This thesis is focused on the problem of growth and integrability in multi-valued dynamics generated by $SL_2 (\mathbb{Z})$ actions. An important example is given by Markov dynamics on the cubic surface $$x^2+ y^2 +z^2 = 3xyz,$$ generating all the integer solutions of this celebrated Diophantine equation, known as Markov triples. To study the growth problem of Markov numbers we use the binary tree representation. This allows us to define the Lyapunov exponents $\Lambda (x)$ as the function of the paths on this tree, labelled by $x \in \mathbb{R}P^1$. We prove that $\Lambda (x)$ is a $PGL_2 (\mathbb{Z})$-invariant function, which is zero almost everywhere but takes all values in $\left[ 0, \ln \varphi \right]$ (where $\varphi$ denotes the golden ratio). We also show that this function is monotonic, and that its restriction to the Markov-Hurwitz set of most irrational numbers is convex in the Farey parametrisation. We also study the growth problem for integer binary quadratic forms using Conway's topograph representation. It is proven that the corresponding Lyapunov exponent $\Lambda_Q(x) = 2 \Lambda(x)$ except for the paths along the Conway river. Finally, we study the tropical version of the Markov dynamics on the tropical version of the Cayley cubic proposed by Adler and Veselov, and show that it is semi-conjugated to the standard action of $SL_2(\mathbb{Z})$ on a torus. This implies the dynamics is ergodic, with the Lyapunov exponent and entropy given by the logarithm of the spectral radius of the corresponding matrix.

abstract: There are many inconsistencies in the literature regarding how to estimate the Lyapunov Exponent (LyE) for gait. In the last decade, many papers have been published using Lyapunov Exponents to determine differences between young healthy and elderly adults and healthy and frail older adults. However, the differences in methodologies of data collection, input parameters, and algorithms used for the LyE calculation has led to conflicting numerical values for the literature to build upon. Without a unified methodology for calculating the LyE, researchers can only look at the trends found in studies. For instance, LyE is generally lower for young adults compared to elderly adults, but these values cannot be correlated across studies to create a classifier for individuals that are healthy or at-risk of falling. These issues could potentially be solved by standardizing the process of computing the LyE. This dissertation examined several hurdles that must be overcome to create a standardized method of calculating the LyE for gait data when collected with an accelerometer. In each of the following investigations, both the Rosenstein et al. and Wolf et al. algorithms as well as three normalization methods were applied in order to understand the extent at which these factors affect the LyE. First, the a priori parameters of time delay and embedding dimension which are required for phase space reconstruction were investigated. This study found that the time delay can be standardized to a value of 10 and that an embedding dimension of 5 or 7 should be used for the Rosenstein and Wolf algorithm respectively. Next, the effect of data length on the LyE was examined using 30 to 1300 strides of gait data. This analysis found that comparisons across papers are only possible when similar amounts of data are used but comparing across normalization methods is not recommended. And finally, the reliability and minimum required number of strides for each of the 6 algorithm-normalization method combinations in both young healthy and elderly adults was evaluated. This research found that the Rosenstein algorithm was more reliable and required fewer strides for the calculation of the LyE for an accelerometer. / Dissertation/Thesis / Appendix A / Doctoral Dissertation Biomedical Engineering 2019

We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. Then, we use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.

The complexity of the interface between the muscular system and the nervous system is still elusive. We investigated how the neuromuscular system functions and how it is influenced by various perturbations. Postural stability was selected as the model system, because this system provides complex output, which could indicate underlying mechanisms and feedback loops of the neuromuscular system. We hypothesized that aging, physical pain, and mental and physical perturbations affect balancing strategy, and based on these observations, we constructed a model that simulates many aspects of the neuromuscular system. Our results show that aging changes the control strategy of balancing from more chaotic to more repetitive. The chaotic elements ensure quick reactions and strong capacity to compensate for the perturbations; this adeptly reactive state changes into a less reactive, slower, probably less mechanically costly balancing strategy. Mental tasks during balancing also decreased the chaotic elements in balancing strategy, especially if the subject experienced chronic pain. Additional motoric tasks, such as tying knots while balancing, were correlated with age but unaffected by chronic pain. Our model competently predicted the experimental findings, and we proceeded to use the model with an external data set from Physionet to predict the balancing strategy of Parkinson’s patients. Our neurological model, comprised of RLC circuits, provides a mechanistic explanation for the neuromuscular system adaptations.

