Smooth foliation

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 ... 

Based on the above, we see that the eigenvectors of the matrix A give us the linear approximation to the stable and unstable manifolds of Mq. The coordinate system which is based on these eigenvectors for Mq changes smoothly to the coordinate system for the flow near Mg. Using this coordinate system, the flow near is expressed in the Fenichel normal form. Moreover, the Fenichel normal form shows the remarkable property that the flow on the stable and unstable manifolds of is foliated [7,26,27], as we now explain. 

Thus, the Fenichel normal form provides us with foliation of the stable and unstable manifolds of the NHIM Me. Moreover, their foliation smoothly depends on the parameter s from E = 0 to a small and positive locally near the NHIM. 

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In the special case when the direction of the final separation is along the unstable foliation of the chaotic advection the corresponding backward trajectories are convergent, S(t — r) lexp(—Ar). Therefore we should replace A with —A in (6.30) and, ignoring the second term since there is no saturation time, we obtain 5C(l) l. Thus, in the direction of the unstable foliation the concentration field is always smooth at every point in space. When b < A the existence of a smooth direction in an otherwise rough concentration field results... 

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. 

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 . 

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