Local unstable manifold

By construction, it is obtained as follows apply the map T to the intersection line of the local unstable manifold fl 17+ with the cross-section... 

On the cross-section 5o, the local unstable manifold of the fixed point M (/x) is a small piece of the invariant curve Zo(/x) through this point. The entire imstable manifold of M (/x) on Sq can be obtained by iterating the local unstable manifold under the action of the map T. Since the domain of the map T is bounded by the surface yo = 0 = fl So, the unstable... 

What cannot be obtained through local bifurcation analysis however, is that both sides of the one-dimensional unstable manifold of a saddle-type unstable bimodal standing wave connect with the 7C-shift of the standing wave vice versa. This explains the pulsating wave it winds around a homoclinic loop consisting of the bimodal unstable standing waves and their one-dimensional unstable manifolds that connect them with each other. It is remarkable that this connection is a persistent homoclinic loop i.e. it exists for an entire interval in parameter space (131. It is possible to show that such a loop exists, based on the... 

Proposition 1. For e sufficiently small the periodic orbit 7(t) of (7) survives as aperiodic orbit 7 (t) = rj(t)+0(e), of (10) having the same stability type as 7(t), and depending on e in a C2 manner. Moreover, the local stable and unstable manifolds Wlsoc( ye(t)) and Wfoc e(t)) of 7 (t) remain also e-close to the local stable and unstable manifolds WfoMt)) and Wfioc( (t)) of ft), respectively. 

Remark 1. The concept for local stable and unstable manifolds becomes clear when one represents the stable and unstable manifolds of the hyperbolic fixed point (periodic orbit) locally. For details see (Wiggins, 1989) or (Wiggins, 1988). 

Remark 7.8. In order to compute the Poincare polynomial, we do not actually need the holomorphic symplectic form on T E. The fact that it is locally isomorphic to T C is enough. Hence the above argument holds and shows Gottsche s formula also for the case of the total space of a holomorphic line bundle over E, not necessarily T E. The only difference is that the unstable manifold W becomes Lagrangian in the case of T E. 

Wiggins et al. [22] pointed out that one can always locally transform a Hamiltonian to the form of Eq. (1.38) if there exists a certain type of saddle point. Examination of the associated Hamilton s equations of motion shows that q = Pn = 0 is a fixed point that defines an invariant manifold of dimension 2n — 2. This manifold intersects with the energy surface, creating a (2m — 3)-dimensional invariant manifold. The latter invariant manifold of dimension 2m — 3 is an excellent example of an NHIM. More interesting, in this case the stable and unstable manifolds of the NHIM, denoted by W and W ,... 

It is well known (see, e.g.. Ref. 13) that the normal form transformations do not converge in the sense that normalization to all orders generally does not yield a meaningful result. However, this is of no consequence for our purposes. We view the technique more as the input to a numerical method for realizing the NHIM, its stable and unstable manifolds, and the TS. In this sense the limitations of machine precision make normalization beyond a certain finite order meaningless. This is a local result valid in the neighborhood of the equilibrium point of center center saddle type. However, once the phase-space structure is established locally, it can be numerically continued outside of the local region. 

This is also the Hamiltonian of the activated complex. We will encounter it in Eq. (23) with the customary symbol H. Regardless of its stability properties or the size of the nonlinearity, Eq. (12) is always an invariant manifold. However, we are interested in the case when it is of the saddle type with stable and unstable manifolds. If the physical Hamiltonian is of the form of Eq. (1), then a preliminary, local transformation is not required. The manifold (12) is invariant regardless of the size of the nonlinearity. Moreover, it is also of saddle type with respect to stability in the transverse directions. This can be seen by examining Eq. (1). On qn = Pn = 0 the transverse directions, (i.e., q and p ), are still of saddle type (more precisely, they grow and decay exponentially). 

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) ... 

Cylindrical Manifolds. There is one big advantage of looking at 2-DOF TS in phase space It puts emphasis on the existence of the tubes that determine the transport of classical probability in phase space. Existence of those tubes has been known for a long time [48]. These tubes are the set of trajectories that constitute the stable/unstable manifolds of PODS. Locally, in the vicinity of P, they immediatly generalize to higher dimensions. They are constructed as follows ... 

