Invariant center local manifold

We have outlined how the conceptual tools provided by geometric TST can be generalized to deterministically or stochastically driven systems. The center-piece of the construction is the TS trajectory, which plays the role of the saddle point in the autonomous setting. It carries invariant manifolds and a TST dividing surface, which thus become time-dependent themselves. Nevertheless, their functions remain the same as in autonomous TST there is a TST dividing surface that is locally free of recrossings and thus satisfies the fundamental requirement of TST. In addition, invariant manifolds separate reactive from nonreactive trajectories, and their knowledge enables one to predict the fate of a trajectory a priori. 

As we have discussed in Section 2 in classical mechanics the transition state is represented by a lower dimensional invariant subsystem, the center manifold. In the quantum world, due to Heisenberg s uncertainty principle, we cannot localize quantum states entirely on the center manifold, so fhere cannot be any invariant quantum subsystem representing the transition states. Instead we expect a finite lifetime for fhe fransition state. The lifetime of fhe fransifion sfate is determined by the Gamov-Siegert resonances, whose importance in the theory of reacfion rafes has been emphasized in fhe liferafure [61, 62]. 

The key methods in our presentation of local bifurcations are based on the center manifold theorem and on the invariant foliation technique (see Sec. 5.1. of Part I). The assumption that there are no characteristic exponents to the right of the imaginary axis (or no multipliers outside the unit circle) allows us to conduct a smooth reduction of the system to a very convenient standard form. We use this reduction throughout this book both in the study of local bifurcations on the stability boundaries themselves and in the study of global bifurcations on the route over the stability boundaries (Chap. 12).These... 

Starting with any (x, y), a trajectory of system (12.4.8) converges typically to an attractor of the fast system corresponding to the chosen value of x. This attractor may be a stable equilibrium, or a stable periodic orbit, or of a less trivial structure — we do not explore this last possibility here. When an equilibrium state or a periodic orbit of the fast system is structurally stable, it depends smoothly on x. Thus, we obtain smooth attractive invariant manifolds of system (12.4.8) equilibrium states of the fast system form curves Meq and the periodic orbits form two-dimensional cylinders Mpo, as shown in Fig. 12.4.6. Locally, near each structurally stable fast equilibrium point, or periodic orbit, such a manifold is a center manifold with respect to system (12.4.8). Since the center manifold exists in any nearby system (see Chap. 5), it follows that the smooth attractive invariant manifolds Meqfe) and Mpo( ) exist for all small e in the system (12.4.7) [48]. 

In the original proof the system under consideration was assumed to be analytic. Later on, other simplified proofs have been proposed which are based on a reduction to a non-local center manifold near the separatrix loop (such a center manifold is, generically, 3-dimensional if the stable characteristic exponent Ai is real, and 4-dimensional if Ai = AJ is complex) and on a smooth linearization of the reduced system near the equilibrium state (see [120, 147]). The existence of the smooth invariant manifold of low dimension is important here because it effectively reduces the dimension of the problem. ... 

In addition to the symmetry assumption, we will also suppose that the linear part of the system near the origin O restricted to the invariant plane z = 0 has a complete Jordan block. Then, the system in the restriction to the center manifold may locally be written in the form... 

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