Stable/unstable manifolds connections

The connection between the observation in Fig. 5 and the observation in Figs. 6b and 7b is unknown here. Now we elucidate this connection. Figure 6b for (Z, ) = (1,1) shows one branch of Wtcm c) and the triple collision orbits on the Poincare section. A remarkable point is that the triple collision curves tc and accumulate at 10 points on 0. As shown in Fig. 9a, these points are the points at which if tcm c) and if TCM d) cross the plane x = 0 and which are, of course, just on 0. We denote these points by PTCM,r=o- It is clear that the number of points of Ptcm,t=o is related to the existence of tori in the Poincare section 3>. If the tori exist, its outer most torus has periodic points. These periodic points have the stable and unstable manifolds. Branches of these stable and unstable manifolds run toward precisely if tcm c) and if TCM d) on 0. This situation was observed in Fig. 8. Therefore, the number of the points of Ptcm,t=o is related to the existence of the tori. At the same time, the number of the points of PrcM,r=Q just corresponds to the winding number xF of f tcm c) or ifTcmid) around the body of the TCM as mentioned in the previous subsection. In Figs. 7b and 9b, the case of (Z, = (1,7) is shown. As... 

Second in our strategy, we ask how the dynamics near NHIMs are connected with each other. Here, intersections between the stable and unstable manifolds of the NHIMs play a major role. 

Suppose that the unstable manifold of a NHIM intersects with the stable manifold of another NHIM (or the same NHIM) such intersections are called heteroclinic (or homoclinic). This means that there exists a path that connects these two NHIMs (or a path that leaves from and comes back to the NHIM). Thus, their intersections offer the information on how the NHIMs are connected. 

We regard the intersections between stable and unstable manifolds as a skeleton of reaction paths. The skeleton has the structure of a network, since one NHIM can be connected with multiple NHIMs. Then, branching in the skeleton will manifest itself as tangency [1]. This observation suggests the importance of tangency in multidimensional reaction dynamics. 

Moreover, the NHIM with a saddle with index 1 can be connected with NHIMs with saddles with indexes larger than 1. To see this possibility, let us count the dimension of the intersections. Suppose we have a saddle with index L. Then, the NHIM of 2N — 2L dimension exists with (2N — L)-dimensional stable and unstable manifolds. In the equi-energy surface, the dimension of the NHIM is 2N — 2L—1, and that of its stable and unstable manifolds is 2N — L — 1. Thus, the dimension of the intersection, if any, between its stable manifold and the unstable manifold of the NHIM with a saddle with index 1 is 2N — L — 2.lf its value is larger than 0, a path exists which connects these two NHIMs. Therefore, the allowed values of L for systems of 3 degrees of freedom (for example) are 1 and 2, when we also take into account the condition that 2N — 2L—1 (i.e., the dimension of the NHIM with a saddle with index L in the equi-energy surface) should not be negative. 

Here, we mention only two possibilities, though we could have other cases. The hrst is that the condition of normal hyperbolicity breaks down for some NHIMs. Then, what happens to those NHIMs Do they bifurcate into other NHIMs, or do they disappear at all The second possibility is that intersections between the stable and unstable manifolds of NHIMs change into tangency. This could lead to bifurcation in the way NHIMs are connected by their stable and unstable manifolds. 

NHIM Ma connected with two other NHIMs, Mi, and Me- There, the unstable manifold W of the NHIM Ma has intersections with the stable manifolds and WJ of the NHIMs Mb and Me, respectively. As initial conditions continuously vary from PI through P2, P3, and P4 to P5 on 1T , its intersections with the stable manifolds change as follows. The transverse intersection with Wl for PI first changes into tangency with Wl at Tab for P2, then no intersection for P3, tangency with Tac for P4, and finally transverse intersection with We for P5. Thus, we expect that whenever branching exists, tangency occurs as initial conditions continuously vary on the unstable manifolds. 

In the previous section, we explained the bifurcation on NHIMs which would result from breakdown of normal hyperbolicity. Here, we speculate on bifurcation in the connections among NHIMs. In Fig. 33, we schematically display how the connections among NHIMs would change. As parameters of the system vary, transverse intersections between the unstable manifold VT of a NHIM Ma and the stable manifold W of a NHIM Mb [see Fig. 33(i)] disappear. Instead, transverse intersections between the unstable manifold W of the NHIM Ma and the stable manifold of another NHIM M [see Fig. 33(ii)] appear. 

In order to make more direct correspondence between tangency and global changes in the dynamical behavior, we propose to use different methods to characterize chaos. The first one focuses attention on how normally hyperbolic invariant manifolds are connected with each other by their stable and unstable manifolds. Then, crisis would lead to a transition in their connections. The second one is to characterize chaos based on how unstable manifolds are folded as they approach normally hyperbolic invariant manifolds. Then, crisis would manifest itself as a change in their folding patterns. Let us explain these ideas in more detail. 

In Sections IV and V, we discussed these two processes from the viewpoint of chaos. In these discussions, the following points are of importance. As for the barrier crossing, intersection (either homoclinic or heteroclinic) between stable and unstable manifolds offers global information on the reaction paths. It not only includes how transition states are connected with each other, but also reveals how reaction paths bifurcate. Here, the concepts of normally hyperbolic invariant manifolds and crisis are essential. With regard to IVR, the concept of Arnold web is crucial. Then, we suggest that coarse-grained features of the Arnold webs should be studied. In particular, the hierarchy of the web and whether the web is uniformly dense or not would play an important role. 

The basins of attraction of the coexisting CA (strange attractor) and SC are shown in the Fig. 14 for the Poincare crosssection oyf = O.67t(mod27t) in the absence of noise [169]. The value of the maximal Lyapunov exponent for the CA is 0.0449. The presence of the control function effectively doubles the dimension of the phase space (compare (35) and (37)) and changes its geometry. In the extended phase space the attractor is connected to the basin of attraction of the stable limit cycle via an unstable invariant manifold. It is precisely the complexity of the structure of the phase space of the auxiliary Hamiltonian system (37) near the nonhyperbolic attractor that makes it difficult to solve the energy-optimal control problem. 

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