The present work is divided in two parts. The first is concerned with the dynamics on the Grassmann manifold of k-dimensional subvector spaces of an n dimensional real or complex vector space induced by a linear invertible transformation A of the vector space into itself. The Grassmann map GA sends p to Ap, and one asks, what are the dynamics of GA? In the second part, I consider dynamics induced by a linear cocycle covering a diffeomorphism of a compact manifold, acting on the Grassmann bundle of k-dimensional linear subspaces of TN. I prove a Kupka-Smale theorem for the space of cocycles covering diffeomorphisms of a compact manifold. The proof of this theorem implies the same type of results for derived cocycles parametrized in the space of diffeomorphisms. The results of the second part can be generalized without effort to cocycles covering endomorphisms of N.

Thesis (Ph.D.)--Tufts University, 2004. / Adviser: Boris Hasselblatt. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves 61-62). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2007. / Title from title screen (site viewed May 20, 2008). PDF text: 77 p. ; 328 K. UMI publication number: AAT 3284240. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (Master)--İzmir Institute of Technology, İzmir, 2004. / Includes bibliographical references (leaves.55-57).

Thesis (Ph. D.)--Lehigh University, 2006. / Includes vita. Includes bibliographical references (leaves 72-76).

For parametrized families of dynamical systems, two major goals are classifying the systems up to topological conjugacy, and understanding the structure of the bifurcation locus. The family Fλ = z^n + λ/z^d gives a 1-parameter, n+d degree family of rational maps of the Riemann sphere, which arise as singular perturbations of the polynomial z^n. This work presents several results related to these goals for the family Fλ, particularly regarding a structure of "necklaces" in the λ parameter plane. This structure consists of infinitely many simple closed curves which surround the origin, and which contain postcritically finite parameters of two types: superstable parameters and escape time Sierpinski parameters. First, we derive a dynamical invariant to distinguish the conjugacy classes among the superstable parameters on a given necklace, and to count the number of conjugacy classes. Second, we prove the existence of a deeper fractal system of "subnecklaces," wherein the escape time Sierpinski parameters on the previously known necklaces are themselves surrounded by infinitely many necklaces.

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Rotating wave solutions to evolution equations have been shown to govern many important biological and chemical processes. Much of the rigorous mathematical investigations of rotating waves rely on the model exhibiting a continuous Euclidean symmetry, which is only present in an idealized situation. Here we investigate the existence of rotationally propagating solutions in a discrete spatial setting, in which typical symmetry methods cannot be applied, thus presenting an unique perspective on rotating waves. Our goal in this thesis is to demonstrate the existence and potential stability of rotating wave solutions to a spatially discretized infinite systems of coupled differential equations. This goal is achieved by considering so-called Lambda-Omega systems, which have frequently been used to model typical oscillatory dynamics. Our work is broken into three major components: 1. An infinite system of coupled phase equations is investigated and we demonstrate that under some mild assumptions the system exhibits a phase-locked rotating wave solution. The phase system is derived from a limiting case of the original Lambda-Omega system, and therefore solutions of the phase equation will be useful in finding rotating wave solutions to the full Lambda-Omega system. 2. We examine the stability of the rotating wave solution found in the coupled phase equations. This is achieved by providing a link with an underlying graph-theoretic geometry endowed by the spatially discretized system. We use results from random walks on infinite graphs to provide a general stability theorem for coupled phase equations. 3. We use the rotating wave solution of the phase equations to extend to a rotating wave solution of the full Lambda-Omega system. This result is achieved using a non-standard Implicit Function Theorem, since we show that typical implicit function arguments cannot be applied to our present situation.

In this dissertation, we developed predictive models for legged and limbless locomotion on dry, homogeneous granular media. The vertical plane Resistive Force Theory (RFT) for frictional granular fluids accurately predicted the reaction forces on intruders (with complex geometries) translating and rotating at low speeds ( < 0.5 m/s). Using RFT and multibody simulation, we predicted the forward moving speed of legged robots. During the locomotion of lightweight robots and animals where instantaneous limb penetration speed can reach values greater than ~0.5 m/s, a Discrete Element Method (DEM) simulation was developed to capture the limb-ground interaction. We demonstrated that hydrodynamic-like forces generated by accelerated particles can balance the robot weight and inertia, and promote the rapid movement on granular media. Forces from the environment can not only determine locomotion dynamics, but also affect the locomotion strategy. We studied and simulated the limbless locomotion of snakes in a heterogeneous environment and demonstrated how touch sensing was used to adjust the movement pattern. In heterogeneous environments, the long-term locomotion dynamics is also poorly understood. We presented a theory for transport and diffusion in such settings.

This Thesis proposes three main objectives: (i) to provide the concept of a new generalized non-standard synthesis model that would provide the framework for incorporating other non-standard synthesis approaches; (ii) to explore dynamic sound modeling through the application of new non-standard synthesis techniques and procedures; and (iii) to experiment with dynamic sound synthesis for the creation of novel sound objects. In order to achieve these objectives, this Thesis introduces a new paradigm for non-standard synthesis that is based in the algorithmic assemblage of minute wave segments to form sound waveforms. This paradigm is called Extended Waveform Segment Synthesis (EWSS) and incorporates a hierarchy of algorithmic models for the generation of microsound structures. The concepts of EWSS are illustrated with the development and presentation of a novel non-standard synthesis system, the Dynamic Waveform Segment Synthesis (DWSS). DWSS features and combines a variety of algorithmic models for direct synthesis generation: list generation and permutation, tendency masks, trigonometric functions, stochastic functions, chaotic functions and grammars. The core mechanism of DWSS is based in an extended application of Cellular Automata. The potential of the synthetic capabilities of DWSS is explored in a series of Case Studies where a number of sound object were generated revealing (i) the capabilities of the system to generate sound morphologies belonging to other non-standard synthesis approaches and, (ii) the capabilities of the system of generating novel sound objects with dynamic morphologies. The introduction of EWSS and DWSS is preceded by an extensive and critical overview on the concepts of microsound synthesis, algorithmic composition, the two cultures of computer music, the heretical approach in composition, non- standard synthesis and sonic emergence along with the thorough examination of algorithmic models and their application in sound synthesis and electroacoustic composition. This Thesis also proposes (i) a new definition for “algorithmic composition”, (ii) the term “totalistic algorithmic composition”, and (iii) four discrete aspects of non-standard synthesis.