The objective of this thesis is to develop a novel methodology of fractional stochastic dynamics to study stochastic stability of viscoelastic systems under stochastic loadings. Numerous structures in civil engineering are driven by dynamic forces, such as seismic and wind loads, which can be described satisfactorily only by using probabilistic models, such as white noise processes, real noise processes, or bounded noise processes. Viscoelastic materials exhibit time-dependent stress relaxation and creep; it has been shown that fractional calculus provide a unique and powerful mathematical tool to model such a hereditary property. Investigation of stochastic stability of viscoelastic systems with fractional calculus frequently leads to a parametrized family of fractional stochastic differential equations of motion. Parametric excitation may cause parametric resonance or instability, which is more dangerous than ordinary resonance as it is characterized by exponential growth of the response amplitudes even in the presence of damping. The Lyapunov exponents and moment Lyapunov exponents provide not only the information about stability or instability of stochastic systems, but also how rapidly the response grows or diminishes with time. Lyapunov exponents characterizes sample stability or instability. However, this sample stability cannot assure the moment stability. Hence, to obtain a complete picture of the dynamic stability, it is important to study both the top Lyapunov exponent and the moment Lyapunov exponent. Unfortunately, it is very difficult to obtain the accurate values of theses two exponents. One has to resort to numerical and approximate approaches. The main contributions of this thesis are: (1) A new numerical simulation method is proposed to determine moment Lyapunov exponents of fractional stochastic systems, in which three steps are involved: discretization of fractional derivatives, numerical solution of the fractional equation, and an algorithm for calculating Lyapunov exponents from small data sets. (2) Higher-order stochastic averaging method is developed and applied to investigate stochastic stability of fractional viscoelastic single-degree-of-freedom structures under white noise, real noise, or bounded noise excitation. (3) For two-degree-of-freedom coupled non-gyroscopic and gyroscopic viscoelastic systems under random excitation, the Stratonovich equations of motion are set up, and then decoupled into four-dimensional Ito stochastic differential equations, by making use of the method of stochastic averaging for the non-viscoelastic terms and the method of Larionov for viscoelastic terms. An elegant scheme for formulating the eigenvalue problems is presented by using Khasminskii and Wedig’s mathematical transformations from the decoupled Ito equations. Moment Lyapunov exponents are approximately determined by solving the eigenvalue problems through Fourier series expansion. Stability boundaries, critical excitations, and stability index are obtained. The effects of various parameters on the stochastic stability of the system are discussed. Parametric resonances are studied in detail. Approximate analytical results are confirmed by numerical simulations.

Résumé The dissertation called The Tarahumara Worldview is the attempt to indicate and clear up the relationship between the language and the culture. The work follows from the mytical arrangement of the Tarahumara natural world living in the Northwestern of Mexico from the description and analyses of some language elements and searching for its language features in the culture. The author puts a few questions: what is position of myth in the worldview - in case there is some?; what is the relation of the Tarahumara language and culture?; does exist any possibility to outcome of the relation of two parts of one culture as it is myth system and language system into thing what the anthropology consider as the worldview?, does myths and language could show in any way a setting of Tarahumara native world setting, their way of thinking?; does the myth shows the base for the cultural creation and how?; even if the question is dificult becouse of the European culture influences we are trying to ask for the origin of Tarahumara culture in the myth - what is the role of the myth and what is the role of the language in Tarahumara worldview. Basicaly I ask for the relation between language and myth in culture environment: where is the edge of this relation, is this relaiton important or it has marginal impact and it...

