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Journal of Mathematical Biology

, Volume 46, Issue 6, pp 537–563 | Cite as

Bifurcation analysis of an orientational aggregation model

  • Edith Geigant
  • Michael Stoll

Abstract.

 We consider an integro-differential equation for the evolution of a function f on the circle, describing an orientational aggregation process. In the first part we analyze generic bifurcations of steady-state solutions when a single eigenvalue changes sign. Lyapunov-Schmidt reduction leads to the bifurcation equation which is solved explicitly by formal power series. We prove that these series have positive radius of convergence. Two examples exhibit forward and backward bifurcations, respectively. In the second part we assume that the first and second eigenvalues become positive. Again we use Lyapunov-Schmidt reduction to arrive at the reduced bifurcation system from which we get the bifurcating branches as power series. We calculate the two most important parameters of the reduced system for two examples; one of them has interesting mode interactions which lead to various kinds of time-periodic solutions.

Keywords

Power Series Aggregation Process Mode Interaction Formal Power Series Bifurcation Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Edith Geigant
    • 1
  • Michael Stoll
    • 2
  1. 1.Abteilung Theoretische Biologie, Universität Bonn, Kirschallee 1, 53 115 Bonn, Germany. e-mail: edith.geigant@uni-bonn.deDE
  2. 2.International University Bremen, P.O.Box 750561, 28725 Bremen, Germany. e-mail: m.stoll@iu-bremen.deDE

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