Center stable manifold

Armed as we are now with the KAM theorem, the Center Manifold theorem, and the Stable Manifold theorem, we can begin to visualize the phase space of reaction dynamics. Returning to our original system (see Uncoupled Reaction Dynamics in Two Degrees of Freedom ), we now realize that the periodic orbit that sews together the half-tori to make up the separatrix is a hyperbolic periodic orbit, and it is not a fixed point of reflection. From our previous visualization of uncoupled phase-space dynamics, we know that the separatrix is completely nontwisted. In the terminology of Poliak and Pechukas, the hyperbolic periodic orbit is a repulsive PODS. ... 

Kelley, A. (1967). The stable, center-stable, center, center-unstable and unstable manifolds. J. Diff. Eqs., 3, 546-70. 

The strongly stable manifold is one of the leaves of a -smooth foliation which is transverse to the center manifold. As we have shown in Chap. 5 the following reduction theorem holds ... 

In the new variables the equation of the center manifold becomes y 0, and the equation of the strongly stable manifold becomes x = 0, The leaves of the strong stable foliation are the surfaces x = const. 

If the first non-zero Lyapunov value is positive and if all non-critical characteristic exponents (71,...,7n) lie to the left of the imaginary axis in the complex plane, then the equilibrium state is a complex saddle-focus, as shown in Fig. 9.3.2(b). Its stable manifold is and the unstable manifold coincides with the center manifold W, The trajectories lying neither in nor pass nearby the equilibrium state. 

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 //. 

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. 

Thus, we have found that the mechanisms of escape from a nonhyperbolic attractor and a quasihyperbolic (Lorenz) attractor are quite different, and that the prehistory of the escape trajectories reflects the different structure of their chaotic attractors. The escape process for the nonhyperbolic attractor is realized via several steps, which include transitions between low-period saddle-cycles coexisting in the system phase space. The escape from the Lorenz attractor consist of two qualitatively different stages the first is defined by the stable and unstable manifolds of the saddle center point, and lies on the attractor the second is the escape itself, crossing the saddle boundary cycle surrounding the stable point attractor. Finally, we should like to point out that our main results were obtained via an experimental definition of optimal paths, confirming our experimental approach as a powerful instrument for investigating noise-induced escape from complex attractors. 

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. 

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. 

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. 

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-... 

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. 

For the problem under consideration, i.e. Eq. (11), the nonlinear function is odd and v = H(u,n) is at least quadratic in u and n. Thus, Eq. (12) restricted to the center manifold will have contribution from the stable equations of the order ( u ), k > 3, and can be neglected in the first approximation. The above equation can be further simplified either by method of averaging or method of normal forms. It may be noted that the averaged and normal form equations can also be obtained directly from Eq. (11) without employing the center manifold theorem as indicated in Sri Namachchivaya and Chow and Mallet-Paret. ... 

Let X be a set of parameter values for which a solution of eqs. (2), referred to as reference state, loses its stability and gives rise to new branches o7 sol uti ons by a bifurcation mechanism. We want to see how the solution of the master equation, eq. (1), behaves under these conditions, and how this behavior depends on small changes of the parameters X around The answer to this question depends on the kind of bifurcation considered, on the nature of the reference state, and on the number of variables involved in the dynamics. The simplest case is, by far, the pitchfork bifurcation occurring as a first transition from a previously stable spatially uniform stationary state. This transition is characterized by a remarkable universality. First, whatever the number of variables present initially, it is always possible to cast the stochastic dynamics in terms of a single, "critical" variable. This is the probabilistic analog of adiabatic elimination or, in more modern terms, of the center manifold theorem [4,8-10]. Second, the stationary probability distribution of the critical variable can be cast in the form (we set 6X = (p-Xg, rstands for the spatial coordinates) ... 

In the general case where there are both stable and unstable characteristic exponents, or stable and unstable multipliers in the spectrum, the local bifurcation problem does not cause any special difficulties, thanks to the reduction onto the center manifold. Consequently, the pictures from Chaps. 9-H will need only some slight modifications where unstable directions replace stable ones, or be added to existing directions in the space. However, the reader must... 

In Sec. 13.5 we consider the bifurcation of the homoclinic loop of a saddle without any restrictions on the dimensions of its stable and imstable manifolds. We prove a theorem which gives the conditions for the birth of a single periodic orbit from the loop [134], and also formulate (without proof) a theorem on complex dynamics in a neighborhood of a homoclinic loop to a saddle-focus. Here, we show how the non-local center manifold theorem (Chap. 6 of Part I) can be used for simple saddles to reduce our analysis to known results (Theorem 13.6). 

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... 

Theorem 9.1. If the equilibrium state is Lyapunov stable in the center manifold then the equilibrium state of the original system (9.1.1) is Lyapunov stable as well Moreover if the equilibrium state is asymptotically stable in the center manifold, then the equilibrium state of the original system is also asymptotically stable. 

If Lk < 0, then for the original multi-dimensional map (10.4.1), the fixed point is also a stable focus. Moreover, its leading manifold coincides with the center manifold. This means that all positive semi-trajectories, excluding those in the non-leading manifold tend to O along spirals which are... 

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,

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