Strong-stable foliation

In the new variables the equation of the center manifold becomes y 0, and the equation of the strongly stable manifold becomes x = 0, The leaves of the strong stable foliation are the surfaces x = const. 

If m = 1, then is a manifold. It is homeomorphic to a torus if m 1 and to a Klein bottle if m = — 1. As already mentioned, this manifold may be smooth or non-smooth, depending on whether intersects the strong-stable foliation transversely everywhere or not. When x and (p are... 

Recall that in the smooth case, the manifold intersects the strong-stable foliation transversely, and each leaf has only one point of intersection with In the generic non-smooth case, some of the leaves have one-sided tangencies to Therefore, there must be leaves in the node region where each leaf has several intersections with Wf . 

Fig. 12.2.4. Geometrical interpretation of the big lobe condition — each leaf of the strong stable foliation F must cut through not fewer than two points.

Moreover, it follows from Theorems 12.5 and 12.6 that chaotic behavior may also be possible if the condition /o(< ) < 1 of Theorem 12.9 is not met. In particular, the big lobe condition (Sec. 12.2) is here equivalent to the existence of a leaf of the strong-stable foliation which intersects at least two connected components of the intersection of with the local cross-section S (p = constant (see Fig. 12.4.2). In terms of the essential map this condition is written as... 

The strongly stable manifold is one of the leaves of a -smooth foliation which is transverse to the center manifold. As we have shown in Chap. 5 the following reduction theorem holds ... 

Here, the center manifold is defined by the equation y = 0. The surfaces x = constant are the leaves of the strong-stable invariant foliation In particular, x = 0 is the equation of the strong-stable manifold of O. At fi — Oj the function g (nonlinear part of the map on W ) has a strict extremum at X = 0. For more definiteness, we assume that it is a minimum, i.e. y(x, 0) > 0 when X 0. Thus, the saddle region on the cross-section corresponds to x > 0, and the node region corresponds to x < 0. Since the saddle-node disappears when /Lt > 0, it follows that y(x,/x) > 0 for all sufficiently small x and for all small positive //. 

Next, let us straighten the strong stable invariant foliation. The leaves of the foliation are given by x Q y], x p), (p = constant where x is the coordinate of intersection of a leaf with the center manifold Q is a C -function (it is C -smooth with respect to y). The straightening is achieved via a coordinate transformation Xh- which brings the invariant foliation to the form x = constant,

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