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 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.

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.

QC 20100414

Networks of neurons, which form central pattern generators (CPGs), are important for controlling animal behaviors. Of special interest are configurations or CPG motifs composed of reciprocally inhibited neurons, such as half-center oscillators (HCOs). Bursting rhythms of HCOs are shown to include stable synchrony or in-phase bursting. This in-phase bursting can co-exist with anti-phase bursting, commonly expected as the single stable state in HCOs that are connected with fast non-delayed synapses. The finding contrasts with the classical view that reciprocal inhibition has to be slow or time-delayed to synchronize such bursting neurons. Phase-locked rhythms are analyzed via Lyapunov exponents estimated with variational equations, and through the convergence rates estimated with Poincar\'e return maps. A new mechanism underlying multistability is proposed that is based on the spike interactions, which confer a dual property on the fast non-delayed reciprocal inhibition; this reveals the role of spikes in generating multiple co-existing phase-locked rhythms. In particular, it demonstrates that the number and temporal characteristics of spikes determine the number and stability of the multiple phase-locked states in weakly coupled HCOs. The generality of the multistability phenomenon is demonstrated by analyzing diverse models of bursting networks with various inhibitory synapses; the individual cell models include the reduced leech heart interneuron, the Sherman model for pancreatic beta cells, the Purkinje neuron model and Fitzhugh-Rinzel phenomenological model. Finally, hypothetical and experiment-based CPGs composed of HCOs are investigated. This study is relevant for various applications that use CPGs such as robotics, prosthetics, and artificial intelligence.

We consider three problems in this dissertation, all under the unifying theme of random matrix products. The first and second problems are concerned with weak convergence in stochastic matrices and circulant matrices, respectively, and the third is concerned with the numerical calculation of the Lyapunov exponent associated with some random Fibonacci sequences. Stochastic matrices are nonnegative matrices whose row sums are all equal to 1. They are most commonly encountered as transition matrices of Markov chains. Circulant matrices, on the other hand, are matrices where each row after the first is just the previous row cyclically shifted to the right by one position. Like stochastic matrices, circulant matrices are ubiquitous in the literature.In the first problem, we study the weak convergence of the convolution sequence mu to the n, where mu is a probability measure with support S sub mu inside the space S of d by d stochastic matrices, d greater than or equal to 3. Note that mu to the n is precisely the distribution of the product X sub 1 times X sub 2 times and so on times X sub n of the mu distributed independent random variables X sub 1, X sub 2, and so on, X sub n taking values in S. In [CR] Santanu Chakraborty and B.V. Rao introduced a cyclicity condition on S sub mu and showed that this condition is necessary and sufficient for mu to the n to not converge weakly when d is equal to 3 and the minimal rank r of the matrices in the closed semigroup S generated by S sub mu is 2. Here, we extend this result to any d bigger than 3. Moreover, we show that when the minimal rank r is not 2, this result does not always hold.The second problem is an investigation of weak convergence in another direction, namely the case when the probability measure mu's support S sub mu consists of d by d circulant matrices, d greater than or equal to 3, which are not necessarily nonnegative. The resulting semigroup S generated by S sub mu now lacking the nice property of compactness in the case of stochastic matrices, we assume tightness of the sequence mu to the n to analyze the problem. Our approach is based on the work of Mukherjea and his collaborators, who in [LM] and [DM] presented a method based on a bookkeeping of the possible structure of the compact kernel K of S.The third problem considered in this dissertation is the numerical determination of Lyapunov exponents of some random Fibonacci sequences, which are stochastic versions of the classical Fibonacci sequence f sub (n plus 1) equals f sub n plus f sub (n minus 1), n greater than or equal to 1, and f sub 0 equal f sub 1 equals 1, obtained by randomizing one or both signs on the right side of the defining equation and or adding a "growth parameter." These sequences may be viewed as coming from a sequence of products of i.i.d. random matrices and their rate of growth measured by the associated Lyapunov exponent. Following techniques presented by Embree and Trefethen in their numerical paper [ET], we study the behavior of the Lyapunov exponents as a function of the probability p of choosing plus in the sign randomization.

published_or_final_version / abstract / Electrical and Electronic Engineering / Master / Master of Philosophy

We investigate how the formation of heterogeneity and structures in flows through explicit porous microstructures depends upon the geometric and topological observables of the porous medium. Using direct numerical simulations of single-phase, isothermal, laminar fluid flow through realistic three-dimensional stochastically generated pore structures, hereafter referred to as pore spaces, the characteristics of the resulting steady state velocity fields are related to physical characteristics of the pore spaces. The results suggest that the spatially variable resistance offered by the geometry and topology of the pore space induces a highly heterogeneous fluid velocity field therein. Focus is placed on three different length scales: macroscopic (cm), mesoscopic (mm), and microscopic (microns). At the macroscopic length scale, volume averaging is used to relate porosity, mean hydraulic radius, and their product to the permeability of the pore space. At the mesoscopic scale, the effect of a medium's porosity on fluid particle trajectory attributes, such as passage time and tortuosity, is studied. At the final length scale, that of the microscopic in-pore fluid dynamics, finite time Lyapunov exponents are used to determine expanding, contracting, and hyperbolic regions in the flow field, which are then related to the local structure of the pore space. The results have implications to contaminant transport, mixing, and how chemical reactions are induced at the pore-scale. A description of the adopted numerical methods to simulate flow and generate the pore space are provided as well.