This thesis explores the dynamics of two bipedal, passive-walker models that are free to move in a three-dimensional environment. Specifically, two rigid-bodied walkers that can sustain anthropomorphic gait down an inclined plane with gravity being the only source of energy were studied using standard dynamical systems methods. This includes calculating the stability of periodic orbits and varying the system parameter to create bifurcation diagrams and to address the persistence of a periodic solution under specific parameter variations. These periodic orbits are found by implementing the Newton-Raphson root solving scheme. The dynamical systems associated with these periodic orbits are not completely smooth. Instead, they include discontinuities, such as those produced due to forces at foot contact points and during knee hyper-extension. These discontinuities are addressed in the stability calculations through appropriate discontinuity mappings. The difference between the two walker models is the number of degrees of freedom (DOF) at the hip. Humans possess three DOF at each hip joint, one DOF at each knee joint, and at least two DOF at each ankle joint. The first walker model studied had revolute joints at the hips and knees and completely locked ankles. To make the walking motion more anthropomorphic, additional degrees of freedom were added to the hip. Specifically, the second walker model has ball joints at the hips. Two control algorithms are used for controlling the local stability of periodic motions for both walker models. The methods, reference and delay feedback control, rely on the presence of discontinuities in the system. Moreover, it is possible to predict the effects of the control strategy based entirely on information from the uncontrolled system. Control is applied to both passive walker models to try and stabilize an unstable periodic gait by making small, discrete, changes in the foot orientation during gait. Results show that both methods are successful in stabilizing an unstable walking motion for a 3D model with one DOF in each hip and to reduce the instability of the walking motions for the model having more mobility in the hip joints. / Master of Science

Neurological and developmental disorders (NDDs) impose an enormous burden of disease on children throughout the world. Two of the most common are autism spectrum disorder (ASD) and epilepsy. ASD has recently been estimated to affect 1 in 68 children, making it the most common neurodevelopmental disorder in children. Epilepsy is also a spectrum disorder that follows a developmental trajectory, with an estimated prevalence of 1%, nearly as common as autism. ASD and epilepsy co-occur in approximately 30% of individuals with a primary diagnosis of either disorder. Although considered to be different disorders, the relatively high comorbidity suggests the possibility of common neuropathological mechanisms. Early interventions for NDDs lead to better long-term outcomes. But early intervention is predicated on early detection. Behavioral measures have thus far proven ineffective in detecting autism before about 18 months of age, in part because the behavioral repertoire of infants is so limited. Similarly, no methods for detecting emerging epilepsy before seizures begin are currently known. Because atypical brain development is likely to precede overt behavioral manifestations by months or even years, a critical developmental window for early intervention may be opened by the discovery of brain based biomarkers. Analysis of brain activity with EEG may be under-utilized for clinical applications, especially for neurodevelopment. The hypothesis investigated in this dissertation is that new methods of nonlinear signal analysis, together with methods from biomedical informatics, can extract information from EEG data that enables detection of atypical neurodevelopment. This is tested using data collected at Boston Children’s Hospital. Several results are presented. First, infants with a family history of ASD were found to have EEG features that may enable autism to be detected as early as 9 months. Second, significant EEG-based differences were found between children with absence epilepsy, ASD and control groups using short 30-second EEG segments. Comparison of control groups using different EEG equipment supported the claim that EEG features could be computed that were independent of equipment and lab conditions. Finally, the potential for this technology to help meet the clinical need for neurodevelopmental screening and monitoring in low-income regions of the world is discussed.

This dissertation addresses the question, how and why do 'flying' snakes (Chrysopelea) undulate through the air? Instead of deploying paired wings or wing-like surfaces, flying snakes jump, splay their ribs into a bluff-body airfoil, and undulate through the air. Aerial undulation is the dominant feature of snake flight, but its effects on locomotor performance and stability are unknown. Chapter 2 describes a new non-equilibrium framework to analyze gliding animals and how the pitch angle affects their translational motion. Chapter 3 combines flying snake glide experiments and detailed dynamic modeling to address what is aerial undulation and how each kinematic component affects rotational stability and translational performance. Chapter 4 combines the kinematic data of Chapter 3, with elements of the non-equilibrium framework of Chapter 2, to examine the kinematics of snake flight in greater detail. This chapter also tests if our current understanding of flying snake aerodynamics is sufficient to explain the observed center of mass motion. / Ph. D. / Flying snakes can move unlike any other flying or slithering animal. Flying snakes have evolved the ability to glide safely to the ground by undulating through the air. Flapping flyers, like birds, bats, and insects, move through the air using wings and powerful flight muscles. In contrast, gliding animals fall through the air, using gravity to increase their speed, and air resistance to produce lift and drag forces such that they move over the ground. Flying snakes glide by jumping, flattening their bodies (similar to a cobra hooding), and undulating through the air using an ‘S’-shaped body. This dissertation addresses the question, how and why do flying snakes undulate through the air? First, I describe a new mathematical analysis of gliding animals, which provides a framework to understand how an animal’s size and orientation to the oncoming airflow affect its glide performance. Second, I describe glide experiments where the bodies of flying snakes were measured as they flew through a large indoor glide arena. From these measurements, we quantified how the body bends in the horizontal and vertical directions. Next, I describe a detailed mathematical model used to test how the different body configurations we measured affect glide performance and flight stability. The model result indicate that flying snakes likely use aerial undulation to stabilize their rotational motion. Third, I tested if our current measurements of the lift and drag properties of flying snakes, based on the quasi-steady assumption, can account for the trajectories we recorded. The force analysis suggests that flying snakes produce more force than the quasi-steady assumption can account for, and that future work is needed to understand unsteady aerodynamic mechanisms relevant for snake flight.

