Strongly stable invariant manifold

In the new coordinates, the strong-stable invariant manifold Wff is the surface x = 0 the node region C/ now corresponds to small negative x and the saddle region t/ corresponds to small positive x. 

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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Figure 3.25(a) shows a case where a strong correlation exists between processes of crossing two neighboring barriers. This correlation results from direct intersection between the stable and unstable manifolds of the normally hyperbolic invariant manifolds located on the tops of neighboring barriers. Then, some of the orbits starting from one of the normally hyperbolic invariant manifolds directly reach the other one without falling into the potential well. Some other orbits may fall into the well and would take some time to reach the other manifold. The ratio of these two types of orbits depends on how steep the intersection is. 

Transversely to the center manifold, another invariant manifold passes through the point 0(0,0). It is called strongly stable and, as usual, we denote it by Its equation is given by a = y), where (y) vanishes at... 

We can now describe the behavior of trajectories in a small neighborhood of the periodic trajectory L to which the fixed point O of the Poincare map corresponds. In the two-dimensional case the behavior of trajectories is shown in Fig. 10.2.4, and a higher-dimensional case in Fig. 10.2.5. The invariant strongly stable manifold Wff (the imion of the trajectories which start from the points of Wq on the cross-section) partitions a neighborhood of L into a node and a saddle region. In the node region all trajectories wind towards L... 

When the separatrix returns to O, it lies in the stable manifold y = 0. If the system has order more then three, we will assume that F does not belong to the strong stable manifold Recall that is a smooth invariant... 

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