The model is the three-dimensional vector field

with parameters
and
. this
system has two equilibria, one at the origin and one at *B =
*(*, 0, 0*).

We take
* = 3.2* and
* =
2*. For these parameter values, the point *B* is unstable
due to a Hopf bifurcation at *= * *= 2*. The attracting periodic orbit
that appears in this Hopf bifurcation becomes of saddle type due to a
period-doubling bifurcation. This means that one of its Floquet
multipliers moved through *-1* and the unstable manifold is,
therefore, non-orientable. In fact, also the stable manifold of this
periodic orbit is topologically a Möbius strip. The animations
below show the stable and unstable manifolds.

The two-dimensional unstable manifold of the equilibrium (, 0, 0) accumulates on the attracting period-doubled periodic orbit. An impression of the structure of this manifold can be obtained by considering a Poincaré section and rotating this section over 360 degrees.

Accumulation | Two stills showing the period-doubled attractor and the
unstable manifold of B accumulating on it. |

Heteroclinic intersection |
The unstable manifold of B forms a generic
heteroclinic intersection with the non-orientable stable manifold of
the periodic orbit. |

The animated gif shows the unstable manifold of
B = (,
0, 0) in a rotating Poincaré section. The attractor
appears as a pair of period-two points in the section. Note how the
Poincaré sections "glue" one point of the attractor to the
other after one rotation (215KB). |

Copyright © 1999 by Hinke Osinga

Last modified: Tue Oct 30 16:51:06 2001