(p.374) Appendix D The onset of chaos
(p.374) Appendix D The onset of chaos
Inspection of the phase space structures in Section 7.1 shows that the onset of chaos, as signalled by a scatter of points in the Poincaré section, is localized around the separatrices between different types of motion (e.g. between local and normal modes in Fig. 7.4 or librational and precessional ones in Fig. 7.5). The origin of this scatter lies in the influence of two or more Chirikov resonances, which is related below to the way in which the separatrix structure is lost. Consequences relevant to the quantization question, to intramolecular dynamics (Davis 1985; Davis and Gray 1986) and to chemical reactions (Davis 1987; Skodje and Davis 1988) are also discussed. The reader is referred for more details to the papers by MacKay et al. (1984) and Bensimon and Kadanoff (1984), and to the literature on chaos in dynamical systems for other aspects of the situation (Casati 1985; Berry et al. 1987; Tabor 1989; Ozorio de Almeida 1988; Gutzwiller 1990) .
D.1 Breaking the separatrix
The existence of a separatrix derives from the presence of an unstable periodic trajectory, whose intersection with the Poincaré section constitutes a hyperbolic or unstable fixed point . The example shown in Fig. D.1 derives from the Hamiltonian
with parameter values and . Inward motion towards the hyperbolic point, P, occurs in directions marked H+, which are termed stable manifolds (Tabor 1989), while outward flow occurs along the unstable manifolds, H−. As first recognized by Poincaré (1892), the regularity or otherwise of motions in the neighbourhood depends on the fate of H− as it is integrated outwards from P, or equivalently of H+ as it is integrated backwards in time. One possibility is that H− joins smoothly with H+, as shown in Fig. D.1(a), which is a diagnostic for regularity; no internally generated trajectory could pass outside or vice versa, because to do so it would cross the union of H+ and H−, which is itself a trajectory, and no two trajectories can cross in phase space. (Because ambiguity in the path from the intersection would make it undeterministic.)
Suppose next that H+ fails to join with H− in a weakly chaotic situation with islands of regularity around the separatrix, as in Fig. D.1(b). Since H− cannot cross either of the bounding regular orbits, it must eventually tangle up with itself, and similarly for the backward integration of H+ . Moreover, the nature of this tangling (p.375) is quite specific. At some time the unstable manifold H+ must cross H−, at what is termed a homoclinic point X, as shown in Fig. D.2(a)—a crossing that is allowed by the deterministic principle because are defined by separate equations of motion. Poincaré (1892) was the first to recognize that such homoclinic intersections have remarkable consequences. To see this, consider the fate of neighbouring points and (p.376) , when integrated in the appropriate time directions, indicated by the arrows in Fig. D.2(a). will appear at on H− and at on H+, but what happens to the homoclinic point X? It can only appear both on H− ahead of (in the arrow direction) and on H+ ahead of if there is a second intersection at TX, as shown in Fig. D.2(b). Repetition of the argument implies the existence of an infinite family of further homoclinic intersections , etc., of which the first is sketched in Fig. D.2 (c). The story is not finished, however, because the area-preserving nature of the dynamics ensures that the two shaded areas are equal to each other and to the areas of all succeeding loops. On the other hand the density of homoclinic intersections on H+ increases without limit towards the fixed point; hence, by area preservation, the tendrils of H− must become ever longer and thinner. Similar arguments apply to the loops and tendrils of the stable manifold H+.
The nature of this homoclinic tangle and some consequences for the chaotic dynamics are illustrated in Figs. D.3 and D.4, by use of a two-dimensional area-preserving map, which offers a computationally efficient alternative to a full trajectory study for pedagogic purposes (Tabor 1989) . The map employed is due to Henon (1969), and details of the algorithm employed are given in Section D.2. Figure D.3 illustrates an overview of the map for a particular parameter value, cos . The five islands are interconverted, one to the next, by each iteration of the map, so that each island has a central elliptic fixed point equivalent to the intersection of a 5:þinspace n stable periodic orbit in a normal Poincar é section. Similarly the five hyperbolic points belong to a single unstable periodic orbit .
The magnified view of one of the islands in Fig. D.4 shows the beginnings of the homoclinic tangle and charts the steps involved in a typical chaotic escape from the central region. One unusual feature is that the two hyperbolic points, A and B, are directly linked by one branch of the separatrix because the unstable manifold of A (p.377) joins smoothly with the stable manifold of B. Consequently the inner region of the complete map (Fig. D.3) contains only regular orbits because the smooth manifolds cut off any connection with the islands. The more common situation is that A and B would both generate stable and unstable homoclinic tangles, such as those shown in Fig. D.4(a). A central island is also included to indicate the bounds of clearly regular motion.
