Date of Award
Fall 2020
Degree Type
Open Access Dissertation
Degree Name
Mathematics, PhD
Program
Institute of Mathematical Sciences
Advisor/Supervisor/Committee Chair
Marina Chugunova
Dissertation or Thesis Committee Member
Asuman Aksoy
Dissertation or Thesis Committee Member
Ali Nadim
Terms of Use & License Information
This work is licensed under a Creative Commons Attribution-No Derivative Works 4.0 License.
Rights Information
© Copyright Casey Johnson, 2020 All rights reserved
Keywords
Differential Equations, HPA axis, Spectral Analysis, Stieltjes strings
Subject Categories
Applied Mathematics | Mathematics
Abstract
The spectrum of any differential equation or a system of differential equations is related to several important properties about the problem and its subsequent solution. So much information is held within the spectrum of a problem that there is an entire field devoted to it; spectral analysis. In this thesis, we perform spectral analysis on two separate complex dynamical systems. The vibrations along a continuous string or a string with beads on it are the governed by the continuous or discrete wave equation. We derive a small-vibrations model for multi-connected continuous strings that lie in a plane. We show that lateral vibrations of such strings can be decoupled from their in-plane vibrations. We then study the eigenvalue problem originating from the lateral vibrations. We show that, unlike the well-known one string vibrations case, the eigenvalues in a multi-string vibrating system do not have to be simple. Moreover we prove that the multiplicities of the eigenvalues depend on the symmetry of the model and on the total number of the connected strings [50]. We also apply Nevanlinna functions theory to characterize the spectra and to solve the inverse problem for a discrete multi-string system in a more general setting than it was done in [71],[73], [22], [69]-[72]. We also represent multi-string vibrating systems using a coupling of non-densely defined symmetric operators acting in the infinite dimensional Hilbert space. This coupling is defined by a special set of boundary operators acting in finite dimensional Krein space (the space with indefinite inner product). The main results of this research are published in [50]. The Hypothalamic Pituitary Adrenal (HPA) axis responds to physical and mental challenge to maintain homeostasis in part by controlling the body’s cortisol level. Dysregulation of the HPA axis is implicated in numerous stress-related diseases. For a structured model of the HPA axis that includes the glucocorticoid receptor but does not take into account the system response delay, we first perform rigorous stability analysis of all multi-parametric steady states and secondly, by construction of a Lyapunov functional, we prove nonlinear asymptotic stability for some of multi-parametric steady states. We then take into account the additional effects of the time delay parameter on the stability of the HPA axis system. Finally we prove the existence of periodic solutions for the HPA axis system. The main results of this research are published in [51].
ISBN
9798557034265
Recommended Citation
Johnson, Casey Lynn. (2020). Spectral Analysis of Complex Dynamical Systems. CGU Theses & Dissertations, 257. https://scholarship.claremont.edu/cgu_etd/257.
