Strongly unstable manifold

If A > 0, then there is a strongly unstable manifold which divides the neighborhood of O into a node region where all trajectories diverge from O, and a saddle region where there is a single stable separatrix entering O as t 4-00 and the other trajectories bypass O. 

W, Recall (Secs. 2.6 and 2.7) that the unstable manifold of a saddle has a special subset — an invariant smooth strong unstable manifold It is... 

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. 

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. 

Structure is interpreted based on the concept of free energy, that is, the concept of equihbrium statistical mechanics. However, if the folding processes take place far from equilibrium, the concept of equilibrium statistical mechanics cannot be applied. On the other hand, from our point of view, the funnel-like structure would give a typical example where the strong dynamical correlation exists. If we can analyze how stable and unstable manifolds intersect in the folding processes, we can have an alternative explanation concerning the funnel, which is a future project from our approach. 

Fig. 12.2.2. A nontransverse tangency of the unstable and strong-stable manifolds of a saddle-node fixed point may be obtained by a small time-periodic perturbation of the system with an on-edge homoclinic loop to a saddle-node equilibrium state, as shown in Fig. 12.1.4.

Theorem C.4. The stable manifold of an unstable, hyperbolic rest point of a monotone dynamical system cannot contain two points that are related by the strict inequality stable manifold cannot contain two distinct points that are related by stable manifold is unordered. 

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