Stable Manifold theorem

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.)... 

Trajectories initiated close to an elliptic fixed point behave in a qualitatively different manner from those near a hyperbolic fixed point. For one thing, elliptic fixed points are invariably surrounded by invariant tori, with frequency ratios not far from that of the fixed point all motion on each torus stays on the same torus. Hyperbolic fixed points may or may not be surrounded by tori. However, they are always associated with a single unique set of manifolds composed of motion asymptotic to them in positive and negative time. These manifolds are called stability manifolds or separatrix manifolds, and their continued existence in the presence of a coupling term is guaranteed by the Stable Manifold theorem. The nature of the asymptotic manifolds will be seen to be of special interest and importance, and we discuss them at length in the following section. 

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. ... 

Consider the space state model R deflned by Eq.(52), showing an equilibrium point such that the matrix of the linearized system at this point has a real negative eigenvalue A and a pair of complex eigenvalues a j/3, j = /—1) with positive real parts 0.. In this situation, the equilibrium point has onedimensional stable manifold and two-dimensional unstable manifold. If the condition A < a is verified, it is possible that an homoclinic orbit appears, which tends to the equilibrium point. This orbit is very singular, and then the Shilnikov theorem asserts that every neighborhood of the homoclinic orbit contains infinite number of unstable periodic orbits. 

While the one-dimensional case may seem too simple, even trivial, it presents a good opportunity to put forward some very general concepts. These concepts, like the existence of barriers in phase space and the stable/unstable manifolds theorem, are best introduced here, having in mind that most interesting applications will come later on. Also, the one-dimensional case has been employed in less trivial ways, by reducing all rapid DOFs to some adiabatic approximation allowing nonlinear one-dimensional TST to be applied [34]. 

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) ... 

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. 

Let p = x(0) be an arbitrary initial point with Jr,(0) > 0. Then the initial data do not belong to either stable manifold. Hence w p) is not equal to either Eq or E2, but it does lie on E = 0. Since it is invariant and since every solution of (5.2) on E = 0 converges to an equilibrium, u(p) contains an equilibrium. By the Butler-McGehee theorem, Eq uIp) since M (Eq) is unbounded. If o p) contains E2, then o p) also contains either Eq or an unbounded orbit, again by the Butler-McGehee theorem (see Figure 5.2). Since this is impossible, E must be in 0 p). However, is a local attractor, so u p) = Ey This completes the proof. ... 

Proof. Assertion (1) is just Theorem C.4. The assertion concerning M (xf) follows from the Perron-Frobenius theory (Theorem A.5) and the monotonicity of the time-reversed system (6.3). If J is the Jacobian matrix of / at Xq, then (6.2) implies that —J satisfies the hypotheses of Theorem A.5. It follows that r = —s —J) < 0 is an eigenvalue of J corresponding to an eigenvector u > 0. Because M (Xo) is tangent at Xq to the line through Xq in the direction v, the local stable manifold of Xq is totally ordered. Since M X()) is the extension of the local stable manifold by the order-preserving backward (or time-reversed) system, it follows... 

Proof. Note that M (Eq), the stable manifold of Eq, is either the p axis if El exists or the x -p plane if Ei does not exist. The manifold M E2) is the X2 p plane less the p axis if E exists, M (Ei) is the Xi p plane less the p axis. Since (Xi(0), X2(0), p(0)) does not belong to any of these stable manifolds, its omega limit set (denoted by w) cannot be any of the three rest points. Moreover, w cannot contain any of these rest points by the Butler-McGehee theorem (see Chapter 1). (By arguments that we have used several times before, if w did then it would have to contain Eq or an unbounded orbit.) If w contains a point of the boundary of then, by the invariance of w, it must contain one of the rest points Eq,Ei,E2 or an unbounded trajectory. Since none of these alternatives are possible, CO must lie in the interior of the positive cone. This completes the proof. 

Theorem C.4. The stable manifold of an unstable, hyperbolic rest point of a monotone dynamical system cannot contain two points that are related by the strict inequality stable manifold cannot contain two distinct points that are related by stable manifold is unordered. 

