Arnold model

In the following, we consider two examples where the Melnikov integral is explicitly calculable. The first example serves as a standard case for the calculation of the integral. The results will be used in the later sections where we study the Arnold model and tangency. The second one serves as a prototype for molecular systems under time-dependent electric fields. 

Bearing these problems in mind, we will study the Arnold model as an example of the Melnikov integral. This section will also serve as an introduction leading to the argument on tangency in the next section. 

The Arnold model is a time-dependent Hamiltonian with two degrees of freedom. 

In the next section, we will also smdy the Arnold model with 0 = 0 from a different point of view. We show that the model exhibits tangency and that the condition for tangency can be derived using the Melnikov integral. 

The Arnold model with ft = 0 corresponds to scattering processes. In the planar Coulomb three-body problem, the asymptotic limit where one of the three bodies goes to infinity corresponds to the Arnold model with 0 = 0 [35]. For three-body clusters interacting with van der Waals potential, the Arnold model with 0 = 0 also arises when one of the three bodies goes to infinity [37]. 

We can also regard the Arnold model with O = 0 as a system with three resonances. In other words, it models the dynamics around resonance intersections. This is because the condition O = 0 is regarded as a resonance condition, where the movement 0 becomes slow. Then, the term cos 0 cannot be averaged out but must be kept in the analyses. Thus, in addition to the term V cos q, the model has another resonance term cos 0. Therefore, the Arnold model with H = 0 can be regarded as a model of a resonance intersection. 

Thus, the Arnold model with H = 0 belongs to a universal and important class of dynamical systems. 

This model is the same as the Arnold model except that Eq. (93) has two small parameters s and p. In the following analysis we will consider s and p separately. 

T. Konishi, Slow Dynamics in Multidimensional Phase Space Arnold Model Revisited, Adv. Chem. Phys. Part B 130, 423 (2005). 

SLOW DYNAMICS IN MULTIDIMENSIONAL PHASE SPACE ARNOLD MODEL REVISITED... 

In this chapter we call the following model the Arnold model ... 

Figure 2. Schematic figure of unstable and stable manifolds in the Arnold model, Eq. (8).

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