# Browsing by Subject "Dynamical Systems"

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Item Open Access A Multi-Disciplinary Systems Approach for Modeling and Predicting Physiological Responses and Biomechanical Movement Patterns(2017) Mazzoleni, MichaelIt is currently an exciting time to be doing research at the intersection of sports and engineering. Advances in wearable sensor technology now enable large quantities of physiological and biomechanical data to be collected from athletes with minimal obstruction and cost. These technological advances, combined with an increased public awareness of the relationship between exercise, fitness, and health, has created an environment where engineering principles can be integrated with biomechanics, exercise physiology, and sports science to dramatically improve methods for physiological assessment, injury prevention, and athletic performance.

The first part of this dissertation develops a new method for analyzing heart rate (HR) and oxygen uptake (VO2) dynamics. A dynamical system model was derived based on the equilibria and stability of the HR and VO2 responses. The model accounts for nonlinear phenomena and person-specific physiological characteristics. A heuristic parameter estimation algorithm was developed to determine model parameters from experimental data. An artificial neural network (ANN) was developed to predict VO2 from HR and exercise intensity data. A series of experiments was performed to validate: 1) the ability of the dynamical system model to make accurate time series predictions for HR and VO2; 2) the ability of the dynamical system model to make accurate submaximal predictions for maximum heart rate (HRmax) and maximal oxygen uptake (VO2max); 3) the ability of the ANN to predict VO2 from HR and exercise intensity data; and 4) the ability of a system comprising an ANN, dynamical system model, and heuristic parameter estimation algorithm to make submaximal predictions for VO2max without requiring VO2 data collection. The dynamical system model was successfully validated through comparisons with experimental data. The model produced accurate time series predictions for HR and VO2 and, more importantly, the model was able to accurately predict HRmax and VO2max using data collected during submaximal exercise. The ANN was successfully able to predict VO2 responses using HR and exercise intensity as system inputs. The system comprising an ANN, dynamical system model, and heuristic parameter estimation algorithm was able to make accurate submaximal predictions for VO2max without requiring VO2 data collection.

The second part of this dissertation applies a support vector machine (SVM) to classify lower extremity movement patterns that are associated with increased lower extremity injury risk. Participants for this study each performed a jump-landing task, and experimental data was collected using two video cameras, two force plates and a chest-mounted single-axis accelerometer. The video data was evaluated to classify the lower extremity movement patterns of the participants as either excellent or poor using the Landing Error Scoring System (LESS) assessment method. Two separate linear SVM classifiers were trained using the accelerometer data and the force plate data, respectively, with the LESS assessment providing the classification labels during training and evaluation. The same participants from this study also performed several bouts of treadmill running, and an additional set of linear SVM classifiers were trained using accelerometer data and gyroscope data to classify movement patterns, with the LESS assessment again providing the classification labels during training and evaluation. Both sets of SVM's performed with a high level of accuracy, and the objective and autonomous nature of the SVM screening methodology eliminates the subjective limitations associated with many current clinical assessment tools.

Item Open Access Atlas Simulation: A Numerical Scheme for Approximating Multiscale Diffusions Embedded in High Dimensions(2014) Crosskey, Miles MartinWhen simulating multiscale stochastic differential equations (SDEs) in high-dimensions, separation of timescales and high-dimensionality can make simulations expensive. The computational cost is dictated by microscale properties and interactions of many variables, while interesting behavior often occurs on the macroscale with few important degrees of freedom. For many problems bridging the gap between the microscale and macroscale by direct simulation is computationally infeasible, and one would like to learn a fast macroscale simulator. In this paper we present an unsupervised learning algorithm that uses short parallelizable microscale simulations to learn provably accurate macroscale SDE models. The learning algorithm takes as input: the microscale simulator, a local distance function, and a homogenization scale. The learned macroscale model can then be used for fast computation and storage of long simulations. I will discuss various examples, both low- and high-dimensional, as well as results about the accuracy of the fast simulators we construct, and its dependency on the number of short paths requested from the microscale simulator.

