Strong stable invariant foliation

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,

[Pg.286]

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