Tori, KAM

The KAM theorem demonstrates the existence of KAM tori when the perturbations to the motion are small. What happens when a nearly integrable Hamiltonian is strongly perturbed For example, with increasing perturbation strength, what is the last KAM torus to be destroyed and how should we characterize the phase space structures when all KAM tori are destroyed Using simple dynamical mapping systems, which can be regarded as Poincare maps in Hamiltonian systems with two DOFs, MacKay, Meiss, and Percival [8,9] and Bensimon and Kadanoff [10] showed that the most robust KAM curve... 

Davis and Gray also demonstrated the existence of a series of intramolecular energy transfer bottlenecks, each corresponding to the breakup of a KAM torus. For example, for I2 in the vibrational state v = 20 they found intramolecular bottlenecks associated with frequency ratios equal to (3 + g) and up. However, Davis and Gray found that the last golden mean torus to be broken up is the most effective bottleneck to intramolecular energy transfer and is therefore... 

Meiss and Ott [36] have developed these arguments further and proposed a model based on the Markov tree structure. They have also assumed that partial barriers are formed and cantori with small gaps divide the region around the outermost KAM torus into infinitely many states. An essential point in their argument is to give a labeling scheme to these states and to specify transition probabilities between adjacent states. In particular, each state is viewed as the node of tree, and the transition between these states is assumed to obey the Markov process. 

Bounded motions on KAM tori, or Nekhoroshev-type long-time stability could, however, hardly explain such variety of time scales, because the trajectory on a KAM torus is confined on /V-dimensional subspaces in... 

The dimension of the phase space is 2N, as we have N / s and N (p s. If we still have global conservation laws (such as conservation of energy), the net dimension of the phase space is (/phasespace = 2N — (number of conservation laws). The problem is whether a KAM torus can divide the phase space (or energy surface) into two disjoint parts or not. The necessary condition for global drift of chaotic motion is... 

Random phase approximation D = (K2 + b2)/2 works well for strong coupling (e.g., b > 10) and strong nonlinearity (e.g., K = 4.0), that waits for higher-order correction by Fourier path method [11,15,16], With K > 1.0, diffusion coefficients approach some constants as b —> 0 due to the breakup of the last KAM torus of each standard map, while with K < 0.9 and smaller b, they are expected to be evaluated by the stochastic pump or three resonance model and their extensions [12,17,18]. 

Superexponential stability is in a sense the outcome of the combination of perturbation methods. The simplest case to be considered is, again, that of an elliptic equilibrium or of the neighborhood of an invariant KAM torus. For definiteness, let us consider the latter case. 

Froeschle, C. and Lega, E. (1996). On the measure of the structure around the last KAM torus before and after its break-up. Celest. Mech. and Dynamical Astron., 64 21 31. 

Lega, E. and Froeschle, C. (1996). Numerical investigations of the structure around an invariant KAM torus using the frequency map analysis. Physica D, 95 97-106. 

Figure 20 Schematic drawing of two cylindrical manifolds within isomer A in the weak-coupling limit (refer to Figure 8 for an explanation of the symbols). The two-dimensional cylinders will intersect each other along one-dimensional lines. These lines are two homoclinic orbits. The small, thin tube spanning both isomers corresponds to a reactive KAM torus. Note that although we have stopped drawing the cylinders beyond a certain point for clarity, in reality the cylinders continue to wind about and explore the entire accessible region of chaotic phase space. Reprinted with permission from Ref. 108.

The first task is to construct semiclassical wave function in the classically allowed region. As is well known, the KAM torus exists according to the Kolmogorov-Arnold-Moser (KAM) [57,58] and this integral system can be quantized by the Einstein-Brillouin-Keller (EBK) quantization rule [58] as... 

FIGURE 3.1 Coordinate space (x, y) near one of the potential wells. Distorted rectangle represents the KAM torus projected on this space. (I), (II), and (III) specify the regions to be treated separately in Maslov s theory. Tunneling region is divided into two kinds C and I. A typical trajectory that reflects back at the right canstics is depicted with the branch numbers 1 and 4. (Taken from Reference [30] with permission.)... 

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