Also: Manifolds of equilibria
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Manifolds of periodic orbits

The zeta^3 model is the three-dimensional vector field

with parameters alpha and beta . this system has two equilibria, one at the origin and one at B = ( <em>alpha</em> , 0, 0).

We take alpha = 3.2 and beta = 2. For these parameter values, the point B is unstable due to a Hopf bifurcation at alpha = beta = 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 animated gif shows the unstable manifold rotating about the z-axis, centered at B (2.8MB).

The animated gif shows how the stable manifold grows, while it rotates about the y-axis, centered at B (766KB).

Both manifolds rotating about the z-axis, centered at B (2.8MB).

The two-dimensional unstable manifold of the equilibrium ( alpha , 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 = ( alpha , 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).

Next: Twisted homoclinic bifurcation
Also: Manifolds of equilibria

	The animated gif shows the unstable manifold rotating about the z-axis, centered at B (2.8MB).
	The animated gif shows how the stable manifold grows, while it rotates about the y-axis, centered at B (766KB).
	Both manifolds rotating about the z-axis, centered at B (2.8MB).

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).