Center manifold

Because in an autonomous system many of the invariant manifolds that are found in the linear approximation do not remain intact in the presence of nonlinearities, one should expect the same in the time-dependent case. In particular, the separation of the bath modes will not persist but will give way to irregular dynamics within the center manifold. At the same time, one can hope to separate the reactive mode from the bath modes and in this way to find the recrossing-free dividing surfaces and the separatrices that are of importance to TST. As was shown in Ref. 40, this separation can indeed be achieved through a generalization of the normal form procedure that was used earlier to treat autonomous systems [34]. 

For time-dependent Hamiltonian systems we chose in Section IVB to use a normal form that decouples the reactive mode from the bath modes, but does not attempt a decoupling of the bath modes. This procedure is always safe, but in many cases it will be overly cautious. If it is relaxed, the dynamics within the center manifold is also transformed into a (suitably defined) normal form. This opens the possibility to study the dynamics within the TS itself, as has been done in the autonomous case, for example in Ref. 107. One can then try to identify structures in the TS that promote or inhibit the transport from the reactant to the product side. 

The center manifold approach of Mercer and Roberts (see the article Mercer and Roberts, 1990 and the subsequent article by Rosencrans, 1997) allowed to calculate approximations at any order for the original Taylor s model. Even if the error estimate was not obtained, it gives a very plausible argument for the validity of the effective model. This approach was applied to reactive flows in the article by Balakotaiah and Chang (1995). A number of effective models for different Damkohler numbers were obtained. Some generalizations to reactive flows through porous media are in Mauri (1991) and the preliminary results on their mathematical justification are in Allaire and Raphael (2007). 

In the article Balakotaiah and Chang (1995) the surface reactions are much faster and do not correspond to our problem. In order to compare two approaches we will present in the paragraph from Section 3.4 computations with our technique for the timescale chosen in Balakotaiah and Chang (1995) and we will see that one gets identical results. This shows that our approach through the anisotropic singular perturbation reproduces exactly the results obtained using the center manifold technique. 

The goal of this subsection is to compare our approach with the center manifold technique from Balakotaiah and Chang (1995). We study the 2D variant of the model from Balakotaiah and Chang (1995, pp. 58-61), and we keep the molecular diffusion. Then the corresponding analog of the problem (6)-(10), with K— +co, is... 

A more sophisticated form of reduction is obtained when the so-called center manifold theorem is invoked. This says essentially that a subspace of lower dimension than the whole state-space gives a true representation of the essential features of the system and one that can be built on to give a yet more accurate picture. We shall not attempt to go into this here to see the method in action, the reader cannot do better than to read C. Chang and... 

V. Balakotaiah s treatment of Dispersion in Chemical Solutes in Chromatographs and Reactors 4 Carr s Applications of Center Manifold Theory5 is the standard text. 

Carr, J., Applications of Center Manifold Theory . Springer, Berlin (1981). 

Moreover, the intersection of the center manifold with an energy shell yields an NHIM. The NHIM, which is a (2n — 3)-dimensional hypersphere, is the higher-dimensional analog of Pechukas PODS. Because this manifold is normally hyperbolic, it will possess stable and unstable manifolds. These manifolds are the 2n — 2)-dimensional analogs of the separatrices. The NHIM is the edge of the TS, which is a (2n — 2)-dimensional hemisphere. 

Center Sector. Corresponding to the n 1 pairs of imaginary eigenvalues, it is possible to build a center manifold spanned by the corresponding eigenvectors, of dimension 2n — 2. Its intersection with the energy level 4>( ) is precisely the NHIM. 

It must be underlined that the central manifold theorem, extending the linear center manifold into the nonlinear regime, is way less powerful than its stable/ unstable counterpart. There is no limit t —> oo and even no unicity of nonlinear center manifolds. Consequently, it is not well known how this whole beautiful stmcture bifurcates and disappears as E > E. There has been virtually no study of the bifurcation stmcture (see, however, Ref. 55), and the transition from threshold behavior to far-above-threshold behavior is an open question, as far as I am aware. 

