Dynamical Systems Commons^™

Stability Analysis Of Thermohaline Convection With A Time-Varying Shear Flow Using The Lyapunov Method, Kalin Kochnev

Honors Scholar Theses

This work applies the Lyapunov method to identify instabilities and compute the growth rate of a linear time-varying system. The linear system studied describes cold fresh water on top of hot salty water with a periodically time-varying background shear flow. A time-dependent weighting matrix is employed to construct a Lyapunov function candidate. The resulting linear matrix inequalities are discretized in time using the forward Euler method. As the number of temporal discretization points increases, the growth rate predicted by the Lyapunov method or Floquet theory, used for comparison, will converge to the same value obtained from numerical simulations. Furthermore, the …

Universal Systems Simulation Via Constraint Hypergraphs With Applications To Digital Twins, John Morris

All Dissertations

The characterization of systems encompasses a variety of modeling frameworks designed to capture specific behaviors and components of various system domains. Whatever the framework, the core elements of a system representation are the information of the system and a description of how that information is related. The relations in deterministic systems are functions, which, when composed to form executable processes, can be used to simulate system data. A declarative modeling framework is one that encodes mechanisms for preparing these simulations within the model structure, allowing an external agent to form the execution processes required for a given context. To date, …

Modeling Synaptic Dysfunction As Neural Contagion: A Graph-Based Sedr Framework For Simulating Signal Spread, Michelle Marfo, Dr. Padmanabhan Seshaiyer, Alonso Ogueda-Oliva

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Modeling The Cancer Cell Growth Predictions Based On Classical Mathematical Models With Physics-Informed Neural Network, Widodo Samyono

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Robust Spacecraft Autonomy For Deep Space Exploration In Special Euclidean Group Se(3), Matthew Wittal

Doctoral Dissertations and Master's Theses

Over the past half-century, humanity has gained extensive experience conducting manned spaceflight near Earth. Arguably, "near Earth" could even include the Moon — the most distant destination humans have reached. However, "near" in this work primarily refers low Earth orbit (LEO). One could argue that we have not truly left Earth since the Apollo, as spacecraft in some LEOs remain subject to atmospheric drag thus emphasizing their continued connection to Earth's immediate environment. Reflecting on this, it becomes clear that humanity has largely remained bound to Earth’s immediate vicinity since the Apollo missions reached the Moon. However, that is set …

The Computational Search For Unidentified Central Configurations Of The Newtonian N-Body Problem, Hannah G. Havel

CURE Proceedings

The N-body problem is a field of study in mathematics and physics that involves predicting the motion of particles moving under their mutual gravitational attraction. It is vital in celestial mechanics, such as planning collision-free satellite orbit trajectories. When beginning to understand the N-body problem, we can start by looking at equal masses of these particles or celestial bodies. As particles move, their position and velocity change, both energy and angular momentum are conserved. Sets of constant energy and angular momentum, known as integral manifolds, are higher-dimensional figures that represent constraints of movement to a system. Integral manifolds are described …

Reducing Uncertainty In Sea-Level Rise Prediction: A Spatial-Variability-Aware Approach, Subhankar Ghosh, Shuai An, Arun Sharma, Jayant Gupta, Shashi Shekhar, Aneesh Subramanian

I-GUIDE Forum

Given multi-model ensemble climate projections, the goal is to accurately and reliably predict future sea-level rise while lowering the uncertainty. This problem is important because sea-level rise affects millions of people in coastal communities and beyond due to climate change's impacts on polar ice sheets and the ocean. This problem is challenging due to spatial variability and unknowns such as possible tipping points (e.g., collapse of Greenland or West Antarctic ice-shelf), climate feedback loops (e.g., clouds, permafrost thawing), future policy decisions, and human actions. Most existing climate modeling approaches use the same set of weights globally, during either regression or …

Machine Learning-Based Data And Model Driven Bayesian Uncertanity Quantification Of Inverse Problems For Suspended Non-Structural System, Zhiyuan Qin

All Dissertations

Inverse problems involve extracting the internal structure of a physical system from noisy measurement data. In many fields, the Bayesian inference is used to address the ill-conditioned nature of the inverse problem by incorporating prior information through an initial distribution. In the nonparametric Bayesian framework, surrogate models such as Gaussian Processes or Deep Neural Networks are used as flexible and effective probabilistic modeling tools to overcome the high-dimensional curse and reduce computational costs. In practical systems and computer models, uncertainties can be addressed through parameter calibration, sensitivity analysis, and uncertainty quantification, leading to improved reliability and robustness of decision and …

Bifurcation Levels Of The Integral Manifolds Of The Newtonian N-Body Problem, Hannah G. Havel