In constructing the stable manifold of the NHIM we follow, backward in time, the normal directions of the NHIM with negative local Lyapunov exponents. For the unstable manifold we follow forward in time the normal directions of the NHIM with positive local Lyapunov exponents. 

Suppose we have a saddle with index 1. Then, a NHIM of 2N — 2 dimension exists above it in the phase space, with two directions that are normal to it. Along these normal directions, with negative and positive Lyapunov exponents, 2N — 1)-dimensional stable and unstable manifolds exist, respectively. The normal directions of the saddle correspond to the degree of freedom that is the reaction coordinate near the saddle, and they describe how the reaction proceeds locally near the NHIM. 

These NHIMs are of the largest dimension in the phase space. In the 2N — 1)-dimensional equi-energy surface, the dimension of these NHIMs is 2N — 3, and that of their stable and unstable manifolds is 2N — 2. Therefore, their stable and unstable manifolds separate the equi-energy surface locally into two regions. This separation corresponds to separating the equi-energy surface locally into the reactant and product sides. 

For the case of normal hyperbolicity, the theorem proved by Fenichel and independently by Hirsch et al. guarantees the following For a small and positive E, there exists a NHIM with stable and unstable manifolds, and VF , respectively. The NHIM varies smoothly with respect to the parameter . Moreover, and W also vary smoothly with respect to the parameter s at least locally near the NHIM Mg-. 

Thus, the Fenichel normal form provides us with foliation of the stable and unstable manifolds of the NHIM Me. Moreover, their foliation smoothly depends on the parameter s from E = 0 to a small and positive locally near the NHIM. 

Until now we have discussed local aspects of the dynamics near NHIMs with saddles with index 1. In the second stage of our strategy, we are interested in how the dynamics near these NHIMs are connected with each other. The information on this feature of the dynamics is offered by the intersections between the stable and unstable manifolds of NHIMs. 

In our study, NHIMs and their stable and unstable manifolds, at least locally near the NHIM, smoothly depend on parameters of the system. This is guaranteed by the theorem of Fenichel and Hirsch et al. Moreover, transverse intersections between stable and unstable manifolds are structurally stable. This is because their existence and characteristics do not change as the parameters vary by sufficiently small amounts. Thus, aU of the features except possibly for tangency are structurally stable. 

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. 

The behavior of orbits of P near a fixed point x can be described in the case where x is a hyperbolic fixed point, that is, when no eigenvalue (multiplier) of the Jacobian of P at x has modulus equal to 1. In this case there exist (local) stable and unstable manifolds M (x) and M (x) (respectively) containing the point x which are tangent to the stable (resp. unstable) subspace of the Jacobian of P at x. (The stable (unstable) subspace... 

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... 

The stable and unstable sets correspond to the stable and unstable manifolds introduced for rest points and periodic orbits in Chapter 1. Unfortunately, if the attractors are more complex than rest points or periodic orbits, the question of the existence of stable and unstable manifolds becomes a difficult topological problem. In the applications that follow, these more complicated attractors do not appear, so one can simply deal with the stable manifold theorem. The Butler-McGehee lemma (used in Chapter 1) played a critical role in the first uses of persistence. The following lemma is a generalization of this work. It can be found (with slightly different hypotheses) in [BW], [DRS], and [HaW]. (In particular, the local compactness is not needed if a stronger condition - asymptotic smoothness - is placed on the semidynamical system.)... 

Figure 12.2.3 suggests that the Henon attractor is Cantor-like in the transverse direction, but smooth in the longitudinal direction. There s a reason for this. The attractor is closely related to a locally smooth object—the unstable manifold of a saddle point that sits on the edge of the attractor. To be more precise, Benedicks and Carleson (1991) have proven that the attractor is the closure of a branch of the unstable manifold see also Simo (1979). 

Figure 7.4 Snapshots of the spatial distribution for the autocatalytic model (7.1) in the open blinking vortex-sink flow at time intervals equal to the flow period for a supercritical Damkohler number, Da = 7.0. Note, that after a transient time a time-periodic asymptotic state is reached where the autocatalytic growth, localized on the fractal unstable manifold, is balanced by the loss of product due to the outflow from the mixing region, in this case through the point sinks.

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