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.

Creating Common Ground: Architecture For Tactical Learning and Creative Convergence

Certain environments can inhibit learning and stifle enthusiasm, while others enhance learning or stimulate curiosity. Furthermore, in a world where technological change is accelerating we could ask how might architecture connect resource abundant and resource scarce innovation environments? Innovation environments developed out of necessity within urban villages and those developed with high intention and expectation within more institutionalized settings share a framework of opportunity for addressing change through learning and education. This thesis investigates formal and informal learning environments and how architecture can stimulate curiosity, enrich learning, create common ground, and expand access to education. The reason for this thesis exploration is to better understand how architects might design inclusive environments that bring people together to build sustainable infrastructure encouraging innovation and adaptation to change for years to come. The context of this thesis is largely based on Colin McFarlane’s theory that the “city is an assemblage for learning” The socio-spatial perspective in urbanism, considers how built infrastructure and society interact. Through the urban realm, inhabitants learn to negotiate people, space, politics, and resources affecting their daily lives. The city is therefore a dynamic field of emergent possibility. This thesis uses the city as a lens through which the boundaries between informal and formal logics as well as the public and private might be blurred. Through analytical processes I have examined the environmental devices and assemblage of factors that consistently provide conditions through which learning may thrive. These parameters that make a creative space significant can help suggest the design of common ground environments through which innovation is catalyzed.