Numerical Analysis of equivariant evolution equations
11-117 Wolf-Jürgen Beyn, Thorsten Hüls.
Continuation and collapse of homoclinic tangles
By a classical theorem transversal
homoclinic points of maps lead to shift dynamics on a maximal
invariant set, also referred to as a homoclinic tangle. In this
paper we study the fate of homoclinic tangles in parameterized
systems from the viewpoint of numerical continuation and
bifurcation theory. The bifurcation result shows that the
maximal invariant set near a homoclinic tangency, where two
homoclinic tangles collide, can be characterized by a system of
bifurcation equations that is indexed by a symbolic sequence.
For the H\'enon family we investigate in detail the bifurcation
structure of multi-humped orbits originating from several
tangencies. The homoclinic network found by numerical
continuation is explained by combining our bifurcation result
with graph-theoretical arguments.
