Piecewise Linear Models for the Quasiperiodic Transition to Chaos

We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode-locking and the quasi-periodic transition to chaos. For instance, for these families, we obtain complete solutions to several questions still largely unanswered for families of smooth circle maps. Our main results describe (1) the sets of maps in these families having some prescribed rotation interval; (2) the boundaries between zero and positive topological entropy and between zero length and non-zero length rotation interval; and (3) the structure and bifurcations of the attractors in one of these families. We discuss the interpretation of these maps as low-order spline approximations to the classic ``sine-circle'' map and examine more generally the implications of our results for the case of smooth circle maps. We also mention a possible connection to recent experiments on models of a driven Josephson junction.

Computer-Mediated Communication and Ecclesiological Challenges To and From the Reformed Tradition

Communities of faith have appeared online since the inception of computer -­ mediated communication (CMC)and are now ubiquitous. Yet the character and legitimacy of Internet communities as ecclesial bodies is often disputed by traditional churches; and the Internet's ability to host the church as church for online Christians remains a question. This dissertation carries out a practical theological conversation between three main sources: the phenomenon of the church online; ecclesiology (especially that characteristic of Reformed communities); and communication theory. After establishing the need for this study in Chapter 1, Chapter 2 investigates the online presence of Christians and trends in their Internet use, including its history and current expressions. Chapter 3 sets out an historical overview of the Reformed Tradition, focusing on the work of John Calvin and Karl Barth, as well as more contemporary theologians. With a theological context in which to consider online churches in place, Chapter 4 introduces four theological themes prominent in both ecclesiology and CMC studies: authority; community; mediation; and embodiment. These themes constitute the primary lens through which the dissertation conducts a critical-­confessional interface between communication theory and ecclesiology in the examination of CMC. Chapter 5 continues the contextualization of online churches with consideration of communication theories that impact CMC, focusing on three major communication theories: Narrative Theory; Interpretive Theory; and Speech Act Theory. Chapter 6 contains the critical conversation between ecclesiology and communication theory by correlating the aforementioned communication theories with Narrative Theology, Communities of Practice, and Theo-­Drama, and applying these to the four theological themes noted above. In addition, new or anticipated developments in CMC investigated in relationship to traditional ecclesiologies and the prospect of cyber-­ecclesiology. Chapter 7 offers an evaluative tool consisting of a three-­step hermeneutical process that examines: 1) the history, tradition, and ecclesiology of the particular community being evaluated; 2) communication theories and the process of religious-­social shaping of technology; and 3) CMC criteria for establishing the presence of a stable, interactive, and relational community. As this hermeneutical process unfolds, it holds the church at the center of the process, seeking a contextual yet faithful understanding of the church.