Computation and visualization of invariant manifolds

In this thesis, we start with the basic concepts of dynamical systems. Then we introduce the general types of problems that the well-known software package AUTO solves. AUTO uses a boundary value algorithm with Gauss collocation and pseudo-arclength continuation. These two features distinguish AUTO from other general ODE solvers for dynamical systems. In order to compute 2D solution manifolds, AUTO uses orbit continuation. With these tools, we study two famous problems, the Lorenz system and the Circular Restricted Three-Body Problem (CR3BP). We briefly discuss the basic bifurcation and stability analysis of general ODE systems. The numerical analysis of the two problems leads to the newest algorithm to compute the 2D stable manifold of the origin of the Lorenz system and the 2D unstable manifold of appropriate periodic orbits of the CR3BP. We utilize Python for the flow control of AUTO. We also implement two visualization packages, QTPlaut and MATPlaut. They make possible the processing of large quantities of AUTO solution data with the OpenGL graphical library, dynamic memory allocation and interpolation methods. We conclude with prospect for future research

Classical and quantum mechanics with chaos

This thesis is concerned with the study, classically and quantum mechanically, of the square billiard with particular attention to chaos in both cases. Classically, we show that the rotating square billiard has two regular limits with a mixture of order and chaos between, depending on an energy parameter, E. This parameter ranges from -2w(^2) to oo, where w is the angular rotation, corresponding to the two integrable limits. The rotating square billiard has simple enough geometry to permit us to elucidate that the mechanism for chaos with rotation or curved trajectories is not flyaway, as previously suggested, but rather the accumulation of angular dispersion from a rotating line. Furthermore, we find periodic cycles which have asymmetric trajectories, below the value of E at which phase space becomes disjointed. These trajectories exhibit both left and right hand curvatures due to the fine balance between Centrifugal and Coriolis forces. Quantum mechanically, we compare the spectral analysis results for the square billiard with three different theoretical distribution functions. A new feature in the study is the correspondence we find, by utilising the Berry-Robnik parameter q, between classical E and a quantum rotation parameter w. The parameter q gives the ratio of chaotic quantum phase volume which we can link to the ratio of chaotic phase volume found classically for varying values of E. We find good correspondence, in particular, the different values of q as w is varied reflect the births and subsequent destructions of the different periodic cycles. We also study wave packet dynamics, necessitating the adaptation of a one dimensional unitary integration method to the two dimensional square billiard. In concluding we suggest how this work may be used, with the aid of the chaotic phase volumes calculated, in future directions for research work

NASA Workshop on Distributed Parameter Modeling and Control of Flexible Aerospace Systems

Although significant advances have been made in modeling and controlling flexible systems, there remains a need for improvements in model accuracy and in control performance. The finite element models of flexible systems are unduly complex and are almost intractable to optimum parameter estimation for refinement using experimental data. Distributed parameter or continuum modeling offers some advantages and some challenges in both modeling and control. Continuum models often result in a significantly reduced number of model parameters, thereby enabling optimum parameter estimation. The dynamic equations of motion of continuum models provide the advantage of allowing the embedding of the control system dynamics, thus forming a complete set of system dynamics. There is also increased insight provided by the continuum model approach

The origin, evolution, and function of the myth of the white goddess in the writings of Robert Graves

This is a study of the development of the myth of the White Goddess in the work of Robert Graves, a subject related to the wider field of the place of myth in modern culture. It begins by looking at the conditions which promoted Graves' interest in myth, principally his experience of the Great War. The responses of other writers are examined to provide a context for understanding Graves' transition from Georgianism to myth, as reflected in his early poetry, autobiography and writings on psychology. Before looking at how Graves' myth was formed, the history of the concept of myth is examined, from primitive peoples to civilized religion. Focus is centred upon the dual tendency of myth to reinforce and to undermine authority. Some of the figures behind Graves' interest in myth and anthropology are subject to scrutiny. An account of the relations between myth, literature and psychology permits the survey of Graves' gradual transition from psychological theory to mythographic speculation. The gradual emergence in his poetry of devotion to a Love Goddess can also be traced. Detailed interpretation of The White Goddess, its arguments and procedures, brings to light Graves' theories of the single poetic theme and the primitive matriarchy, both of which can then be evaluated and set in the context of his dedication to non-rational forms of thought. This leads into a close reading of Graves' major mythological poems, followed by reflections upon the myth's application in his critical writings and cultural commentaries. Finally, consideration is given to Graves' later writings, especially his attraction to Orphism and the adoption of mythic personae in his verse. The influence of the Black Goddess of Wisdom over these later works is interpreted and assessed

Fighting for the mantle of science : the epistemological foundations of neoliberalism, 1931-1951

