Mobius manifold

The set of all points of the phase space whose trajectories converge to L as t —> +00 (—cx)) is called the stable (unstable) manifold of the periodic orbit. They are denoted by W and W , respectively. In the case where m = n, the attraction basin of L is Wf. In the saddle case, W is (m + l)-dimensional if m is the number of multipliers inside the unit circle, and is (p + 1)-dimensional where p is the number of multipliers outside of the unit circle, p = n — m — 1. In the three-dimensional Cctse, Wl and are homeomorphic either to two-dimensional cylinders if the multipliers are positive, or to the Mobius bands if the multipliers are negative, as illustrated in Fig. 7.5.1. In the general case, they are either multi-dimensional cylinders diffeomorphic to X S, or multi-dimensional Mobius manifolds. 

Fig 7.5.1. Saddle periodic orbit in R are distinguished by the topology of the stable and unstable invariant manifolds which may be homeomorphic to a cylinder (left) or a Mobius band (right). 

As for the original map (10.3.1) the fixed point O is asymptotically stable when Ik < 0 and is a saddle when Ik > 0. In the latter case the stable and unstable manifolds of O are the manifolds and, respectively. In terms of the Poincare map of the system of differential equations, the corresponding periodic trajectory L is stable when Ik < 0, or a saddle when Ik > 0. Note that in the saddle case the two-dimensional unstable manifold W L) is, in a neighborhood of the periodic trajectory, a Mobius band. 

If all Lyapunov values are equal to zero and the system is analytic, then the center manifold is also analytic, and all points on it, except O, are periodic of period two. This means that for the system of differential equations there exists a non-orientable center manifold which is a Mobius band with the cycle L as its median and which is filled in by the periodic orbits of periods close to the double period of L (see Fig. 10.3.2). 

Fig. 10.3.2. The center manifold of the primary periodic orbit L of an analytic system is a Mobius strip filled densely by periodic orbits of double period when all Lyapunov coefficients vanish.

The remarkable feature of this bifurcation in the case of periodic orbits of autonomous systems of differential equations is that the center manifold of the periodic orbit L corresponding to the fixed point O of the Poincare map is a Mobius band. The orbit itself is the mean line of the Mobius band, and consequently a new orbit that bifurcates from L must wind twice around L as shown in Fig. 11.4.5. It is quite clear that the period of the new orbit is nearly the double period of L. Consequently, this bifurcation is called... 

It follows from the form of the Poincare map that the invariant center manifold, corresponding to y = 0, is a Mobius band in this case, with the periodic trajectory as its median line. At e > 0, another, double-round periodic trajectory appears which inherits the stability and attracts all nearby trajectories, see Fig. 14.2.2. 

This is the same as Case 2 but with l > 0. The instability occurs because a period-two saddle periodic trajectory merges with a stable periodic orbit. When e > 0, the latter becomes a saddle so that its unstable manifold is homeomorphic to a Mobius band. 

Let us describe the essential bifurcations in this system on the path 6 = 2 as fjL increases. On the left of the curve AH, the equilibrium state 0 is stable. It undergoes the super-critical Andronov-Hopf bifurcation on the curve AH. The stable periodic orbit becomes a saddle through the period-doubling bifurcation that occurs on the curve PD. Figure C.6.7 shows the unstable manifold of the saddle periodic orbit homeomorphic to a Mobius band. As a increases further, the saddle periodic orbit becomes the homoclinic loop to the saddle point 0(0,0,0,) at a 5.545. What can one say about the multipliers of the periodic orbit as it gets closer do the loop Can the saddle periodic orbit shown in this figure get pulled apart from the double stable orbit after the fiip bifurcation In other words, in what ways are such paired orbits linked in in R ... 

In terms of the flow, this means that for the parameter values from an exponentially narrow region in the parameter space, which adjoins to the point A = 0 on HS from the side of A < 0, there exists a Lorenz-like attractor containing infinitely many saddle periodic orbits whose stable and unstable manifolds are homeomorphic to a Mobius band. 

---