CURE Proceedings

The N-body problem, first proposed by Isaac Newton, is a field of study in mathematics and physics that involves predicting the motion of particles moving under their mutual gravitational attraction. It has significance to many areas of science, including physics and computer science, and is crucial in understanding how the universe works. In fact, it was a primary motivation for Newton's development of calculus. An important application of the N-body problem is within celestial mechanics and involves how planets and other celestial bodies move with mutual gravitational attraction. It is important in developing how satellites behave in space using complicated …

Machine Learning To Predict Warhead Fragmentation In-Flight Behavior From Static Data, Katharine Larsen

Doctoral Dissertations and Master's Theses

Accurate characterization of fragment fly-out properties from high-speed warhead detonations is essential for estimation of collateral damage and lethality for a given weapon. Real warhead dynamic detonation tests are rare, costly, and often unrealizable with current technology, leaving fragmentation experiments limited to static arena tests and numerical simulations. Stereoscopic imaging techniques can now provide static arena tests with time-dependent tracks of individual fragments, each with characteristics such as fragment IDs and their respective position vector. Simulation methods can account for the dynamic case but can exclude relevant dynamics experienced in real-life warhead detonations. This research leverages machine learning methodologies to …

Lecture 08: Partial Eigen Decomposition Of Large Symmetric Matrices Via Thick-Restart Lanczos With Explicit External Deflation And Its Communication-Avoiding Variant, Zhaojun Bai

Mathematical Sciences Spring Lecture Series

There are continual and compelling needs for computing many eigenpairs of very large Hermitian matrix in physical simulations and data analysis. Though the Lanczos method is effective for computing a few eigenvalues, it can be expensive for computing a large number of eigenvalues. To improve the performance of the Lanczos method, in this talk, we will present a combination of explicit external deflation (EED) with an s-step variant of thick-restart Lanczos (s-step TRLan). The s-step Lanczos method can achieve an order of s reduction in data movement while the EED enables to compute eigenpairs in batches along with a number …

An Update On The Computational Theory Of Hamiltonian Period Functions, Bradley Joseph Klee

Graduate Theses and Dissertations

Lately, state-of-the-art calculation in both physics and mathematics has expanded to include the field of symbolic computing. The technical content of this dissertation centers on a few Creative Telescoping algorithms of our own design (Mathematica implementations are given as a supplement). These algorithms automate analysis of integral period functions at a level of difficulty and detail far beyond what is possible using only pencil and paper (unless, perhaps, you happen to have savant-level mental acuity). We can then optimize analysis in classical physics by using the algorithms to calculate Hamiltonian period functions as solutions to ordinary differential equations. The simple …

Period Estimation And Noise In A Neutrally Stable Stochastic Oscillator, Kevin R. Sanft, Ben F. M. Intoy

Spora: A Journal of Biomathematics

The periods of the orbits for the well-mixed cyclic three-species Lotka-Volterra model far away from the fixed point are studied. For finite system sizes, a discrete stochastic approach is employed and periods are found via wavelet analysis. As the system size is increased, a hierarchy of approximations ranging from Poisson noise to Gaussian noise to deterministic models are utilized. Based on the deterministic equations, a mathematical relationship between a conserved quantity of the model and the period of the population oscillations is found. Exploiting this property we then study the deterministic conserved quantity and period noise in finite size systems.

Introducing The Fractional Differentiation For Clinical Data-Justified Prostate Cancer Modelling Under Iad Therapy, Ozlem Ozturk Mizrak

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Physical Applications Of The Geometric Minimum Action Method, George L. Poppe Jr.

Dissertations, Theses, and Capstone Projects

This thesis extends the landscape of rare events problems solved on stochastic systems by means of the \textit{geometric minimum action method} (gMAM). These include partial differential equations (PDEs) such as the real Ginzburg-Landau equation (RGLE), the linear Schroedinger equation, along with various forms of the nonlinear Schroedinger equation (NLSE) including an application towards an ultra-short pulse mode-locked laser system (MLL).

Additionally we develop analytical tools that can be used alongside numerics to validate those solutions. This includes the use of instanton methods in deriving state transitions for the linear Schroedinger equation and the cubic diffusive NLSE.

These analytical solutions are …

Distributed Evolution Of Spiking Neuron Models On Apache Mahout For Time Series Analysis, Andrew Palumbo

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Fractal Analysis Of Dna Sequences, Christian G. Arias, Pedro Antonio Moreno Phd, Carlos Tellez

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Dynamics Of The Fitzhugh-Nagumo Neuron Model, Zechariah Thurman

Physics

In this paper, the dynamical behavior of the Fitzhugh-Nagumo model is examined. The relationship between neuron input current and the firing frequency of the neuron is characterized. Various coupling schemes are also examined, and their effects on the dynamics of the system is discussed. The phenomenon of stochastic resonance is studied for a single uncoupled Fitzhugh-Nagumo neuron.