Invariant center manifold

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]. 

Theorem 9.3. If all Lyapunov values are equal to zero, then the associated analytic system has an analytic invariant (center) manifold which is filled with closed trajectories around the origin, as shown in Fig. 9.3.3. On the center manifold the system has a holomorphic integral of the type... 

From (10.2.5)-(10.2.7) we can conclude that the analytic curve y = (t> x) consists entirely of fixed points and, therefore, is an invariant (center) manifold. 

It follows from the form of the Poincare map that the invariant center manifold, corresponding to y = 0, is a Mobius band in this case, with the periodic trajectory as its median line. At e > 0, another, double-round periodic trajectory appears which inherits the stability and attracts all nearby trajectories, see Fig. 14.2.2. 

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]. 

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. 

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]. 

Center Manifolds. The Center Manifold Theorem (see Carr (1981)) states that all branches of stationary and periodic states in a neighborhood of a bifurcation point are embedded in a sub-manifold of the extended phase space X M that is invariant with respect to the flow generated by the ODE (2.1). All trajectories starting on this so-called center manifold remain on it for all times. All trajectories starting from outside of it exponentially converge towards the center manifold. Specifically, static bifurcations are embedded in a two dimensional center manifold, whereas center manifolds for Hopf bifurcations are three dimensional. Figures 2.1 and 2.2 summarize the geometric properties of the flows inside a center manifold in the case of saddle-node and Hopf bifurcations, respectively. 

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... 

As shown in Chap. 5, the above critical equilibrium state lies in an invariant C -smooth center manifold defined by an equation of the form y = (a ), where (x) vanishes at the origin along with its first derivative. 

Transversely to the center manifold, another invariant manifold passes through the point 0(0,0). It is called strongly stable and, as usual, we denote it by Its equation is given by a = y), where (y) vanishes at... 

In a neighborhood of the point O there exists a -smooth change of variables which straightens both the invariant foliation and the center manifold so that the system in the new variables assumes the following standard form... 

It follows from formula (9.2.9) that if the right-hand side of the system (9.2.6) is analytic, and if all Lyapunov values vanish, then g[x,ip x)) = 0. Hence, since y = (p x) is the solution of the system (9.2.7), it follows that the curve y = ip x) is filled out by the equilibrium states of the system (9.2.6). Thus, it is an invariant manifold of this system. Since it is tangent to y = 0 at O, it is the center manifold by definition. It follows that for the case imder consideration, the system has an analytic center manifold W y = (p(x) which consists of equilibrium states as illustrated in Fig. 9.2.5. 

To resume we note that it may often happen in practice that the equation on the center manifold is such that the governing parameters do not come in the generic way. For example, if a system is invariant with respect to the symmetry X —— X, then the equation of the center manifold admits the same symmetry. 

The above theorem is related to the map on the center manifold. Reconstructing the behavior of trajectories of the original map (11.6.2) is relatively simple. Here, if L < 0, then the fixed point is stable when /i < 0. When /i > 0 it becomes a saddle-focus with an m-dimensional stable manifold (defined by T = 0) and with a two-dimensional unstable manifold which consists of a part of the plane y = 0 bounded by the stable invariant curve C,... 

Here, the center manifold is defined by the equation y = 0. The surfaces x = constant are the leaves of the strong-stable invariant foliation In particular, x = 0 is the equation of the strong-stable manifold of O. At fi — Oj the function g (nonlinear part of the map on W ) has a strict extremum at X = 0. For more definiteness, we assume that it is a minimum, i.e. y(x, 0) > 0 when X 0. Thus, the saddle region on the cross-section corresponds to x > 0, and the node region corresponds to x < 0. Since the saddle-node disappears when /Lt > 0, it follows that y(x,/x) > 0 for all sufficiently small x and for all small positive //. 

Next, let us straighten the strong stable invariant foliation. The leaves of the foliation are given by x Q y], x p), (p = constant where x is the coordinate of intersection of a leaf with the center manifold Q is a C -function (it is C -smooth with respect to y). The straightening is achieved via a coordinate transformation Xh- which brings the invariant foliation to the form x = constant,

[Pg.286]

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. ... 

The bifurcation of a periodic orbit with three multipliers +1. On the center manifold we introduce the coordinates (x y z jp) where is the angular coordinate and (x, y, z) are the normal coordinates (see Sec. 3.10), Assuming that the system is invariant under the transformation x,y) —> (—X, — y), the normal form truncated up to second order terms is given by... 

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... 

The geometric version of TST laid out in Section II is centered around the NHIM that defines the dividing surface and its stable and unstable manifolds that act as separatrices. The NHIMs at different energies are in turn organized by the saddle point. It forms a fixed point of the dynamics—that is it is itself an invariant object—and it provides the Archimedean point in which the geometric phase-space structure is anchored. 

From a geometric point of view, the autonomous fixed point is the organizing center for the hierarchy of invariant manifolds. From a technical point of view, it is also the expansion center around which all Taylor series expansions are carried out. If the TS trajectory is to take over the role of the fixed point, this observation suggests that it be used as a time-dependent coordinate origin. We therefore introduce the relative coordinates... 

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. 

First we should determine the characteristic exponents at the origin. It is easy to see that there is a pair of zero exponents and one equal to —1. The eigenspace corresponding to the zero pair is given by z = 0. The center invariant manifold, tangent to this plane at the origin, is written as... 

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