Dynamical Systems Commons^™

The Global Phase Space Of The Three-Vortex Interaction System And Its Application To Vortex-Dipole Scattering, Atul Anurag

Dissertations

This dissertation presents a global reduction of the classical three-vortex problem that is free from coordinate singularities, enabling a comprehensive analysis of the system's dynamics across all circulation regimes.

To achieve this, a two-step symplectic reduction procedure is developed. The first step introduces Jacobi coordinates adapted to the symplectic structure of the vortex system, and the second applies a Lie-Poisson reduction to the resulting system. This formulation eliminates the non-physical singularities associated with collinear vortex configurations and facilitates a global phase space analysis, including a detailed and novel investigation of bifurcations.

Within this reduced framework, all relative fixed points are …

Bright Light Therapy And Depression: Assessing Suitability Using Entrainment Maps, Charles A. Mainwaring

Theses

Bright Light Therapy has been shown to be efficacious to mood disorders including Major Depression. Researchers use the Jewett-Forger-Kronauer model of the circadian rhythm with the Unified Model of melatonin including a mathematical term implementing feedback from the melatonin system into the circadian system to quantify the effects of bright light. Early investigations into intrinsic period, light sensitivity, and the circadian pacemaker's sensitivity to blood melatonin concentration may be indicators of subsets of patients with long intrinsic periods exhibiting symptoms of depression.

Stationary Probability Distributions Of Stochastic Gradient Descent And The Success And Failure Of The Diffusion Approximation, William Joseph Mccann

Theses

In this thesis, Stochastic Gradient Descent (SGD), an optimization method originally popular due to its computational efficiency, is analyzed using Markov chain methods. We compute both numerically, and in some cases analytically, the stationary probability distributions (invariant measures) for the SGD Markov operator over all step sizes or learning rates. The stationary probability distributions provide insight into how the long-time behavior of SGD samples the objective function minimum.

A key focus of this thesis is to provide a systematic study in one dimension comparing the exact SGD stationary distributions to the Fokker-Planck diffusion approximation equations —which are commonly used in …