Applications Of Functional-Analytic Methods In Nonlocal And Dynamical System Problems
2020-05
Loading...
View/Download File
Persistent link to this item
Statistics
View StatisticsJournal Title
Journal ISSN
Volume Title
Title
Applications Of Functional-Analytic Methods In Nonlocal And Dynamical System Problems
Authors
Published Date
2020-05
Publisher
Type
Thesis or Dissertation
Abstract
This paper is concerned with functional analytic techniques in problems related to dynamical systems and contains two parts. In the first part we show bifurcation of localized spike solutions from spatially constant states in systems of nonlocally coupled equations in the whole space. The main assumptions are a generic bifurcation of saddle-node or transcritical type for spatially constant profiles, and a symmetry and second moment condition on the convolution kernel. Rather than relying on center manifolds methods, we pursue a more direct approach, deriving leading order asymptotics and Newton corrections for error terms. We are able to extend well-known results for spots, spikes, and fronts in locally coupled systems on the real line, and for radially symmetric profiles in higher space dimensions. In the second part, we revisit the classical problem of determining the asymptotic expansion of the solution near the passage of a fold point in a singularly perturbed system, where the theory of normally hyperbolic invariant manifold cannot be directly applied. While the standard remedy is the blow-up method first demonstrated by Krupa and Szmolyan, we will show how one can use a functional analytic approach to achieve the same goal.
Description
University of Minnesota Ph.D. dissertation.May 2020. Major: Mathematics. Advisors: Arnd Scheel, Daniel Spirn. 1 computer file (PDF); iv, 90 pages.
Related to
Replaces
License
Collections
Series/Report Number
Funding information
Isbn identifier
Doi identifier
Previously Published Citation
Suggested citation
Tao, Tianyu. (2020). Applications Of Functional-Analytic Methods In Nonlocal And Dynamical System Problems. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/215093.
Content distributed via the University Digital Conservancy may be subject to additional license and use restrictions applied by the depositor. By using these files, users agree to the Terms of Use. Materials in the UDC may contain content that is disturbing and/or harmful. For more information, please see our statement on harmful content in digital repositories.