Strongly stable manifold

C) there exists no separatrix of a saddle-node which belongs to its strongly stable manifold, as shown in Fig. 8.1.3 ... 

Fig. 8.1.3. A nontransverse homociinic loop F to a saddle node. The separatrix enters the equilibrium state along its strongly stable manifold.

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 ... 

In the new variables the equation of the center manifold becomes y 0, and the equation of the strongly stable manifold becomes x = 0, The leaves of the strong stable foliation are the surfaces x = const. 

If the first non-zero Lyapunov value is negative and has an odd index number, i.e. < 0, fc = 2p+l, then the equilibrium state is stable. All trajectories tend to O as t -foo. Moreover, the trajectories which do lie on the strong stable manifold converge to O along as shown... 

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... 

R+ X S. The orbit L also has a strong-stable manifold which divides a neighborhood of L into a saddle region and a node region. When /x > 0, the saddle-node disappears and all orbits leave its small neighborhood. Note that the time necessary to pass through the neighborhood tends to infinity as /X —> -l-0. 

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 //. 

The non-smooth case appears, for example, when Wq touches the strong-stable manifold Wq, as shown in Fig. 12.2.2. The latter, in turn, may be detected via a small time-periodic perturbation of a system with an on-edge homoclinic loop to a saddle-node (see the previous section). Generically, the non-transversality of with respect to is also preserved under small smooth perturbations (say, if the tangency between and the corresponding leaf of is quadratic). 

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... 

Note that >1 > 0 by assumption, and that x > 0 because x is close to (the coordinate of the point = F n 5o) which is positive. Recall that x" " / 0 because F does not lie in the strong stable manifold by assumption. 

The heteroclinic cycles including the saddles whose unstable manifolds have different dimensions were first studied in [34, 35]. This study mostly focused on systems with complex dynamics. Let us, however, discuss here a case where the dynamics is simple. Let a three-dimensional infinitely smooth system have two equilibrium states 0 and O2 with real characteristic exponents, respectively, 7 > 0 > Ai > A2 and 772 > 1 > 0 > (i.e. the unstable manifold of 0 is onedimensional and the unstable manifold of O2 is two-dimensional). Suppose that the two-dimensional manifolds (Oi) and W 02) have a transverse intersection along a heteroclinic trajectory To (which lies neither in the corresponding strongly stable manifold, nor in the strongly unstable manifold). Suppose also that the one-dimensional unstable separatrix of Oi coincides with the one-dimensional stable separatrix of ( 2j so that a structurally unstable heteroclinic orbit F exists (Fig. 13.7.24). The additional non-degeneracy assumptions here are that the saddle values are non-zero and that the extended unstable manifold of Oi is transverse to the extended stable manifold of O2 at the points of the structurally unstable heteroclinic orbit F. 

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