Pour un réel ξ qui n’est pas algébrique de degré ≤ 2, on peut définir plusieurs exposants diophantiens qui mesurent la qualité d’approximation du vecteur (1, ξ, ξ² ) par des sous-espaces de ℝ³ définis sur ℚ de dimension donnée. Cette thèse s’inscrit dans l’étude de ces exposants diophantiens et des questions relatives à la détermination de leur spectre. En utilisant notamment les outils modernes de la géométrie paramétrique des nombres, nous construisons une nouvelle famille de réels – appelés nombres de type sturmien – et nous déterminons presque complètement le 3-système qui leur est associé. Comme conséquence, nous en déduisons la valeur de leurs exposants diophantiens et certaines informations sur les spectres. Nous considérons également le problème plus général de l’allure d’un 3-système associé à un vecteur de la forme (1, ξ, ξ ²), en formulant entre autres certaines contraintes qui n’existent pas pour un vecteur (1, ξ, η) quelconque, et en explicitant les liens qu’il entretient avec la suite des points minimaux associée à ξ. Sous certaines conditions de récurrence sur la suite des points minimaux nous montrons que nous retrouvons les 3-systèmes associés aux nombres de type sturmien. / Given a real number ξ which is not algebraic of degree ≤ 2 one may defineseveral diophantine exponents which measure how “well” the vector (1, ξ, ξ ²) can be approximated by subspaces of fixed dimension defined over ℚ. This thesis is part of the study of these diophantine exponents and their spectra. Using the parametric geometry of numbers, we construct a new family of numbers – called numbers of sturmian type – and we provide an almost complete description of the associated 3-system. As a consequence, we determine the value of the classical exponents for numbers of sturmian type, and we obtain new information on their joint spectra. We also take into consideration a more general problem consisting in describing a 3-system associated with a vector (1, ξ, ξ²). For instance we formulate special constraints which do not exist for a general vector (1, ξ, η) and we also clarify connections between a 3-system which represents ξ and the sequence of minimal points associated to ξ. Under a specific recurrence relation hypothesis on the sequence of minimal points, we show that the previous 3-system has the shape of a 3-system associated to a number of sturmian type.

Four aspects of the dynamics of continuous-time dynamical systems are studied in this work. The relationship between the Lyapunov exponents of the original system and the Lyapunov exponents of induced Poincare maps is examined. The behavior of these Poincare maps as discriminators of chaos from noise is explored, and the possible Poissonian statistics generated at rarely visited surfaces are studied.

Let f be a C2 local diffeomorphism, of a closed surface M without zero Lyapunov exponents. We have proved that the number of ergodic hyperbolic measures of f with SRB property is less than equal to the number of homoclinic equivalence classes. We use an adaptation of Katok closing lemma for endomorphisms and prove ergodic criterion, introduced in [HHTU], for endomorphisms. We also prove some folklore results on uniqueness of SRB measures, in the presence of topological transitivity / Seja f um endomorfismo C2 non-singular (difeomorfismo local), de uma superfície fechada M e µ uma medida probabilidade Borel f-invariante e ergódica com expoentes de Lyapunov Não nulo. Nós provamos que o número de medidas hiperbólicas com propriedade SRB é para f so menor ou igual ao número de classes equivalentes homoclínicos. Usamos uma adaptaão do closing lema de Katok por endomorfismos e provamos critrio ergódico, introduzido em [HHTU], para endomorfismos. Também provamos alguns resultados folclóricos em unicidade de medidas SRB, na presena de transitividade topológica vii

O objetivo deste trabalho é analisar o comportamento dos expoentes críticos do modelo de Oito Vértices através de cotas superior e inferior para sua função de partição na vizinhança de modelos solúveis. O método é ilustrado pelo modelo de Heisenberg quântico unidimensional também denominado modelo XYZh. Aplica-se igualmente ao modelo de Ising bidimensional (com interação quártica e segundos vizinhos). Assim, propomos um modo alternativo de abordar universalidade nos modelos de Heisenberg unidimensional quântico e Ising bidimensional clássico por desigualdades satisfeitas por suas funções de partição. Dentre os métodos que utilizamos para a obtenção das cotas destacam-se: a interação Gaussiana nas variáveis reais e nas variáveis de Grassmann; o mapeamento de um modelo unidimensional em um bidimensional através do auxílio da fórmula Trotter; a representação da função de partição pelo Pfaffiano de uma matriz; e, para a obtenção da cota superior, a técnica de positividade por reflexão, estendida ao acaso de variáveis que anti-comutam. / The aim of this work is to analyze the behavior of critical exponents in the eight-vertex model starting from the upper and lower bound obtained for its partition function. We studied the quantum onedimensional Heisenberg model also denominated XYZh model. We propose na alternative way of approaching universality in Heisenberg and Ising models using inequalities satisfied for their partition functions.Among the methods that we used in the solutions of the models atand out the integration on the Grassmann variables, the mapping of a onedimensional model in a two-dimensional one through the aid of the Trotter formula and, finally, the representation of the partition function as Pfaffian of a matrix. To obtain na upper bound, the positivity reflection technique was used, extended to the case of variables that, anticomute, and the method of thechess board estimate.