Cette thèse examine la genèse intellectuelle du néolibéralisme au prisme de son épistémologie. Elle interroge le développement de ses arguments concernant la production et la diffusion de la connaissance, guidée par l’hypothèse que la formulation d’une position épistémologique commune a été cruciale pour la consolidation de son programme idéologique. Je propose que le néolibéralisme, en provoquant une rupture avec le libéralisme classique, a opéré un recodage des principes libéraux à l’intérieur d’un cadre épistémologique basé sur le conventionnalisme, à l’aide de prémisses tirées des sciences naturelles, de la théorie économique, et de la philosophie des sciences. Afin d’obtenir un panorama contextuel de son émergence, cette thèse fournit une reconstruction des débats intellectuels des années 1930 en Angleterre sur deux plans principaux : le débat sur la planification de la science, et celui sur la planification de l’économie. Dans un climat propice aux idées planistes, perçues comme davantage rationnelles et scientifiques, les néolibéraux précoces s’attelèrent à montrer la portée limitée de la science positive pour orienter les décisions politiques. La montée du totalitarisme contribua à donner à leur discours une urgence singulière, puisqu’il expliquait le recours au collectivisme étatique par la prégnance d’opinions scientifiques erronées. Pendant la Seconde Guerre mondiale, la formation d’un réseau néolibéral déclencha une fertilisation croisée entre ces différents penseurs, dont l’agenda commun avait été défini au moment du Colloque Walter-Lippmann en 1938. En développant leurs intuitions sur le fonctionnement interne de la science et de l’économie comme modèles de coopération libre, Michael Polanyi, Friedrich Hayek, Louis Rougier, Walter Lippmann, et Karl Popper, élaborèrent une théorie sociale cohérente, qui supportaient les idéaux libéraux sur de nouvelles bases épistémologiques. Pour eux, le désir de mener une politique « scientifique » relevait d’un aveuglement méthodologique issu d’une mécompréhension de la nature de la connaissance et du travail scientifique, ainsi que d’une conception disproportionnée de leur potentiel. En reliant de manière analogique la liberté scientifique, avec celle garantie par le marché ou la règle de droit, la position de la pensée et de la connaissance dans la société est devenue leur préoccupation principale. Ce recodage met en lumière la forme particulière de l’idéologie néolibérale : la compétition et les marchés sont redéfinis comme procédures de découverte, les traditions sont perçues comme des réservoirs de connaissance tacite, et les institutions sont conçues comme les préconditions et les résultats d’ordres spontanés. L’institutionnalisation de ce collectif de pensée fragmentaire lors de la fondation de la Société du Mont-Pèlerin en 1947 révéla à la fois l’ambition idéologique de ce projet et ses limites immédiates.This dissertation examines the intellectual genesis of neoliberalism through the prism of its epistemology. It interrogates the development of its arguments regarding the production and diffusion of knowledge, guided by the hypothesis that formulating a common epistemological stance was crucial for the consolidation of its ideological program. I propose that early neoliberalism, by provoking a rupture with classical liberalism, recoded liberal principles into an epistemological framework based on conventionalism, with premises drawn from the natural sciences, economic theory, and the philosophy of science. To achieve a contextual picture of its emergence, the dissertation provides a reconstruction of the intellectual debates of the 1930s in England on two major fronts: the debate on planning in science, and the debate on planning in the economy. Amidst a general enthusiasm for planning ideas perceived as being more rational and scientific, early neoliberals warned of the limited value of positive science in guiding policy decisions. The rise of totalitarianism gave their discourse a dramatic urgency as it explicitly linked faulty scientific views with the rise of state collectivism. During the Second World War, the formation of a neoliberal network triggered a cross-fertilization between these early neoliberal thinkers, whose common agenda had been defined at the Walter-Lippmann Colloquium in 1938. Drawing from their intuitions about the inner workings of science and the economy held as models of free cooperation, Michael Polanyi, Friedrich Hayek, Louis Rougier, Walter Lippmann, and Karl Popper, cemented a coherent social theory which vindicated liberal ideals on new epistemological grounds. To them, the aspiration towards ‘scientific’ politics denoted a methodological delusion built on a misunderstanding of the nature of knowledge and of scientific work, as well as on a hubristic conception of their potential. By linking analogically the freedom experienced by the scientist, to the one guaranteed by the market or by the rule of law, the position of thought and knowledge in society became their core concern. Paying attention to this recoding process sheds light on the peculiar shape of neoliberal ideology: competition and markets were redefined as discovery procedures, traditions were seen as receptacles of tacit knowledge, and institutions were conceived as the preconditions and results of dynamic evolutionary orders. The institutionalization of this fragmentary thought collective at the foundation of the Mont-Pèlerin Society in 1947 revealed both the novelty of this project and its immediate limits, in particular the tensions between its scientific ambition and its ideological projection

Metaphors in the construction of theory: Ramus, Peirce and the American mind

This study argues for the mutual impenetration of logical, legal and scientific metaphors and attempts to determine the role played by them in the construction of theory. Specifically it attempts to discover the impact which the metaphors of topical logic may have had on the construction of American ideology. Chapter 1 offers a brief discussion of logical metaphors and their relation to the social and intellectual settings which generate them. Chapter 2 extends that discussion to principles of positive law and political order as they developed in the unstable atmosphere of 16th Century Europe. Chapters 3, 4 and 5 attempt to relate the metaphors defined in Chapters 1 and 2 to the development of the scientific models which emerged during the scientific revolution of the 17th Century. These chapters proceed in the context of a discussion of the interaction of Aristotelian, Cartesian and Ramean paradigms. Chapter 6 argues for the crucial importance of topical metaphors in the establishment of order in the American colonies. Chapters 7 and 8 carry the argument for a New England Mind into a national setting and discuss how Ramean metaphors contributed to the construction of American conceptions of political order and physical law. These chapters attempt to identify a controlling metaphor of continuity which operated at the base of American models. Chapter 9 claims this metaphor of continuity as the logical ground of pragmatic thought, transmitted to C. S. Peirce through the German logical tradition via Leibniz and Wolff. Chapter 10 extends that discussion to a specific investigation of Peirce\u27s Illustrations of the Logic of Science, considered here as representative of a fundamental commitment on Peirce\u27s part to a methodology which would underwrite the rest of his thought. Chapter 11 laments the failure of James, Dewey and Royce to appreciate the power of Peirce\u27s model and discusses the effect which their fragmentation of his continuous reality had on American philosophy. Peirce\u27s logic of science emerges as a fundamental expression of an American mind with roots sunk deep in a Ramean logical paradigm