Science: Physics: The Structure of Chaos in a Stochastic Layer
A phenomenon of At the end of the 19-th century a French
mathematician and physicist Henri Poincaré first noted that a motion of
planetary bodies may become unpredictable. In a modern terminology he predicted
a chaotic behavior of dynamical systems. He realized that the problem lay not
with the universal laws of motion, but with the specification of the initial
conditions, "… Later a number of distinguished
mathematicians, Emile Borel, Andrei N. Kolmogorov and followers proved that for
a vast majority of dynamical systems any small error in initial conditions will
be fast growing, and in general exponentially, that the prediction of results
will be practically impossible (see Figure 1). These dynamical systems called
as chaotic systems exhibit very sensitive dependence on initial conditions.
During the second half of the 20-th century many mathematicians and
physicists made enormous contributions for understanding and description of
this phenomenon in different areas of natural and engineering sciences, even
economical sciences [1,2]. A dynamical chaos occurs in wide-range problems of
physics, astronomy, chemistry, biology, and ecology. For instance, a particle
motion in accelerators, magnetic field lines in magnetically confinement fusion
devices, motion of planetary bodies in a Solar system, etc. In
this short paper I would like to describe a simple example, which shows the
onset of chaotic motion in so-called Hamiltonian systems. This example also
demonstrates some new features of chaotic motion that has been recently found
(see Ref. [3] and references therein). For many fundamental models of physical
systems whenever dissipation is negligible Newton's equations of motion can be
formulated as a set of ordinary differential equations determined only by one
master function. This function and the corresponding equations of motion are
called
Consider a one-dimensional motion of a
charged particle in a field of two monochromatic electromagnetic waves
propagating along the axis x. The motion of particle is determined by the
Newton's equation
where
The first wave with the amplitude _{0 }along the axis x, while the second wave with the amplitude
E is running with the velocity W_{1}_{1}. Suppose that the amplitude of the first wave, E is
much larger than the amplitude of the second wave, _{0 }E, i.e._{1} E
>> _{0}E. The parameter _{1}c in the equation (2) is a phase difference
between waves. To study the motion of particle it is
convenient to change a coordinate system
where w The equation (3) describes also the motion
of a pendulum whose a suspension point is oscillating with the frequency W and the amplitude e (see Fig. 2). Then the frequency w Consider first the case of the fixed
suspension point when e = 0. The full energy of pendulum is then
conserved
where It is convenient to describe a motion on
the (
For a given initial condition ( n, p=0),
(n= 0, ±1, ±2,
. . .). There are
two kinds of the fixed points. The points with even numbers (n=2s, (s= 0, ±1, ±2, . . .) are so-called elliptic fixed points where the potential
function U(q) reaches its minimum value U_{min}(q_{2s}) = -w_{0}^{2} .
These points are also called O- points
since trajectories around them form closed elliptic curves (see Figure 4). The second kind of the fixed points with
odd numbers ( As was mentioned above for Consider a motion of particle near the
where
Consider the effect of the second wave
with the amplitude Suppose that the amplitude of perturbation
is small, i.e., e <<1. The
perturbation disturbs the orbits of the unperturbed pendulum. The disturbance
depends on how the orbits are close to the separatrix. The trapped and
non-trapped orbits located sufficiently far from the separatrix are only
slightly deformed. But orbits, which are close to the separatrix, are affected
drastically by the perturbation. A typical behavior of orbits located close to
and far from the separatrix is shown in Figure 6. Orbits close to the O-point
are only slightly deformed: the distance between the perturbed and the
unperturbed orbits with the same initial conditions does not grow (orange curve
in Figure 6). The orbits near the separatrix become very
sensitive to the slight change of initial conditions. The distance between two
orbits located near the unperturbed separatrix with very close initial
coordinates exponentially grows. A typical example of this is shown in Figure
6. Although, each orbit is uniquely determined by it's initial coordinates,
even very small difference in initial conditions leads to the dramatic change
in a final state of the orbit. Therefore, behavior of the systems near the separatrix
becomes unpredictable. Such a phenomenon in dynamical systems is called
Poincaré introduced a powerful tool to
study dynamical systems. This tool is based on displaying the coordinates of
the orbit ( q(t),
_{k}p(t)) is
known a Poincaré section. The relation between two consecutive points (_{k}q) and (_{k}, p_{k}q_{k}_{+1}, p_{k}_{+1})
or the projection of (q) to
(_{k}, p_{k}q_{k+}_{1},
p_{k}_{+1}):
is called a
The Poincaré map is a convenient tool to visualize a behavior of
dynamical systems, especially, in a chaotic case. Poincaré section of the
periodically driven pendulum described by Eq. (3) is displayed in Figure 7.
