Separatrix splitting

Let be a smooth manifold which we identify with the configuration space (position space) of a certain Hamiltonian system and let be a cotangent 

Let the Hamiltonian H be periodic in t with the period 2x and depend on a certain parameter e, that is, is equal to H (x,, t,e). Suppose that for = 0 the Hamiltonian ff(x, t,0) = ATo(x, ) does not depend on time and satisfies the following four conditions  

2) By A+ (respectively, A ) we denote a stable (respectively, unstable) separated manifold of the critical point (x-., -. ) ( respectively, (x, f-)), that is, a 

Along stable separatrix surfaces, integral trajectories tend to the critical point (with increasing time), while along unstable separatrix surfaces they move away from the critical point (with increasing time). Condition 2 consists in the fact that A-i- = A., that is, the stable separatrix surface A+ of the point (x+, +) coincides with the unstable separatrix surface A- of the point (x, ). FVom this it follows, in particular, that ffo( + +) = o( - -)  

3) Suppose that in a configuration space there exists a domain D = 

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In the paper [209], the theorem formulated above is proposed for A = m = 1. The concluding part of the proof makes use of the separatrix splitting method, in line with [167], but as is seen from the further analysis (see [210]), this step in the proof requires additional motivations. 

Definition 5.1.2 The phenomenon described above is called the splitting of separatrix surfaces (or the separatrix splitting). 

As was noted by Poincar, the separatrix splitting turns out to be an obstacle for integrability of a perturbed Hamiltonian system (viewed in the neighbourhood of an integrable unperturbed system). 

It b just the separatrix splitting that allows us to prove nonintegrahility of the equations of motion of a dynamically nonsymmetric rigid body with a fixed point in the neighbourhood of the integrable Euler case (see subsection 1.2 of the present section). In the problem of fast motion of a nonsymmetric rigid body the Hamiltonian b the form if = where... 

The proof of Theorem 5.1.6 also rests upon the separatrix splitting phenomenon. To this end, one should represent the Kirchhoff equations as perturbations of integrable equations. Introduce a small parameter e replacing in the Kirchhoff equations e by ee. Then on the fixed four-dimensional level surface of two integrals... 

Kozlov, V. V. Separatrix splitting in the perturbed Euler-Poisson problem. Vestnik Mosk. Gos. Univers., ser. mat. mekh.. No. 6 (1976), 99-104. 

The term a[p) governs the splitting of the separatrix loop (see Fig. 13.2.3) it follows from (13.2.2) that a[p) is equal to the difference between the y-coordinates of the points and TiM (the latter is the point where the xmstable separatrix F]" first intersects Sq). By assumption, the separatrix splits inwards for /x > 0, and outwards for p <0. Thus, sign a p) = sign p. 

Nitromethane shows the simplest residue curve map with one unstable curved separatrix dividing the triangle in two basic distillation regions. Methanol and acetonitrile give rise two binary azeotropic mixtures and three distillation regions that are bounded by two unstable curved separatrices. Water shows the most complicated residue curve maps, due to the presence of a ternary azeotrope and a miscibility gap with both the n-hexane and the ethyl acetate component. In all four cases, the heteroazeotrope (binary or ternary) has the lowest boiling temperature of the system. As it can be seen in Table 3, all entrainers except water provide the n-hexane-rich phase Zw as distillate product with a purity better than 0.91. Water is not a desirable entrainer because of the existence of ternary azeotrope whose n-hexane-rich phase has a water purity much lower (0.70). Considering in Table 3 the split... 

When passing through the critical value a, the torus splits into two tori, Ti g and T2, . The separatrix diagram P. goes from the critical surface and, when descending, meets the torus along two circles 71 and 72 (see above). Lemmas... 

Denote by Ki and K2 both rings, into the union of which the circles 71 and 72 split the torus Construct a new surface Pi by adding to the separatrix... 

If a critical torus has the dimension n, then it is either the set of the local minimum or of the local maximum of the energy H. In this case, either two close nonsingular Liouville tori flow into one torus or the torus T splits into two tori r. Let P2 = P2 T ) and = P T ) be, respectively, in and out separatrix diagrams of the critical submanifold... 

The splitting of these separatrix surfaces b studied in [164], [165. The behaviour of the solutions of a perturbed problem was studied using computer in the interesting paper [166]. In the schemes obtained in the calculations and showing the behaviour of integral trajectories it b clearly seen that the invariant curves of the unperturbed problem become chaotic in the neighbourhood of separatrices. Thb... 

Figure 2.4. Trajectory bundles under infinite reflux for (a) three-component ideal and (b) azeotropic mixtures. xd(x) xb(V),xd(2) - a b(2), possible splits solid lines, trajectories dotty line, separatrix under infinite reflux.

The original oversimplified view on feasible azeotropic mixtures splits consists of the following the feed point and product points have to belong to one distillation region (xd e Reg° and Xb e Reg°° if Xp e Reg° ). This view is quite accurate if the separatrix of distillation regions is linear. In a general case, at curvilinear separatrixes, the feed point can lie in one distillation region at infinite reflux and... 

Let s note that the top product point cannot be located at sides 2-3 and 1-3 because Eq. (5.15) is not vahd for these sides (i.e., Xo i [2-3] and xd [1-3]). Let s also note that ihc separatrix sharp split region of section trajectories bundle... 

Figure 5.24 shows the evolution of separatrix sharp split region of top section... 

Figure 5.25. The evolution of separatrix trajectory bundle and separatrix sharp split re-...

At sharp split separatrix sharp split region Af+ s... 

Flgore 534. (a) The taDgeDtial pinch in rectifying section for the acetone(l)-benzene(2)-chloroform(3)-tolnene(4) mixtnre for the split 1,3 2,4, and (b) natnral projection. Separatrix sharp spht region for rectifying section 5 — Sj — =... 

The analysis of dimensionaUty of sections trajectory separatrix bundles shows that for splits with one distributed component trajectory of only one section in the mode of minimum reflux goes through corresponding stationary point or (there is one exception to this rule, it is discussed below). The dimensionality of bundle 5 - A4+ is equal to A - 2, that of bundle — iV+ is equal to n — A — 1. The total dimensionality is equal to n - 3. Therefore, points x/ i and Xf cannot belong simultaneously to minimum reflux bundles at any value of LlV)r. If only one of the composition points at the plate above or below the feed cross-section belongs to bundle 5 - A + and the second point belongs to bundle 5 - 5 - A+, then the total dimensionality of these bundles will become equal n - 2 therefore, such location becomes feasible at unique value oi(LjV)r (i.e., in the mode of minimum reflux). 

L/ K) on D,2- Separatrix sharp split region for rectifying section Reg shaded. 

For four-component mixtures at nim = 3 and at two components in the bottom product (Fig. 6.9b), the conditions of joining in the case of bottom control feed are defined by the dimensionality of trajectory bundles N - Sm(d = 1) and (d = 1) and are similar to those of joining of sections trajectories of two-section column in the mode of minimum reflux at intermediate split (see Section 5.6). Point X/-1 should lie on the separatrix min-reflux region Re (N - Sm) and point Xf should lie on the separatrix min-reflux region Re ( - N ). 

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