Local and global analysis of the quasi-integral

The characteristic property of an integral of motion is its constancy along trajectories in phase space. As mentioned earlier, one cannot expect this behaviour for a formal integral of a nonintegrable system like the magnetic bottle discussed here. But in line with reference [28] (and many others) we still expect convergence of in regular regions of phase space.

Let us define the quasi-integral
of order as that
approximation to
that contains only monomials of degree
and less:

In order to check the convergence of as a function of , we choose a point of the Poincaré surface of section and evaluate the quasi-integral for with being the phase flow of the system (33):

Figure 4 demonstrates that it depends both on the energy (tantamount to the ``degree of chaoticity'' of the system) and on the starting point whether convergence of the quasi-integral is observed or not. At the low energy most yield fast convergence (figure 4a). At the energy divergence occurs almost everywhere, but it is typical that for the first few values of one has ``pseudo-convergence'' before the expected divergence takes over (figure 4b).

The above analysis is *local* in the sense that one has to specify a
single point
for which
ist evaluated. We
now turn
to a kind of *global* analysis and show how to obtain a qualitative
picture of the convergence properties of
in a larger region
of phase space.

In order to have a (though somewhat strong)
criterion for convergence we
say that
is *convergent* at
if

The next step is to define a convergence function
by setting

(30) |

In figure 5 we present a convergence plot for the
magnetic bottle (33) at the energy . This picture
should be compared with figure 3b. On a
grid we have marked with black all points where
indicates
convergence of the quasi-integral.
It is interesting to see that although the condition
(39) is quite strong there are large convergent regions
in . Many of the details of the Poincaré plot
3b show up in the convergence plot as well. As expected,
convergence is most frequent in the centre of the picture. Furthermore,
the *hyperbolic* periodic points of the two dominant Birkhoff chains
(of period four and six, respectively) in figure 3b are
clearly represented in figure
5 by clouds of black marks. This is
surprising, because in the neighbourhoods of these hyperbolic points
one would expect distinct divergent behaviour, caused by the chaotic
dynamics in a heteroclinic tangle scenario.

As an aside we remark that the convergence of the quasi-integral for the magnetic bottle considered here is worse than the convergence for the Hénon-Heiles system discussed in [28]. This seems to be due to the fact that for the latter system the accessible region of phase space is bounded, whereas for the magnetic bottle the dynamical region extends infinitely along the -axis.

In order to get refined information about the convergence properties of
the
as a function of we now modify the rule for
marking points in the convergence plot.
As a measure for the deviation of
from its mean value
we calculate the standard deviation

and for normalization (and for comparability of results that belong to different and thus to different mean values of the quasi-integral) we divide by :

This quantity will play a key role in the following. We can take

as new necessary criterion for convergence. This criterion is similar, but not equal to (39). With (41) one can say more about the divergent quasi-integrals: For each of them there is an such that for all . A convergent (in the sense of (41)) quasi-integral is characterized by (the highest order of approximation). If divergence occurs from the beginning, i.e. if the deviation of the quasi-integral from its mean value grows for all available , then we have . Intermediate values are possible as well. A typical situation (with ) is shown in figure 6.

Again we can mark the points of the surface of section of the magnetic
bottle, this time according to the value of
. The result
can be seen in figure 7 which is to be compared
with the Poincaré plots of figure 3. The dark regions
correspond to larger values of
, thus indicating -- *cum grano salis* -- convergence, while points in the light grey or white
areas have small
, which means that
increases quite from the beginning.
The Poincaré plot 3a at a very low energy shows
regular motion, and figure 7a accordingly indicates
convergence in large regions of .
The light grey spots in the centre of the figure must not be mistaken as
indicating divergent behaviour. On the contrary, convergence is excellent
around the origin, such that very small values of
are being compared, and values of
larger than two are due
to the limited accuracy of the numerical calculation and round-off.
Convergence deteriorates with
increasing (und thus increasing chaoticity) as the comparison of
figures 3b,c and 7b,c shows.
Especially figure 7b reproduces the content of the
Kaluza-Robnik-type picture 5 very well and even
adds on much more information about the convergent regions.
We conclude that the
-plots are considerably better
suited than the
-plots for the analysis of the convergence
properties of the quasi-integral.

It is important to keep in mind that we are using quite a special definition of ``convergence'' here. While this is useful for the present discussion, comparison with rigorous theoretical results about the divergence mechanism [31,32] is delicate. Generally, (pseudo-) convergence (in the usual sense) is expected within a disc, which is compatible with figures 5 and 7a. But figure 7b seems to indicate that convergence has spread into the region between the island chains of period four and six, at variance to the theoretical prediction. The reason being that or 4 reflects only the behaviour for the first few orders and is no direct indicator for true convergence. So the values of can be taken only as a vague phenomenological hint towards true convergence or divergence.

Figure 7b already reproduces reasonably well the features of
the corresponding Poincaré plot, but only in regions not too far away
from the origin. In an attempt to enlarge the region that is accessible
for the analysis we make the following exponential *ansatz* for the
normalized standard deviations as functions of :

It is one of the advantages of the -method that one obtains a continuous spectrum of values of , as opposed to the discrete spectra of and . This becomes apparent in figure 9 where we show again , now shaded according to . The lighter the grey, the larger and thus the faster the divergence of the quasi-integral. The central portion of the picture is similar to the one of figure 7b, but now the outer regions show some structure, too. Comparing with the corresponding Poincaré plot (figure 3b) one sees that the -plot clearly marks the third and fourth largest Birkhoff chains as well. And even more structure can be detected by more careful analysis of the picture. So the ansatz (42) seems to be justified.