Chirikov overlap criterion

In the case of the kicked rotor we were able to predict the critical perturbation strength by applying the Chirikov overlap criterion. This criterion does two things for us. First, it provides us with an excellent physical picture which explains qualitatively the mechanism of the chaos transition secondly, it provides us with an analytical estimate for the critical field. 

To estimate the critical value of the external filed strength ecr we use Chirikov s resonance overlap criterion (Zaslavsky, 1988 Jensen, 1984), which can be written as ... 

Thus we have treated the chaotic dynamics of the quarkonium in a time periodic field. Using the Chirikov s resonance overlap criterion we obtain estimates for the critical value of the external field strength at which chaotization of the quarkonium motion will occur. The experimental realization of the quarkonium motion under time periodic perturbation could be performed in several cases in laser driven mesons and in quarkonia in the hadronic or quark-gluon matter. 

The plan of Chapter 5 is the following. In order to get a feeUng for the dynamics of the kicked molecule, we approximate it by a one-dimensional schematic model by restricting its motion to rotation in the x, z) plane and ignoring motion of the centre of mass. In this approximation the kicked molecule becomes the kicked rotor, probably the most widely studied model in quantum chaology. This model was introduced by Casati et al. in 1979. The classical mechanics of the kicked rotor is discussed in Section 5.1. Section 5.2 presents Chirikov s overlap criterion, which can be applied generally to estimate analytically the critical control parameter necessary for the onset of chaos. We use it here to estimate the onset of chaos in the kicked rotor model. The quantum mechanics of the kicked rotor is discussed in Section 5.3. In Section 5.4 we show that the results obtained for the quantum kicked rotor model are of immediate... 

The numerical results give us the confidence to attempt an analytical calculation of critical ionization fields. This can be done with some success by computing the widths of the resonances apparent in Fig. 7.5 and using the widths as input to Chirikov s overlap criterion as discussed in Section 5.2. The analytical method allows us to compute critical ionization fields for many initial conditions no and field parameters and u without the need to inspect a sequence of Poincare sections in each particular case. Presently, however, the available analytical methods are not very accurate. For rough estimates of classical critical ionization fields, however, the currently available analytical techniques are very useful. 

We discuss now an analytical procedure for the calculation of critical ionization fields that is based on Chirikov s overlap criterion. For a given microwave frequency u we first compute the locations I m of all 1/M type resonances. We then determine a list of fields, m i defined as the critical fields of overlap between the 1/M type resonance and the 1/(M — 1) type resonance, i.e. m is determined from the condition... 

Fig. 7.6. Numerically computed scaled critical fields (bullets) and experimental 10% threshold fields (squares) (from Moorman and Koch (1992)) as a function of scaled frequency. The full line represents critical fields computed on the basis of Chirikov s overlap criterion. (From Blumel (1994b).)...

According to Chirikov [23J, the onset of chaos is associated with the overlap of neighboring nonlinear resonances. The overlap criterion, which bears the qualitative significance, uses the model of isolated resonances. Each resonance is characterized by its width, the maximum distance (in the action variable) from the elliptic fixed point The overlap means that the sum of the widths of two neighboring resonances is equal to the distance between two fixed points of these isolated resonances. We start with the pendulum Hamiltonian, which describes an isolated 1 N resonance under the periodic perturbation of frequency Q ... 

---