Smooth invariant foliation

The key methods in our presentation of local bifurcations are based on the center manifold theorem and on the invariant foliation technique (see Sec. 5.1. of Part I). The assumption that there are no characteristic exponents to the right of the imaginary axis (or no multipliers outside the unit circle) allows us to conduct a smooth reduction of the system to a very convenient standard form. We use this reduction throughout this book both in the study of local bifurcations on the stability boundaries themselves and in the study of global bifurcations on the route over the stability boundaries (Chap. 12).These... 

In a neighborhood of the point O there exists a -smooth change of variables which straightens both the invariant foliation and the center manifold so that the system in the new variables assumes the following standard form... 

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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Already we have stated that the invariant foliation is C -smooth moreover, it can be shown for the case of the saddle-node that the foliation is, in fact, C -smooth everywhere except on Wff at p 0 [140]. The coordinate transformation that reduces (12.2.4) to (12.2.7) has the same smoothness. 

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