The underlying mathematical model of many simulation models is graph dynamical systems (GDS). This dynamical system, its implementation, and analyses on each will be the focus of this paper. When using a simulation model to answer a research question, it is important to describe this underlying mathematical model in which we are operating for verification and validation. In this paper we discuss analyses commonly used in simulation models. These include sensitivity analyses and uncertainty quantification, which provide motivation for stability and structure-to-function research in GDS. We review various results in these areas, which contribute toward validation and computationally tractable analyses of our simulation model. We then present two new areas of research - stability of transient structure with respect to update order permutations, and an application of GDS in which a time-varying generalized cellular automata is implemented as a simulation model. / Master of Science / There are many systems in our society which are vital, and require quantitative analysis. These include population dynamics, transportation, and energy. To answer research questions about these systems, one may construct a mathematical model of the system and conduct simulations. It is important to define both the mathematical model and the simulation model in order to better understand the source of errors, or to be confident in the validity of the models. One source of error may be in parameters of our simulation model. It can be difficult to gather reliable and precise data, especially in massively interacting systems. Thus we would like to know that there is a range of values which will result in similar outcomes. Stability results can give us this assurance. This paper mainly focuses on stability results in graph dynamical systems (GDS), which is the underlying mathematical model of many simulation models, especially ones with a networked structure.

In this dissertation we focus on the application of thermodynamic formalism to non-autonomous and random dynamical systems. Specifically we use the thermodynamic formalism to investigate the dimension of various fractal constructions via the, now standard, technique of Bowen which he developed in his 1979 paper on quasi-Fuchsian groups. Bowen showed, roughly speaking, that the dimension of a fractal is equal to the zero of the relevant topological pressure function. We generalize the results of Rempe-Gillen and Urbanski on non-autonomous iterated function systems to the setting of non-autonomous graph directed Markov systems and then show that the Hausdorff dimension of the fractal limit set is equal to the zero of the associated pressure function provided the size of the alphabets at each time step do not grow too quickly. In trying to remove these growth restrictions, we present several other systems for which Bowen's formula holds, most notably ascending systems. We then use these various constructions to investigate the Hausdorff dimension of various subsets of the Julia set for different large classes of transcendental meromorphic functions of finite order which have been perturbed non-autonomously. In particular we find lower and upper bounds for the dimension of the subset of the Julia set whose points escape to infinity, and in many cases we find the exact dimension. While the upper bound was known previously in the autonomous case, the lower bound was not known in this setting, and all of these results are new in the non-autonomous setting. We also use transfer operator techniques to prove an almost sure invariance principle for random dynamical systems for which the thermodynamical formalism has been well established. In particular, we see that if a system exhibits a fiberwise spectral gap property and the base dynamical system is sufficiently well behaved, i.e. it exhibits an exponential decay of correlations, then the almost sure invariance principle holds. We then apply these results to uniformly expanding random systems like those studied by Mayer, Skorulski, and Urbanski and Denker and Gordin.

Associated to any finite simple graph Γ is the <i>chromatic polynomial </i>PΓ(q) whose complex zeros are called the <i>chromatic zeros </i>of Γ. A hierarchical lattice is a sequence of finite simple graphs {Γ_n}∞_<i>n</i>-0 built recursively using a substitution rule expressed in terms of a generating graph. For each <i>n</i>, let <i>μn</i> denote the probability measure that assigns a Dirac measure to each chromatic zero of Γ_<i>n</i>. Under a mild hypothesis on the generating graph, we prove that the sequence <i>μn</i> converges to some measure <i>μ</i> as <i>n</i> tends to infinity. We call <i>μ</i> the limiting measure of <i>chromatic zeros</i> associated to {Γ_n}∞_<i>n-</i>0. In the case of the Diamond Hierarchical Lattice we prove that the support of <i>μ</i> has Hausdorff dimension two.<div><br></div><div>The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove anew equidistribution theorem that can be used to relate the chromatic zeros of ahierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.<br></div>

