Bifurcation analysis of an orientational aggregation model

 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. Received: 23 April 2001 / Revised version: 29 October 2002 / Published online: 28 February 2003 Key words or phrases: Actin – Cytoskeleton – Orientational Aggregation – Bifurcation Analysis – Mode Interaction – Power Series Expansion