Continuation and Collapse of Homoclinic Tangles

Date: 
Thursday, March 6, 2014 - 2:30pm
Location: 
MSB 111
Speaker: 
W.-J. Beyn (University of Bielefeld and Miller Scholar)

By classical results of Birkhoff, Smale, and Shilnikov, transversal homoclinic points of a map lead to chaotic shift dynamics on a maximal invariant set, also referred to as a homoclinic tangle. In our work we use numerical analysis and bifurcation theory to analyze the fate of two homoclinic tangles which collide at a homoclinic tangency in a parameterized dynamical system. The main bifurcation result shows that the maximal invariant set near a homoclinic tangency can be characterized by a set of bifurcation equations that is indexed by a symbolic sequence. For the Henon family we investigate in detail the bifurcation structure of multi-humped homoclinic orbits originating from several tangencies. The emerging homoclinic network is explained by combining our bifurcation result with graph-theoretical arguments. This is joint work with Thorsten Huels.

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