Projects per year
Abstract
The goal of this paper is to establish the applicability of the Lyapunov–Schmidt reduction and the Centre Manifold Theorem (CMT) for a class of hyperbolic partial differential equation models with nonlocal interaction terms describing the aggregation dynamics of animals/cells in a one-dimensional domain with periodic boundary conditions. We show the Fredholm property for the linear operator obtained at a steady-state and from this establish the validity of Lyapunov–Schmidt reduction for steady-state bifurcations, Hopf bifurcations and mode interactions of steady-state and Hopf. Next, we show that the hypotheses of the CMT of Vanderbauwhede and Iooss (Center manifold theory in infinite dimensions. In: Jones, C., Kirchgraber, U., Walther, H.O. (eds.) Dynamics Reported, vol. 1, pp. 125–163. Springer, Berlin, 1992) hold for any type of local bifurcation near steady-state solutions with SO(2) and O(2) symmetry. To put our results in context, we review applications of hyperbolic partial differential equation models in physics and in biology. Moreover, we also survey recent results on Fredholm properties and Centre Manifold reduction for hyperbolic partial differential equations and equations with nonlocal terms.
Original language | English |
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Title of host publication | Mathematical sciences with multidisciplinary applications |
Subtitle of host publication | in honor of Professor Christiane Rousseau. And in recognition of the Mathematics for Planet Earth Initiative |
Editors | Bourama Toni |
Publisher | Springer International Publishing |
Pages | 29-59 |
Number of pages | 31 |
Volume | 157 |
ISBN (Electronic) | 9783319313238 |
ISBN (Print) | 9783319313214 |
DOIs | |
Publication status | Published - 2016 |
Publication series
Name | Springer proceedings in mathematics & statistics |
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Publisher | Springer International Publishing |
Volume | 157 |
ISSN (Print) | 2194-1009 |
Keywords
- Hyperbolic PDE
- Animal aggregation
- Centre Manifold reduction
- Lyapunov–Schmidt reduction
- Symmetry Fredholm property
Fingerprint
Dive into the research topics of 'Lyapunov-Schmidt and Centre Manifold reduction methods for nonlocal PDEs modelling animal aggregations'. Together they form a unique fingerprint.Projects
- 1 Finished
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Mathematical Investigation into the Role of Cell-cell Communication Pathways on Collective Cell Migration (First Grant Scheme)
Eftimie, R. (Investigator)
Engineering and Physical Sciences Research Council
1/11/13 → 31/10/15
Project: Research
Profiles
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Eftimie, Raluca
- Science and Engineering Office - Honorary Professor
- Mathematics - Associate Staff
Person: Associate Staff, Honorary