabolic set point” persisting through subsequent weight loss and that this modification may involve a persistent change in hepatic AMP-activated protein kinase (AMPK), a key energy-sensing enzyme in the body. To test these hypotheses, we tracked metabolic perturbations through this period, measuring changes in hepatic AMPK. To further understand the role of AMPK we used AICAR, an AMPK activator, following DIO. Our findings established a more dynamic metabolic model of DIO and subsequent weight loss. We observed hepatic AMPK elevation following weight loss, but AICAR administration without similar dieting was unsuccessful in improving metabolic dysregulation. Our findings provide an approach to modeling DIO and subsequent dieting that can be built upon in future studies and hopefully contribute to more effective long-term treatments of obesity.