Proof. If Xq is a hyperbolic rest point then B = M xo), the stable manifold of Xq. Since Xq is hyperbolic and unstable, M (xo) has empty interior (see the proof of Theorem F.l). It follows that M Xq) cannot contain two points Xi and Xj satisfying x [Pg.271]

It is well known that the stable manifold A (x, 0) of a hyperbolic, unstable rest point (x, 0) has Lebesgue measure zero. This follows from Sard s theorem (see Appendix E) and the fact that the stable manifold is the image of a smooth one-to-one map of into K" x K ", where /, is the dimension of the stable subspace of the linearization of (F.l) about (x, 0) and consequently li[Pg.296]

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. ... 

The large maxima of the electron density are expected and are found at the nuclear positions Ra. These points are m-limits for the trajectories of Vp(r), in this sense they are attractors of the gradient field although they are not critical points for the exact density because the nuclear cusp condition makes Vp(Ra) not defined. The stable manifold of the nuclear attractors are the atomic basins. The non-nuclear attractors occur in metal clusters [59-62], bulk metals [63] and between homonu-clear groups at intemuclear distances far away from the equilibrium geometry [64]. In the Quantum Theory of Atoms in Molecules (QTAIM) an atom is defined as the union of a nucleus and of the electron density of its atomic basin. It is an open quantum system for which a Lagrangian formulation of quantum mechanics [65-70] enables the derivation of many theorems such as the virial and hypervirial theorems [71]. As the QTAIM atoms are not overlapping, they cannot share electron pairs and therefore the Lewis s model is not consistent with the description of the matter provided by QTAIM. 

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 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). 

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 ... 

Let us examine next the bifurcations of the system (11.5.1) in the multidimensional case. If Li < 0 (Fig. 11.5.4), then when // < 0, the equilibrium state O is stable (rough focus when p < 0, and a weak focus aX p = 0) and it attracts all trajectories in a small neighborhood of the origin. When > 0 the point O becomes a saddle-focus with a two-dimensional unstable manifold and an m-dimensional stable manifold. The edge of the unstable manifold is the stable periodic orbit which now attracts all trajectories, except those in the stable manifold of O. One multiplier of the periodic orbit was calculated in Theorem 11.1, this is po p) = 1 — 47r /a (0) -h o p). To find the others we... 

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,... 

The situation which we consider here is a particular case of Theorem 13.9 of the next section. It follows from this theorem (applied to the system in the reversed time) that a single saddle periodic orbit L is born from a homoclinic loop it has an m-dimensional stable manifold and a two-dimensional unstable manifold. This result is similar to Theorem 13.6. Note, however, that in the case of a negative saddle value the main result (the birth of a unique stable limit cycle) holds without any additional non-degeneracy requirements (the leading stable eigenvalue Ai is nowhere required to be simple or real). On the contrary, when the saddle value is positive, a violation of the non-degeneracy assumptions (1) and (2) leads to more bifurcations. We will study this problem in Sec. 13.6. 

Consider the case of the one>dimensional stable manifold in Theorem 13.9 and make a reversal of time. After that, the conditions (1) and (2) of the theorem will coincide with the two nondegeneracy assumptions above. 

Note that at // = 0 and ry 0, the separatrix Fi forms a homoclinic loop, approaching one of the two components of Wj c ioc depending on the sign of Tj, Since the non-degeneracy conditions of Theorem 13.7 are satisfied for 17 7 0, the Poincare map T has a smooth invariant curve through the point M" "(0,7 , li" ), transverse to the stable manifold. When restricted to this curve, the map T assumes the form... 

So far, the discussion of the dynamics and the associated phase-space geometry has been restricted to the linearized Hamiltonian in eq. (5). However, in practice the linearization will rarely be sufficiently accurate to describe the reaction dynamics. We must then generalize the discussion to arbitrary nonlinear Hamiltonians in the vicinity of the saddle point. Fortunately, general theorems of invariant manifold theory [88] ensure that the qualitative features of the dynamics are the same as in the linear approximation for every energy not too high above the energy of the saddle point, there will be a NHIM with its associated stable and unstable manifolds that act as separatrices between reactive and nonreactive trajectories in precisely the manner that was described for the harmonic approximation. 

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