Neural synchrony in the brain at rest is usually variable and intermittent, thus intervals of predominantly synchronized activity are interrupted by intervals of desynchronized activity. Prior studies suggested that this temporal structure of the weakly synchronous activity might be functionally significant: many short desynchronizations may be functionally different from few long desynchronizations, even if the average synchrony level is the same. In this thesis, we use computational neuroscience methods to investigate the effects of (i) spike-timing dependent plasticity (STDP) and (ii) noise on the temporal patterns of synchronization in a simple model. The model is composed of two conductance-based neurons connected via excitatory unidirectional synapses. In (i) these excitatory synapses are made plastic, in (ii) two different types of noise implementation to model the stochasticity of membrane ion channels is considered. The plasticity results are taken from our recently published article, while the noise results are currently being compiled into a manuscript.<br><br>The dynamics of this network is subjected to the time-series analysis methods used in prior experimental studies. We provide numerical evidence that both STDP and channel noise can alter the synchronized dynamics in the network in several ways. This depends on the time scale that plasticity acts on and the intensity of the noise. However, in general, the action of STDP and noise in the simple network considered here is to promote dynamics with short desynchronizations (i.e. dynamics reminiscent of that observed in experimental studies) over dynamics with longer desynchronizations.

<p>Recent innovations in combinatorial dynamical systems permit them to be studied with algorithmic methods. One such method from topological data analysis, called persistent homology, allows one to summarize the changing homology of a sequence of simplicial complexes. This dissertation explicates three methods for capturing and summarizing changes in combinatorial dynamical systems through the lens of persistent homology. The first places the Conley index in the persistent homology setting. This permits one to capture the persistence of salient features of a combinatorial dynamical system. The second shows how to capture changes in combinatorial dynamical systems at different resolutions by computing the persistence of the Conley-Morse graph. Finally, the third places Conley's notion of continuation in the combinatorial setting and permits the tracking of isolated invariant sets across a sequence of combinatorial dynamical systems. </p>

As famílias Anosov foram introduzidas por P. Arnoux e A. Fisher, motivados por generalizar a noção de difeomorfismo de Anosov. A grosso modo, as famílias Anosov são sequências de difeomorfismos (fi)i&#8712Z definidos em uma sequencia de variedades Riemannianas compactas (Mi)i&#8712Z, em que fi: Mi ->Mi+1 para todo i &#8712 Z, tal que a composição fi+no· · ·ofi, para n >=1, tem comportamento assintoticamente hiperbólico. Esta noção é conhecida como um sistema dinâmico não-estacionário ou um sistema dinâmico não-autônomo. Sejam M a união disjunta de cada Mi, para i &#8712 Z, e Fm(M) o conjunto consistente das famílias de difeomorfismos (fi)i&#8712Z de classe Cm definidos na sequência (Mi)i&#8712Z. O propósito principal deste trabalho é mostrar algumas propriedades das famílias Anosov. Em particular, mostraremos que o conjunto destas famílias é aberto em Fm(M), em que Fm(M) é munido da topologia forte (ou topologia Whitney); a estabilidade estrutural de certa classe de famílias Anosov, considerando conjugações topológicas uniformes; e várias versões para os Teoremas de variedades estáveis e instáveis. Os resultados que serão apresentados aqui generalizam alguns outros resultados obtidos em Sistemas Dinâmicos Aleatórios, os quais serão mencionados ao longo do trabalho. Além do anterior, será introduzida a entropia topológica para elementos em Fm(M) e mostraremos algumas das suas propriedades. Provaremos que esta entropia é contínua em Fm(M) munido da topologia forte. Porém, ela é descontínua em cada elemento de Fm(M) munido da topologia produto. Também apresentaremos um resultado que pode ser uma ferramenta de muita utilidade no estudo da continuidade da entropia topológica de difeomorfismos definidos em variedades compactas. Finalizaremos o trabalho dando uma lista de problemas que surgiram ao longo desta pesquisa e que serão analisados em um trabalho futuro. / Anosov families were introduced by P. Arnoux and A. Fisher, motivated by generalizing the notion of Anosov dieomorphisms. Roughly, Anosov families are sequences of dieomorphisms (fi)i&#8712Z dened on a sequence of compact Riemannian manifolds (Mi)i&#8712Z, where fi: Mi -> Mi+1 for all i &#8712 Z, such that the composition fi+n o · · · o fi, for n >=1, has asymptotically hyperbolic behavior. This notion is known as a non-stationary dynamical system or a non-autonomous dynamical system. Let M be the disjoint union of each Mi, for each i &#8712 Z, and Fm(M) the set consisting of families of Cm-dieomorphisms (fi)i&#8712Z dened on the sequence (Mi)i&#8712Z. The main goal of this work is to explore some properties of Anosov families. In particular, we will show that the set consisting of these families is open in Fm(M), where Fm(M) is endowed with the strong topology (or Whitney topology); the structural stability of a certain class of Anosov families, considering uniform topological conjugacies; and some versions of stable and unstable manifold theorems. The results that will be presented here generalize some results obtained in Random Dynamical Systems, which will be mentioned throughout the work. In addition to the above mentioned theorems, the topological entropy for elements in Fm(M) will be introduced, and we will show some of its properties. We will prove that this entropy is continuous on Fm(M) endowed with strong topology. However, it is discontinuous at each element of Fm(M) endowed with the product topology. We will also present a result that can be a very useful tool in the study of the continuity of the topological entropy of dieomorphisms dened on compact manifolds. We will nish the work by giving a list of problems that have arisen throughout this research and that will be analyzed in a future work.

