Global stable manifold

The bursting dynamics ends in a different type of process, referred to as a global (or homoclinic) bifurcation. In the interval of coexisting stable solutions, the stable manifold of (or the inset to) the saddle point defines the boundary of the basins of attraction for the stable node and limit cycle solutions. (The basin of attraction for a stable solution represents the set of initial conditions from which trajectories asymptotically approach the solution. The stable manifold to the saddle point is the set of points from which the trajectories go to the saddle point). When the limit cycle for increasing values of S hits its basin of attraction, it ceases to exist, and... 

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) ... 

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. ... 

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. 

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 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. 

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. 

A totally diflFerent situation becomes possible in the case where the system does not have a global cross-section, and when is not a manifold. In this case (Sec. 12.4), the disappearance of the saddle-node periodic orbit may, under some additional conditions, give birth to another (unique and stable) periodic orbit. When this periodic orbit approaches the stability boundary, both its length and period increases to infinity. This phenomenon is called a hlue-sky catastrophe. Since no physical model is presently known for which this bifurcation occurs, we illustrate it by a number of natural examples. 

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