Lower bounds for the chaotic zone near discrete Bogdanov-Takens bifurcation
by
Vassili Gelfreich
Mathematics Institute, University of Warwick

We consider a two-parametric analytic family of diffeomorphisms near the Bogdanov-Takens bifurcation. This bifurcation has exactly the same normal form as the classical one. In particular, the bifurcation diagram contains a line, which corresponds to a homoclinic bifurcation. In the discrete case, the bifurcation normal form generically diverges and a very narrow homoclinic zone is observed instead of the homoclinic line. This zone contains chaotic trajectories of the map and its width is known to be exponentially small compared to a natural parameter. We derive an asymptotic formula for the width of the homoclinic zone. An analytic invariant associated with a parabolic fixed point is an important ingredient of this formula.

Date received: May 9, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calh-21.