Item Open Access From Spectral Theorem to Spectral Statistics of Large Random Matrices with Spatio-Temporal Dependencies(2023) Naeem, Muhammad AbdullahHigh dimensional random dynamical systems are ubiquitous, including-but not limited to- cyber-physical systems, daily return on different stocks of S\&P 1500 and velocity profile of interacting particle systems around McKeanVlasov limit. Mathematically speaking, observed time series data can be captured via a stable $n-$ dimensional linear transformation `$A$' and additive randomness. System identification aims at extracting useful information about underlying dynamical system, given a length $N$ trajectory from it (corresponds to an $n \times N$ dimensional data matrix). We use spectral theorem for non-Hermitian operators to show that spatio-temperal correlations are dictated by the \emph{discrepancy between algebraic andgeometric multiplicity of distinct eigenvalues} corresponding to state transition matrix. Small discrepancies imply that original trajectory essentially comprises of multiple \emph{lower dimensional random dynamical systems living on $A$ invariant subspaces and are statistically independent of each other}. In the process, we provide first quantitative handle on decay rate of finite powers of state transition matrix $\|A^{k}\|$ . It is shown that when a stable dynamical system has only one distinct eigenvalue and discrepancy of $n-1$: $\|A\|$ has a dependence on $n$, resulting dynamics are \emph{spatially inseparable} and consequently there exist at least one row with covariates of typical size $\Theta\big(\sqrt{N-n+1}$ $e^{n}\big)$ i.e., even under stability assumption, covariates can \emph{suffer from curse of dimensionality }.

In the light of these findings we set the stage for non-asymptotic error analysis in estimation of state transition matrix $A$ via least squares regression on observed trajectory by showing that element-wise error is essentially a variant of well-know Littlewood-Offord problem and(can be extremely sensitive to dimension of the state space and number of iterations). We also show that largest singular value of the data matrix can be cursed by dimensionality even when state-transition matrix is stable. Overarching theme of this thesis is new theoretical results on spectral theorem for non-Hermitian operators, non-asymptotic behavior of high dimensional dynamical systems , which we incorporate with the work of Talagrand on concentration of measure phenomenon to better understand behavior of the structured random matrices(data matrix) and subsequently the performance of different learning algorithms with dependent data. Besides, we also show that there exists stable linear Gaussians with process level Talagrands' inequality linear in dimension of the state space(previously an open problem), along with deterioration of mixing times with increase in discrepancy between algebraic and geometric multiplicity of $A$.

Item Open Access Nonlinear Electroelastic Dynamical Systems for Inertial Power Generation(2011) Stanton, SamuelWithin the past decade, advances in small-scale electronics have reduced power consumption requirements such that mechanisms for harnessing ambient kinetic energy for self-sustenance are a viable technology. Such devices, known as energy harvesters, may enable self-sustaining wireless sensor networks for applications ranging from Tsunami warning detection to environmental monitoring to cost-effective structural health diagnostics in bridges and buildings. In particular, flexible electroelastic materials such as lead-zirconate-titanate (PZT) are sought after in designing such devices due to their superior efficiency in transforming mechanical energy into the electrical domain in comparison to induction methods. To date, however, material and dynamic nonlinearities within the most popular type of energy harvester, an electroelastically laminated cantilever beam, has received minimal attention in the literature despite being readily observed in laboratory experiments.

In the first part of this dissertation, an experimentally validated first-principles based modeling framework for quantitatively characterizing the intrinsic nonlinearities and moderately large amplitude response of a cantilevered electroelastic generator is developed. Nonlinear parameter identification is facilitated by an analytic solution for the generator's dynamic response alongside experimental data. The model is shown to accurately describe amplitude dependent frequency responses in both the mechanical and electrical domains and implications concerning the conventional approach to resonant generator design are discussed. Higher order elasticity and nonlinear damping are found to be critical for correctly modeling the harvester response while inclusion of a proof mass is shown to invigorate nonlinearities a much lower driving amplitudes in comparison to electroelastic harvesters without a tuning mass.

The second part of the dissertation concerns dynamical systems design to purposefully engage nonlinear phenomena in the mechanical domain. In particular, two devices, one exploiting hysteretic nonlinearities and the second featuring homoclinic bifurcation are investigated. Both devices exploit nonlinear magnet interactions with piezoelectric cantilever beams and a first principles modeling approach is applied throughout. The first device is designed such that both softening and hardening nonlinear resonance curves produces a broader response in comparison to the linear equivalent oscillator. The second device makes use of a supercritical pitchfork bifurcation wrought by nonlinear magnetic repelling forces to achieve a bistable electroelastic dynamical system. This system is also analytically modeled, numerically simulated, and experimentally realized to demonstrate enhanced capabilities and new challenges. In addition, a bifurcation parameter within the design is examined as a either a fixed or adaptable tuning mechanism for enhanced sensitivity to ambient excitation. Analytical methodologies to include the method of Harmonic Balance and Melnikov Theory are shown to provide superior insight into the complex dynamics of the bistable system in response to deterministic and stochastic excitation.