The next diagram, Fig. D.4(b), shows the first hundred or so iterations of a trajectory initiated at the point . Notice how the trajectory avoids the visible inner loops of H+ and H− during this internal phase of its motion. The next stage, in Fig. D.4(c), which shows the escape phase of the same trajectory, is very illuminating. The motion is carried round clockwise and the first five steps in the lower left-hand part of the figure are seen to be captured by successive inner loops of the stable manifold, H+ from A, the final loop of which initiated the tangle as it propagated backwards in time. Thereafter the trajectory follows the smooth part of H+ until it collides with the fixed point A, and emerges on H− of A. The smooth (p.378) join between H− of A and H+ of B leads to a further collision, now with B, and a switch to H− of B. Finally it appears in successive outer loops of H− until apparently ‘ejected’. An alternative scenario, in the opposite direction, would involve capture by an outside loop of the H+ followed by ultimate transfer to the inner loops of H−.
The unresolvable overlap between the inner tendrils of H+ and H− is immensely complicated, but it is clear by continuity with the above discussion that successive iterations of the map (equivalent to successive points in a Poincaré section) lie in successive closed loops of , wherever such loops are defined; there are, for example, no closed loops of H+ (or H−) in the forward (or backward) time direction of the primary homoclinic point X in Fig. D.4(a). This means that motion in the inner region proceeds by passage from one tendril of H+ to the next, and simultaneously for H− wherever the two sets of tendrils overlap, until escape occurs from the last loop of H+. Hence the first hundred or more iterations shown in Fig. D.4(b) were devoted to patiently unravelling the loops of H+, after which the trajectory was under the control of H− alone. Hence even the final escape would be illusory if the external motion was bounded, because the tendrils of H− would eventually overlap with those of H+, or with those of the stable manifold of another hyperbolic point (a heteroclinic intersection), and the whole process would go into reverse. The Henon (1969) map is, however, dissociative in the external region so that many, but not all, of the trajectories reach infinity.
Two insights from this picture into the discussion in Chapter 7 may be noted. The ‘vague tori’ of Jaffé and Reinhardt (1982) mentioned in Section 7.1 are clearly a manifestation of the long-time trapping illustrated in Fig. D.4(b). Secondly, the tendrils attached to the adiabatically switched torus of Skodje et al. (1985), shown in Fig. 7.11, must correspond to incipient trapping by some stable manifold which may bend and distort the figure out of all recognition, without affecting the area enclosed. In other words there may be a class of chaotic-looking but in fact invariant tori, whose Poincaré section areas cannot be estimated by any direct method, but which can be reached by adiabatic switching from a much simpler invariant torus.
The above picture also casts light on the nature of bottlenecks to molecular dissociation and chemical reaction, in ways that have been described by Davis (1985), Davis and Gray (1986) and Skodje and Davis (1988) .
D.2 Henon map and the separatrix algorithm
It has the inverse
Henon (1969) gives a systematic study of the structure of the map for different values of the parameter cos α. Attention was restricted in the previous section to the value .
The steps in determining the evolution of the broken separatrix are first to locate the hyperbolic fixed points and then to propagate the manifolds H− and H+ by use of eqns (D.2) and (D.3) respectively. Since the five islands in Fig. D.3 are interconverted by successive iterations, five iterations of (D.2) or (D.3) are required to generate a new point in any given island. Hence, if denotes , it is convenient to denote the outcome of five iterations as
The fixed point algorithm then takes the form
which is readily solved by the Newton–Raphson scheme
where I is the unit matrix and the stability matrix A has typical elements
Thus may be approximated as
and similarly for , and . As a test of the step size it may be noted that (D.3) requires that . Convergence to six significant figures was obtained in the present application with a step size of 0.01.
The character of the resulting point is indicated by the eigenvalues of A, which are real for a hyperbolic and complex for an elliptic point, while in both cases their product must equal unity (Percival and Richards 1982). Moreover, the eigenvectors of A provide a linearization of the manifolds in the neighbourhood of the fixed points, in the Poincaré section, with the larger eigenvalue belonging to the unstable manifold . Propagation of H− therefore involves construction of a short ladder, of say N points, spaced along this eigenvector so as to lie on a straight line between some initial point and its image after five iterations of (D.2). Successive five-fold iterations of this manifold map out the tangles of H− shown in Fig. D.4. A similar construction for the stable manifold starts with a ladder along the second eigenvector and proceeds (p.380) by five-fold iterations of the inverse map defined by (D.3). Note that the number of points in each loop or tendril is equal to N, the number of points in the ladder, because each five-fold iteration takes one loop to the next. The larger the initial displacement from the fixed point, the more rapidly but the less accurately will the tangles be drawn. A step size of the order of 0.001 was found convenient for the construction of Fig. D.4.