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Boundary crisis in
quasiperiodically forced systems


This is a joint project with
Ulrike Feudel, University of Potsdam (Germany). We study boundary
crisis in quasiperiodically forced systems using the Hénon map
as a characteristic example. The quasiperiodically forced Hénon
map is defined as
with
^{1} and u, v
. We consider the twoparameter family
of maps in (A, b)space where
is a fixed quasiperiodic
rotation in , and c = 0.1.
A boundary crisis is the sudden disappearance of a chaotic attractor
as it hits an unstable (periodic) set on its basin boundary. For
example, if we take A = 0.2 the attractor becomes chaotic
approximately for b = 0.896 and runs into an invariant circle
of saddle type approximately for b = 1.334.

An animated gif shows how the attractor for A = 0.2 changes
as the parameter b increases from 0.5 to
1.35. The attractor hits the unstable invariant circle for
b in between 1.33 and 1.34 (31KB).



Rotation about the uaxis shows how the attractor is about to
touch its basin boundary. The boundary crisis is induced by the
homoclinic tangency of the stable and unstable manifolds of the circle
on the basin boundary. The parameters are A = 0.2 and
b = 1.33 in this animation (943KB).

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Copyright © 1998 by
Hinke Osinga
Last modified: Fri Sep 8 10:47:37 2000