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.

The Indistinct Edge: Reconnecting Experience in Nature and Architecture

This thesis explores how architectures sense of place is rooted in the natural environment. The built environment has been constructed to protect and sustain human culture from the weathering of nature. Separating experience from the natural environment removes a sense of place and belonging in the natural and reinforces architectural dominance. This separation distinguishes the natural world as an article of spectacle and gives the human experience an unnatural voyeurship to natural changes. By examining the fusion of architectural and natural edges this thesis analyzes how the human experience can reconnect with a naturalistic sense of place through architecture, blending the finite edge where architecture maintains nature, and adapting buildings to the cycles of the environment. Removing dominance of man-made spaces and replacing them with the cohabitation of the edge between built and natural forms.

Rebuilding a Better Response

What is one thing in the world that we have little control over? Mother Nature. We as humans have tried to alter our natural environment countless times, and she has fought back twice as strong, leaving trails of devastation behind. The number of natural disasters has drastically increased, especially in the Southeast Asian region. The aftermath of a tsunami has left a large number of people homeless and lives forever changed. The current recovery process is chaotic and leaves affected individuals stranded without the means of rebuilding for long periods of time. This thesis aims to re-evaluate the recovery relief procedures and provide a better means of rebuilding their lives and community.