Global unstable manifold

M. Dellnitz and A. Hohmann. A subdivision algorithm for the computation of unstable manifolds and global attractors. Numerische Mathematik 75 (1997) 293-317... 

The mechanism of these transitions is nontrivial and has been discussed in detail elsewhere Q, 12) it involves the development of a homoclinic tangencv and subsequently of a homoclinic tangle between the stable and unstable manifolds of the saddle-type periodic solution S. This tangle is accompanied by nontrivial dynamics (chaotic transients, large multiplicity of solutions etc.). It is impossible to locate and analyze these phenomena without computing the unstable, saddle-tvpe periodic frequency locked solution as well as its stable and unstable manifolds. It is precisely the interactions of such manifolds that are termed global bifurcations and cause in this case the loss of the quasiperiodic solution. 

The local and global stable unstable manifold theorems (see, e.g.. Ref. 24, pp. 136-140) tell us the following are the (un)stable manifolds) ... 

In reaction processes for which there is no local equihbrium within the potential well, global aspects of the phase space structure become crucial. This is the topic treated in the contribution of Toda. This work stresses the consequences of a variety of intersections between the stable and unstable manifolds of NHIMs in systems with many degrees of freedom. In particular, tangency of intersections is a feature newly recognized in the phase space structure. It is a manifestation of the multidimensionality of the system, where reaction paths form a network with branches. 

In the next section, these local stability considerations will be shown to lead to corresponding global results. For this analysis, it will be important to approximate the one-dimensional unstable manifold of Ei when both El and "2 exist and (3.10) holds. To this end, we provide information on an eigenvector corresponding to the eigenvalue A] of f. Let x = (xi,Qi,X2,02) denote such an eigenvector. We find that... 

A global property function is usually expressed as the expectation value of an operator or as the derivative of such an expectation value with respect to an internal or external parameter of the system. In the Born-Oppenheimer approximation, the electronic wave function depends parametrically upon the coordinates of the n nuclei, and therefore a set of the 3 -6 linearly independent nuclear coordinates constitutes the natural variables for such a choice of the potential function. However, the manifold M on which the gradient vector field is bound can be defined on a subset of 1R provided q < 3n-6, for example the intrinsic reaction coordinate (unstable manifold of a saddle point of index 1 of the Born-Oppenheimer energy hypersurface) or the reduced reaction coordinate. 

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 general setting of the problem of global bifurcations on the disappearance of a saddle-node periodic orbit is as follows. Assume that there exists a saddle-node periodic orbit and that all trajectories which tend to this periodic orbit as i — 00 also tend to it as -f-oo along some center manifold. In other words, assume that the unstable manifold of the saddle-node returns to the saddle-node orbit from the side of the node region. In this case, either ... 

If the system has a global cross-section (which always exists when we treat a periodically forced autonomous system), the unstable manifold will only be a torus. The intersection of with the cross-section is a closed curve which is invariant under the Poincare map. Consequently, the following two cases are possible ... 

When a system does not have a global cross-section, the unstable manifold of the saddle-node may also be a Klein bottle (if the system is defined in... 

Theorem 12.3. (Afraimovichr-Shilnikov [3, 6]) If the global unstable set of the saddle-node L is a smooth compact manifold a torus or a Klein bottle) at fi = Oy then a smooth closed attractive invariant manifold 7 (fl torus or a Klein bottle, respectively) exists for all small fi. 

Let us now consider the case where the global unstable set of the saddle-node periodic orbit L is not a manifold, but has the structure like shown in Fig. 12.4.1. This means that the integer m which determines the homotopy class of the curve fl jSq in the cross-section 5q x = —d is... 

Since the manifold Mq is a NHIM, it changes continuously, under a small perturbation, into a new NHIM M - Moreover, the separatrix Wq changes, continuously and locally near M, into the stable manifold and the unstable one W" of the NHIM M. Note, however, that, in general, and W no longer coincide with each other to form a single manifold globally. Then, the Lie transformation method brings the total Hamiltonian H x.I, 0) into the Fenichel normal form locally near the manifold M. ... 

In the limit - 0, y(T) changes much more rapidly than x(t) Except near Q = 0 the vector field (x,y) is everywhere nearly horizontal. The two falling sections of the one-dimensional manifold Q 0 are stable, but the middle section is unstable. (We referred to this fact earlier.) For 0 < 6 < 6 and 6 < 6 < /e find that the steady state is globally asymptotically stable (as -> 0). However, under these conditions the system is excitable in the sense described in Chapter IV (pp. 76f) For 6q < 6 < 6 we find an orbitally as3nmptoti-cally stable periodic solution illustrated in Fig. 4. 

---