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Permanent link (DOI): https://doi.org/10.7939/R3RX93K1R

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Hopf Bifurcation from Infinity in Asymptotically Linear Autonomous Systems with Delay Open Access

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Other title
Subject/Keyword
G-space, orthogonal G-representation, ODE, periodic function, locally uniformly asymptotically linear, asymptotic derivative at infinity, branch bifurcating from infinity, characteristic equation at infinity, characteristic root, isolated center at infinity, Sobolev space, Nemytsky operator, Hopf bifurcation from infinity, auxiliary function, Brouwer degree, measure of non-compactness, compact operator,compact field, S^1-equivariant degree, crossing numbers.
Type of item
Thesis
Degree grantor
University of Alberta
Author or creator
Biglands, Adrian
Supervisor and department
Wieslaw Krawcewicz (Mathematics)
Examining committee member and department
Dr. Wieslaw Krawcewicz, Mathematics
Dr. Volker Runde, Mathematics
Dr. Suneeta Vardarajan, Mathematics
Dr. Arno Berger, Mathematics
Dr. Lech Ozimek, Nutrition
Department
Department of Mathematical and Statistical Sciences
Specialization

Date accepted
2009-09-08T17:40:51Z
Graduation date
2009-11
Degree
Master of Science
Degree level
Master's
Abstract
We apply the so-called $S^1$-equivariant degree to study the occurrence of {\it Hopf bifurcations} in a system of nonlinear ordinary differential equations with delay of the form \begin{equation}\label{eqn:0.1} \dot x(t)=f(\alpha,x(t),x(t-r_1),\ldots,x(t-r_m)), \end{equation} where $f:\br\times\br^{(m+1)n}\to\br^n$ is a continuous, locally uniformly asymptotically linear map with respect to $x\in\br^n$. The Hopf bifurcation from infinity occurs in \eqref{eqn:0.1} when the parameter $\alpha$ crosses some ``critical'' value $\alpha_0$ resulting in a branch of large amplitude non-constant periodic solutions to \eqref{eqn:0.1}. In order to apply the $S^1$-equivariant degree theory, the problem of existence of periodic solutions to (\ref{eqn:0.1}) is reformulated as an $S^1$-fixed point problem $u=\mathcal F(u)$ in the appropriate functional space $W:=H^1(S^1;\br^n)$, and then a ``neighborhood'' $\Omega_\infty\subset\br^2\times W$ of the bifurcation point at infinity is constructed. Next, we apply an auxiliary function $\zeta:\overline{\Omega}_{\infty}\to S^1$ to construct $\mathfrak F_{\zeta}(\lambda,u)=(\zeta(\lambda,u),u-\mathcal F(\lambda,u))$, which subsequently allows us to define the {\it $S^1$-equivariant bifurcation invariant from infinity} $\omega(\lambda_o):=S^1\text{-Deg}(\mathfrak F_\zeta,\Omega_\infty)$ $\in A_1(S^1).$ The non-triviality of $\omega(\lambda_o)$ indicates the existence of a Hopf bifurcation from infinity for (\ref{eqn:0.1}).
Language
English
DOI
doi:10.7939/R3RX93K1R
Rights
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
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