Smale horseshoe

We will start by describing the relevant aspects of the classical dynamics, introducing the concepts of the invariant set and the repeller, the Smale horseshoe and its symbolic dynamics, and the bifurcations at their origin [19]. We then turn to the semiclassical quantization based on the Gutzwiller trace formula and the zeta functions. We proceed to show how this new theoretical framework allows us to explain the distribution of resonances in several molecules like Hgl2, CO2, and H3, to calculate their lifetimes, and to provide a synthesis with respect to previous work. 

In the Smale horseshoe and its variants, the repeller is composed of an infinite set of periodic and nonperiodic orbits indefinitely trapped in the region defining the transition complex. All the orbits are unstable of saddle type. The repeller occupies a vanishing volume in phase space and is typically a fractal object. Its construction is based on strict topological rules. All the periodic and nonperiodic orbits turn out to be topological combinations of a finite number of periodic orbits called the fundamental periodic orbits. Symbols are assigned to these fundamental periodic orbits that form an alphabet... 

Figure 12. Formation of a Smale horseshoe with (a) two branches, (b) three branches.

Let us consider here the case of a repeller that is a Smale horseshoe with three branches as described above. The periodic orbits are in one-to-one correspondence with the bi-infinite sequences constructed from the symbols 0,1,2 associated with the three fundamental periodic orbits. A complete list of periodic orbits can be established p) = 0,1,2,01,02,12,.... Here,... 

Figure 15. Three-branch Smale horseshoe in the 2F collinear model of Hgl2 dissociation at the energy E = 600 cm 1 above the saddle in a planar Poincare surface of section transverse to the symmetric-stretch periodic orbit. The Smale horseshoe is here traced out in a density plot of the cumulated escape-time function (4.6).

The bottom of the exit channels is at -3194 cm-1 if the origin corresponds to the saddle of the Karplus-Porter surface. The pair of tangent bifurcations occur at E = 1670 cm 1, which is followed by the subcritical antipitchfork bifurcation at Ea = 2633 cm 1. The bifurcation scenario is thus similar to the CO2 system, and we may expect a three-branch Smale horseshoe in this system as well. 

Also interesting is the dynamical behavior associated with the fixed point at infinity, that is, q,p) = (oo,0). Here we introduce the concept of homoclinic orbit, which is a trajectory that goes to an unstable fixed point in the past as well as in the future. A homoclinic orbit thus passes the intersection between the unstable and stable manifolds of a particular fixed point. Indeed, as shown in Fig. 6, these manifolds generate a so-called homoclinic web. In particular. Fig. 6a displays a Smale horseshoe giving a two-symbol subdynamics, indicating that the fixed point (oo,0) is not a saddle. Nevertheless, it is stUl unstable with distinct stable and unstable manifolds, with its dynamics much slower than that for a saddle. Figure 6b shows an example of a numerical plot of the stable and unstable manifolds. 

Figure 6. (a) Formation of a Smale horseshoe after six iterations starting from the domain... 

The transformation shown in Figure 12.1.3 is normally called a horseshoe map, but we have avoided that name because it encourages confusion with another horseshoe map (the Smale horseshoe), which has very different properties. In particular, Smale s horseshoe map does not have a strange attractor its invariant set is more like a strange saddle. The Smale horseshoe is fundamental to rigorous discussions of chaos, but its analysis and significance are best deferred to a more advanced course. See Exercise 12.1.7 for an introduction, and Gucken-heimer and Holmes (1983) or Arrowsmith and Place (1990) for detailed treatments., ... 

Smale horseshoe) Figure 1 illustrates the mapping known as the Smale horseshoe (Smale 1967). 

Consequently, the stable W Tj Pi,P2)) and mistable W Tj Pi,P2)) manifolds of the KAM tori T.y(Pi, P2) intersect transversely yielding Smale horseshoes on the appropriate 5D level energy surfaces. This implies multiple transverse intersections and the corresponding existence of chaotic djmamics in the perturbed system 7 > 0 and a = 0. 

This map is a canonical example illustrating the chaotic behavior. For certain parameter values the Henon map models the mechanism of the creation of the Smale horseshoe as illustrated in Fig. C.6.3, for the map and for its inverse ... 

The last identifying feature of chaos is the presence of Smale horseshoe maps (Fig. 6.27a). A typical map involves the stretching and folding of a square with itself. Mixing has been promoted in that sense, because the perimeter of the initial square has increased or the striation thickness has... 

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