Non-linear Dynamics Commons^™

Study Of Behaviour Change And Impact On Infectious Disease Dynamics By Mathematical Models, Tianyu Cheng

Electronic Thesis and Dissertation Repository

This thesis uses mathematical models to study human behaviour changes' effects on infectious disease transmission dynamics. It centers on two main topics. The first concerns how behaviour response evolves during epidemics and the effects of adaptive precaution behaviour on epidemics. The second topic is how to build general framework models incorporating human behaviour response in epidemiological modelling.

In the first project, based on the fact that a fraction of the epidemiologically susceptible population is actually susceptible due to precautions, we present a novel perspective on understanding the infection force, incorporating human protection behaviours. This view explains many existing infection force …

Hopf Bifurcation Analysis Of Chaotic Chemical Reactor Model, Daniel Mandragona

Honors Undergraduate Theses

Bifurcations in Huang's chaotic chemical reactor system leading from simple dynamics into chaotic regimes are considered. Following the linear stability analysis, the periodic orbit resulting from a Hopf bifurcation of any of the six fixed points is constructed analytically by the method of multiple scales across successively slower time scales, and its stability is then determined by the resulting final secularity condition. Furthermore, we run numerical simulations of our chemical reactor at a particular fixed point of interest, alongside a set of parameter values that forces our system to undergo Hopf bifurcation. These numerical simulations then verify our analysis of …

On The Existence Of Bogdanov-Takens Bifurcations, Zachary Deskin

MSU Graduate Theses

In bifurcation theory, there is a theorem (called Sotomayor's Theorem) which proves the existence of one of three possible bifurcations of a given system, provided that certain conditions of the system are satisfied. It turns out that there is a "similar" theorem for proving the existence of what is referred to as a Bogdanov-Takens bifurcation. The author is only aware of one reference that has the proof of this theorem. However, most of the details were left out of the proof. The contribution of this thesis is to provide the details of the proof on the existence of Bogdanov-Takens bifurcations.

A Study Of The Effect Of Harvesting On A Discrete System With Two Competing Species, Rebecca G. Clark

Theses and Dissertations

This is a study of the effect of harvesting on a system with two competing species. The system is a Ricker-type model that extends the work done by Luis, Elaydi, and Oliveira to include the effect of harvesting on the system. We look at the uniform bound of the system as well as the isoclines and perform a stability analysis of the equilibrium points. We also look at the effects of harvesting on the stability of the system by looking at the bifurcation of the system with respect to harvesting.

Border-Collision Bifurcations Of Cardiac Calcium Cycling, Jacob Michael Kahle

Masters Theses

In this thesis, we study the nonlinear dynamics of calcium cycling within a cardiac cell. We develop piecewise smooth mapping models to describe intracellular calcium cycling in cardiac myocyte. Then, border-collision bifurcations that arise in these piecewise maps are investigated. These studies are carried out using both one-dimensional and two-dimensional models. Studies in this work lead to interesting insights on the stability of cardiac dynamics, suggesting possible mechanisms for cardiac alternans. Alternans is the precursor of sudden cardiac arrests, a leading cause of death in the United States.

Bifurcation Of Limit Cycles In Smooth And Non-Smooth Dynamical Systems With Normal Form Computation, Yun Tian

Electronic Thesis and Dissertation Repository

This thesis contains two parts. In the first part, we investigate bifurcation of limit cycles around a singular point in planar cubic systems and quadratic switching systems. For planar cubic systems, we study cubic perturbations of a quadratic Hamiltonian system and obtain 10 small-amplitude limit cycles bifurcating from an elementary center, for which up to 5th-order Melnikov functions are used. Moreover, we prove the existence of 12 small-amplitude limit cycles around a singular point in a cubic system by computing focus values. For quadratic switching system, we develop a recursive algorithm for computing Lyapunov constants. With this efficient algorithm, we …