Smooth attractive invariant curve

Figures 13.4.6 Ap > 0) and 13.4.7 Ap < 0) show how the image of 5 by the map T moves when the loop is split. In any case, since the map T is contracting in the a-variables and expanding in y, it follows that it has a smooth attracting invariant curve lo p), transverse to yo = 0 in 5q.

Starting with any (x, y), a trajectory of system (12.4.8) converges typically to an attractor of the fast system corresponding to the chosen value of x. This attractor may be a stable equilibrium, or a stable periodic orbit, or of a less trivial structure — we do not explore this last possibility here. When an equilibrium state or a periodic orbit of the fast system is structurally stable, it depends smoothly on x. Thus, we obtain smooth attractive invariant manifolds of system (12.4.8) equilibrium states of the fast system form curves Meq and the periodic orbits form two-dimensional cylinders Mpo, as shown in Fig. 12.4.6. Locally, near each structurally stable fast equilibrium point, or periodic orbit, such a manifold is a center manifold with respect to system (12.4.8). Since the center manifold exists in any nearby system (see Chap. 5), it follows that the smooth attractive invariant manifolds Meqfe) and Mpo( ) exist for all small e in the system (12.4.7) [48]. 

If the first Lyapunov value Li > 0, then the fixed point of the map (11.6.6) is unstable for sufficiently small /x > 0. When /x < 0 the fixed point is stable its attraction basin is the inner domain of an unstable smooth invariant curve of the form (11.6.7). As p —0, the curve collapses into the fixed point see Fig. 11.6.2). 

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