Abstract/Details

Evans function analysis of the stability of periodic travelling wave solutions associated with the Fitzhugh -Nagumo system


1999 1999

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Abstract (summary)

The difficulty in the problem of determining stability of periodic travelling waves under the most general class of bounded continuous perturbations resides in the fact that the spectrum associated with the linearized operator is composed of bands, or more generally in the non-adjoint case, of “loops”, rather than discrete spectrum as is the case when considering the more restricted case of periodic perturbations. In particular, we are interested in systems exhibiting a singular perturbation structure. We carry out this program for the celebrated Fitz-Hugh Nagumo system, [special characters omitted] where &epsis; is a small parameter and f( u) = u(1 − u)(u a). This equation is a simplified version of the Hodgkin-Huxley equations that model nerve pulse transmission through neurons. It is well known that there is a one-parameter family of periodic solutions for all 0 < &epsis; < &epsis;0, &epsis;<sub>0</sub> small enough. In this work, the precise structure of the continuous spectrum of the linearized operator about these nonlinear waves is obtained (as computed with respect to the space [special characters omitted]) It will be established that this spectral set lies on the (open) left-half plane with the only exception of the eigenvalue at the origin, which is present due to the translation invariance of the waves. From this precise structure of the spectral set, nonlinear stability with respect long-wavelength periodic perturbations will follow.

The technical contributions of this work lie on three different fronts: (1) Matched asymptotic analysis that allow for the computation of the spectrum of the linearized operator about the nonlinear waves near the origin. (2) An extension of the Elephant-Trunk Lemma as studied in connection to the stability analysis of the pulse solutions associated with the Gray-Scott model. (3) The discovery of pole/zero cancellations in the factorization of the Evans Function.

Indexing (details)


Subject
Mechanics;
Electrical engineering
Classification
0346: Mechanics
0544: Electrical engineering
Identifier / keyword
Applied sciences, Evans function, Fitzhugh-Nagumo, Nonlinear waves, Stability, Travelling wave solutions
Title
Evans function analysis of the stability of periodic travelling wave solutions associated with the Fitzhugh -Nagumo system
Author
Eszter, Edgardo Gabriel
Number of pages
157
Publication year
1999
Degree date
1999
School code
0118
Source
DAI-B 60/11, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
9780599530058, 0599530057
Advisor
Gardner, Robert
University/institution
University of Massachusetts Amherst
University location
United States -- Massachusetts
Degree
Ph.D.
Source type
Dissertations & Theses
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
9950150
ProQuest document ID
304515444
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
http://search.proquest.com/docview/304515444
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