As famílias Anosov foram introduzidas por P. Arnoux e A. Fisher, motivados por generalizar a noção de difeomorfismo de Anosov. A grosso modo, as famílias Anosov são sequências de difeomorfismos (fi)i&#8712Z definidos em uma sequencia de variedades Riemannianas compactas (Mi)i&#8712Z, em que fi: Mi ->Mi+1 para todo i &#8712 Z, tal que a composição fi+no· · ·ofi, para n >=1, tem comportamento assintoticamente hiperbólico. Esta noção é conhecida como um sistema dinâmico não-estacionário ou um sistema dinâmico não-autônomo. Sejam M a união disjunta de cada Mi, para i &#8712 Z, e Fm(M) o conjunto consistente das famílias de difeomorfismos (fi)i&#8712Z de classe Cm definidos na sequência (Mi)i&#8712Z. O propósito principal deste trabalho é mostrar algumas propriedades das famílias Anosov. Em particular, mostraremos que o conjunto destas famílias é aberto em Fm(M), em que Fm(M) é munido da topologia forte (ou topologia Whitney); a estabilidade estrutural de certa classe de famílias Anosov, considerando conjugações topológicas uniformes; e várias versões para os Teoremas de variedades estáveis e instáveis. Os resultados que serão apresentados aqui generalizam alguns outros resultados obtidos em Sistemas Dinâmicos Aleatórios, os quais serão mencionados ao longo do trabalho. Além do anterior, será introduzida a entropia topológica para elementos em Fm(M) e mostraremos algumas das suas propriedades. Provaremos que esta entropia é contínua em Fm(M) munido da topologia forte. Porém, ela é descontínua em cada elemento de Fm(M) munido da topologia produto. Também apresentaremos um resultado que pode ser uma ferramenta de muita utilidade no estudo da continuidade da entropia topológica de difeomorfismos definidos em variedades compactas. Finalizaremos o trabalho dando uma lista de problemas que surgiram ao longo desta pesquisa e que serão analisados em um trabalho futuro. / Anosov families were introduced by P. Arnoux and A. Fisher, motivated by generalizing the notion of Anosov dieomorphisms. Roughly, Anosov families are sequences of dieomorphisms (fi)i&#8712Z dened on a sequence of compact Riemannian manifolds (Mi)i&#8712Z, where fi: Mi -> Mi+1 for all i &#8712 Z, such that the composition fi+n o · · · o fi, for n >=1, has asymptotically hyperbolic behavior. This notion is known as a non-stationary dynamical system or a non-autonomous dynamical system. Let M be the disjoint union of each Mi, for each i &#8712 Z, and Fm(M) the set consisting of families of Cm-dieomorphisms (fi)i&#8712Z dened on the sequence (Mi)i&#8712Z. The main goal of this work is to explore some properties of Anosov families. In particular, we will show that the set consisting of these families is open in Fm(M), where Fm(M) is endowed with the strong topology (or Whitney topology); the structural stability of a certain class of Anosov families, considering uniform topological conjugacies; and some versions of stable and unstable manifold theorems. The results that will be presented here generalize some results obtained in Random Dynamical Systems, which will be mentioned throughout the work. In addition to the above mentioned theorems, the topological entropy for elements in Fm(M) will be introduced, and we will show some of its properties. We will prove that this entropy is continuous on Fm(M) endowed with strong topology. However, it is discontinuous at each element of Fm(M) endowed with the product topology. We will also present a result that can be a very useful tool in the study of the continuity of the topological entropy of dieomorphisms dened on compact manifolds. We will nish the work by giving a list of problems that have arisen throughout this research and that will be analyzed in a future work.