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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

Gspace, orthogonal Grepresentation, 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 noncompactness, compact operator,compact field, S^1equivariant 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

20090908T17:40:51Z
 Graduation date

200911
 Degree

Master of Science
 Degree level

Master's
 Abstract

We apply the socalled $S^1$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(tr_1),\ldots,x(tr_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 nonconstant 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 nontriviality 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
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 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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