If the orbit is a regular (non-chaotic)
the set of points ( q, p). An
example of such orbits is plotted in Fig. 7 by pink and black curves. If the
orbit is chaotic the points are scattered on the (q, p) -
plane filling the certain region of the phase space. This region is called a stochastic
(or chaotic) layer. It is shown in Fig. 7 by blue dots formed near the
unperturbed separatrix. The width of the stochastic layer is maximal near the
X-points. The stochastic layer is not uniformly filled with the scattered
points. There are small regions inside the stochastic layer and its boundary
region where orbits have regular (non-chaotic) behavior. These regions called Kolmogorov--Arnold-Moser
(or KAM) stability islands (after the mathematicians who proved a
theorem on existence such a stability of motion) are clearly seen in Fig. 8
where the expanded view of the rectangle region near the X-point plotted Fig. 7
is shown. The motion in the stochastic layer is not
completely chaotic. There exist intervals of time during of which a motion can
be trapped at the border regions of KAM--stability islands. In such time
intervals a particle may move around islands almost regular. Duration of the
trapping time depends on the structure of each island. However, these trapping
events happen occasionally and randomly. One cannot exactly predict a trapping
time or its duration, but one can estimate a probability of trapping time
durations or in general, a statistical description of motion in the stochastic
layer is needed. At the beginning of chaos theory it is expected that a motion
in the stochastic layer can be described as a random walk process (similar to
Brownian motion) with the Gaussian statistics. Later when computational
capabilities were developed, it has been found that in typical chaotic
dynamical systems the Gaussian random process cannot always describe irregular
motion. It turns out that due to stickiness of motion to the KAM--stability
islands the statistics of chaotic motion deviates from the Gaussian one and
depends on the structure of the stochastic layer. Below, we will show that the statistics of
chaotic motion in a stochastic layer significantly depends on the its topological
structure, and its mainly determined by the structure of the stochastic layer
near the X-points since particles spend relatively large intervals of time at
these areas of a phase space.
The width of the stochastic layer is
increased with the perturbation parameter e. The structure of the stochastic layer, i.e., the mutual
positions of the KAM stability islands, also changes with e. However, the change of the structure is
not arbitrary. It has been found that the topological structure of the
stochastic layer near the X-points is a periodical function of the logarithm of
the perturbation parameter e. In
this section we consider this non-trivial property of motion in a stochastic
layer, which has been recently established (see Ref. [3] and references
therein). We will compare the structures of the
stochastic layer for the two sets of perturbation parameter e and its phase c. Let (e
the phase--space topologies of the
stochastic layer near the Moreover the phase--space coordinates ( , i.e., x= q - q, are rescaled according to relation:_{s}, y = p - p_{s}
In
Fig. 9 a, b the coordinates Therefore the structure of the stochastic
layer near
Since a particle motion in a stochastic
layer becomes unpredictable, it does not have a sense to follow each individual
orbit. A statistical description of particle motion in a stochastic layer
becomes more appropriate. To be more specific we consider statistical
properties of chaotic particle motion along the
where the angular bracket <(…)> means a statistical averaging over all
particles:
The quantity s If the chaotic motion of particle in the
stochastic layer could be described as a random walk along the For our model of particle transport in a
stochastic layer the exponent g >
1. It is determined by
the structure of the stochastic layer, mainly near the X-points where a
particle slows down and spends relatively large time intervals. As was shown
above the structure of a stochastic layer near the A numerical simulation really shows such a
log e - periodicity of transport
characteristics, which is shown in Fig. 10 for the dependence of the exponent g. The period of oscillation of g is determined by the rescaling parameter l and equal to log l. It has been also found that the mean square displacement moment s This property shows that the chaotic
transport rate along a stochastic layer is not a monotonically growing function
of the perturbation parameter e, as
it was originally believed. It demonstrates that the structure of a stochastic
layer near the
In this paper I have presented the simple
example of a chaotic system and demonstrated the "non-chaotic"
property of its structure. This property shows that chaotic motion in dynamical
systems is not entirely irregular but it has a certain regular statistical
properties. At the present time a chaotic behavior in
dynamical systems has been found in many branches of physical and engineering
sciences, as well as in socio -economical sciences. It plays an important role
in understanding of complex, irregular behavior of a wide variety of natural
and social phenomena. For readers interested more about a phenomenon of chaos
one can recommend a review article [1] and a book [2].
1.
R.V. Jensen,
2. R.C. Hilborn, 3. S.S. Abdullaev (2000) "Structure of
motion near the saddle points and chaotic transport in Hamiltonian
systems", Physical Review E, 2000,
v. 62, pp. 3508-3528.
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