The Influence of the Kick Amplitude

In this subsection I demonstrate the influence of the kick amplitude
for the case ; similar results can be obtained for and
. The following discussion holds for small values of .
The idea is to derive a *separatrix mapping* that describes the
dynamics of the kicked harmonic oscillator in the vicinity of the
separatrices and allows to estimate the width of the channels of diffusive
dynamics.

The first step is to identify
the
of equation (1.39c) as a
high frequency perturbation to the time averaged Hamiltonian
(1.44).
At kick times
,
, one has

(1.25) |

such that the action-angle variables have been exchanged for the original . now takes the form

which holds for every fourth kick time only. This is a useful feature of equation (1.50), as it facilitates the discussion of the dynamics near

From the last formula it becomes clear that can indeed be treated as a high frequency perturbation of the HARPER Hamiltonian : the time dependence of is given by cosine terms the largest period of which is , whereas the smallest period of is -- as is shown below on page -- and can thus be made as large as desired in the limit .

In [LL73] it is argued
that in a first approximation
the higher frequency terms of
a perturbation expanded as in equation (1.50)
can be neglected, although
they contribute with
roughly
the same
weight
() as the small frequency terms.
Therefore I can drop all cosine terms with and get the
approximant

Now consider
a typical orbit of the web map in the stochastic region of the
web. For some
the corresponding orbit point will be close to and just below of the midpoint of the
HARPER separatrix with ,
which is displayed in the upper right quadrant of
figure 1.13.^{1.7}

In order to determine an approximate expression for
, I first derive an explicit solution of the
HARPER dynamics

on the separatrix for . Such a solution is given by

to .

The rate of change of the value of
during the
-perturbed dynamics can then approximately be
calculated
as

(1.27) |

(1.28) |

which holds for all , , again. Using the solution (1.54) for and , the right hand side of equation (1.57) can be approximated by

(1.30) |

such that by integration along the whole separatrix I finally obtain

Thus, while an exact formula for certainly depends on , in the framework of this first approximation is independent of .

It remains to determine the time interval in equation
(1.52b).
Within the cells, i.e. away from the separatrices, the equations of
motion of the unperturbed HARPER system
can be integrated using
Jacobian
elliptic functions.
With

(1.32) |

(1.33) |

(1.34) |

(1.35) |

For real arguments,
as in the present case,
is periodic with period , being a
complete elliptic integral of the first kind:

is not smaller than , which tends to infinity for . This justifies the above treatment of as a high frequency perturbation of (see pages f).

Approximate expressions for the integral (1.65) can be found
in [AS72] and yield here, in the vicinity of a
separatrix:

when solving the upcoming equation (1.71) it turns out that it is technically more convenient to use here as a replacement for rather than . The - and -dependence in this formula comes from -- see equations (1.52a) and (1.59).

With equations (1.59) and (1.68),
the separatrix mapping (1.52), approximating the dynamics of
near the separatrices, is completely specified.
Note that
the
separatrix mapping can also be obtained in a different way:
up to terms of order , the fourth iterate of the web map
(1.29) is given by

(1.41) |

which finds its application in solid state physics, for example in the theory of electrons in certain magnetic fields [KSD92,Dan95,FGKP95], alongside its unkicked counterpart, the HARPER Hamiltonian . Submitting to manipulations similar to those of the present subsection, formulae (1.59) and (1.68) can be derived once again.

Using the separatrix mapping, I now can proceed
to the estimation of the width of the channels of diffusive dynamics.
As a criterion for the
border between regular dynamics within the meshes
and stochastic dynamics in the channels,

(1.44) |

This expression scales with and, not surprisingly, tends to zero in the limit .

Figure 1.15 illustrates this behaviour: for several
values of stochastic webs are obtained numerically by iterating the
web map for a large number of times;
then the channel widths of these webs are measured
-- by determining the largest distance of any two points of the web
which approximately lie on the line and near the point
-- and compared with the numbers given by the approximating formula
(1.73).
The agreement between the analytical formula

and the numerical data is reasonably good for

and improves for . Figure 1.15 also indicates
that for

the precision of the computer
algorithm
does not suffice any more to produce accurate numerical results,
because in this parameter range the web gets *very* thin
-- the width of the channels
shrinks to less than
here.
Similar results hold for the other values of
.

In this subsection I have discussed how the kick strength of the kicked harmonic oscillator determines the shape of its phase portrait. In particular it has been shown that the periodic kicking acts in a way which is typical for perturbed systems: starting from an integrable system at (here: the harmonic oscillator) the kicking (with ) renders the system nonintegrable, and the area of the phase space region of irregular, chaotic dynamics grows with increasing perturbation parameter .

Having investigated the *shape* and the *size* of the channels of
irregular motion
in this subsection and the previous one, in the next subsection I turn to
the discussion of the most characteristic *dynamical* aspect of the
irregular motion.

- ...HarperDynamics.
^{1.7} - Any other separatrix with and could be considered as well. It depends on the choice of and whether the rotation is clockwise or anticlockwise. The cell centred around the origin of the -plane exhibits anticlockwise rotation, and neighbouring cells have opposite directions of revolution. See figure 1.13.
- ...
obtain
^{1.8} - More explicitly, has to be calculated in two steps: first one integrates along the separatrix from to ; then the perpendicular separatrix is followed from to . A closer investigation shows that this procedure can be replaced by integrating along from up to ; this corresponds to the time integral from to using the on-separatrix solution (1.54), as in equation (1.59).
- ...,
^{1.9} - Note that the period of is twice the period of , .