Circle diffeomorphisms

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. 

Recall that a saddle-node fixed point or periodic orbit has one multiplier equal to +1 and the rest of the multipliers lies inside the unit circle. The diffeomorphism (the Poincare map) near the fixed point may be represented... 

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