Transport. We need now to construct the NHIM, its stable/unstable manifolds, and the center manifold. Let P be the main relative equilibrium point. The first task is to find the short periodic orbits lying above P in energy. These p.o. are unstable. We did so by exploring phase space at energies 4, 10, and 14 cm above E (1 atomic unit = 2.194746 x 10 cm ). It is not possible to go much higher in E, since the center manifold disappears shortly above E + 14cm , because of the structure of the potential energy surface. 

Figure 23. The center manifold of P, for 7 = 8. The thick lines are p. o., and the thin lines serve as a guide for the eyes to visualize the center manifold.

Note This problem can also be solved by a method called center manifold theory, as explained in Wiggins (1990) and Guckenheimer and Holmes (1983).)... 

Moreover, under the assumption of pure point critical spectrum, a center manifold description of the associated meandering, drifting, and resonance phenomena has been achieved see [22], [73], and the survey [21]. We give a brief exposition in section 3.2.1. Specific calculations, however, require disproportionate computational effort to account for the two-dimensional time dependent problem in large (theoretically unbounded) domains. 

Center manifold theory has become an indispensable tool for the study of ODEs. An equilibrium is called hyperbolic if the linearization of the vector field at that equilibrium does not possess spectrum on the imaginary axis. The local dynamics of ODEs near a hyperbolic equilibrium is determined by that linearization. In particular there exist stable and un-... 

Center manifold theory extends to many infinite-dimensional systems, like certain partial differential equations (PDFs). Center manifold reductions can be obtained locally or globally. For local center manifolds of parabolic PDFs see Vanderbauwhede and looss [78]. Dimension reductions via global center manifolds for spatially inhomogeneous planar media have been achieved by Jangle [27, 33] more details will be presented below. 

Instead, we show below how pinning and drifting motions may coexist, within the same center manifold of the same underlying system and at the same parameter values. In fact an ever so slight variation of initial conditions may kick the solution from drifting to pinning mode, or back. The respective initial conditions for these behaviors will be interwoven in a Cantor-like structure. 

Spectral analysis of the linearized semiflow along a periodic solution is called Floquet theory. The eigenvalues p of the linearized period map are called Floquet multipliers. A Floquet exponent is a complex / such that exp(/3r) is a Floquet multiplier of the system, where t denotes the minimal period. A periodic solution is hyperbolic if, and only if, it possesses only the trivial Floquet multiplier p = 1 on the unit circle, and this multiplier has algebraic multiplicity one. Otherwise it is called non-hyperbolic. In ODEs hyperbolic periodic solutions possess stable and unstable manifolds, similarly to the case of hyperbolic equilibria. Non-hyperbolic periodic solutions possess center manifolds. 

The symmetry of equation (3.1), which causes the non-hyperbolic eigenvalue zero to be triply degenerate, gives rise to a three-dimensional invariant manifold of (pt u ). This is the center manifold which coincides with the group orbit of u, namely SE(2)u = pgU, g 6 SE 2). Thus the center manifold of u, is simply the set of all translations and rotations of the initial rotating wave u. See [72] for this result and some generalizations. 

The main mathematical feature of the above invariant center manifold SE 2)u is its normal hyperbolicity. An invariant manifold is called normally hyperbolic if the linearized dynamics in the normal directions are of faster exponential rate than those in tangential direction. Normally hyperbolic invariant manifolds persist under small perturbations [30]. 

We now state the center manifold theorem of the perturbed system ... 

We briefly comment on the basic steps in the proof of theorem 1. For homogeneous lighting of intermediate strength a rigidly rotating spiral wave solution was assumed to be given by u . The manifold is close to the unperturbed normally hyperbolic center manifold SE(2)u given by the... 

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