| Rainer Hegger |
|
Holger Kantz |
|
Thomas Schreiber |
Exercise 1 using TISEAN Nonlinear Time Series
Routines
Exercises using TISEAN
Part I: Exploring chaos in one-dimensional maps
Exercise I helps you to make the first steps with TISEAN and illustrates
some properties of one dimensional chaotic maps. Due to the single phase
space dimension, time series analysis in this case is analysis of
numerical simulations in phase space.
The Hénon map
is a two-dimensional extension of the logistic equation, xn+1=1-axn2.
With b=0 and
0 < a < 2,
the Hénon map creates a time series of the logistic equation,
provided the intial condition
for x is inside the interval [-1,1].
The
routine henon allows you to
generate a time series of the Hénon map of arbitrary length, for
arbitrary parameters, arbitrary initial conditions, and after
discarding transients. Click on henon
to see the html-manual page. and type henon
-h as a command line in a terminal window of your computer
to see the on-line
help. For proper usage of henon ,
you must specify the number of iterates to be
produced by the -l# option, where # has to be
replaced by an integer, say, 5000 for 5000 data points to be
produced. Other options can be used to modify the
defaults, and among them are the parameters of the map (the defaults
correspond to the values originally used by M. Hénon himself, who
"invented" the map).
Use gnuplot for a fast scan through the different scenarios
(if you do not have gnuplot, store the data in output
files and plot them with your favorite plot-program) in, e.g.,
the following way, where you have to specify
a in -Aa of henon in the
plot command:
yourcomputer:> gnuplot
gnuplot> set yrange [-1:1]
gnuplot> plot '< henon -B0 -Aa -l100' using 0:1 with
linespoints.
The plot command combines the successive
iterates with lines and helps to guide the eye, but if you want to plot
more than about 500 points you should use with
points or even with dots, instead.
You should observe this way:
- The period doubling bifurcation around a=0.75 (bifurcation to
period 2), a=1.25 (period 4), a=1.35 (period 8).
- The two-band chaos: a=1.4 to a=1.55 (approximately).
- The intermittency close to the birth of the period three orbit by
tangent bifurcation at a=1.75 (for intermittency, use a=1.7499).
- transient chaos on the repeller outside the period 3 orbit:
a=1.75. Use -x0 in order to not
discard the initial part of the trajectory (the transient, during
which the orbit is supposed to settle down on the attractor), and
vary the initial condition x (using the -Xx option of henon ) and
study the transients before the trajectory settles down on the period
3 orbit.
A not very elegant but simple way to plot a full bifurcation
diagram using gnuplot and the henon routine is to load the file henon.gnu
in gnuplot: load
'henon.gnu', where you can easily include more parameter values.
When a scalar time series is generated by a one-dimensional map, a
time delay embedding of lag one shows the graph of the map
xn+1=f(xn). The routine delay by default produces a
two-dimensional delay embedding with unit time lag. The -d# option sets a different lag.
Use plot '< henon -B0 -Aa -l5000 | delay' with
dots in the following for several values of a.
trouble?
- Convince yourself that when a
is such that the trajectory is
chaotic, you see thus a part of the parabolic graph of the map.
In particular, for
-A2.0 you should see the full parabola.
- The graph of the p-th iterate can be plotted by using
-dp, i.e. by adjustung the time lag
of the embedding to p.
Every intersection of a graph of a 1-d map and the diagonal is a fixed
point of this map. An intersection point of the graph obtained for
-d2
is a fixed point of the second iterate of the map and thus one of the
two points of a period 2 orbit.
Study the orbits of up to period 4 of the logistic map for a=2 by this
method.
Verify that there are 2 fixed points, 4 period-2 points (one non-trivial
orbit and two trivial ones), 8 period-3 points (two non-trivial orbits
and two trivial ones), and 16 period-4 points (what about the
corresponding orbits?).
- The mechanism of intermittency: Plot the time series for a=1.7499 in
time delay coordinates with lag 3 together with the diagonal.
Can you identify the reason why the
trajectory is intermittent? Answer.
The invariant measure:
histogram
produces a histogram of the input data, where several options can be
used for adjusting, e.g., the number of bins.
Compute the histograms of the distribution of the
variable x of the logistic equation for
various parameter values (e.g.:
gnuplot> plot '< henon -B0 -A2 -l10000 | histogram -b100' with
hist). When a sufficiently long transient has been
discarded, such a histogram is the approximation to the invaraint
measure on the bins of the histogram. Verify numerically:
- The measure corresponding to a periodic orbit constists in
equal-height delta-peaks at the locations of the points of this orbit.
- The measure for a=2 fulfills
rho(x) = 1/(pi sqrt(1-x2)).
- The measure for chaotic orbits with a < 2
contains a huge (a countable, infinite) number of
singularities which are images of the singularity at
x=1 .
Lyapunov exponent:
When performing numerical simulations,
Lyapunov exponent(s) should only
be computed by direct iteration in tangent
space, not by time series analysis! Nonetheless,
here we use time series analysis:
Use lyap_k and
lyap_r
to compute the (only) Lyapunov exponent of the logistic equation for a=2 :
mycomputer:>
henon -l10000 -B0 -A2. | lyap_k -M4 -n1000 -s20 -o lyap_k.dat
mycomputer:> henon -l10000 -B0 -A2. | lyap_r -s20 -o lyap_r.dat
mycomputer:> gnuplot
gnuplot>
plot 'lyap_k.dat' with lines, x*log(2.)-8, 'lyap_r.dat' with lines
Can you thus confirm the precise value lambda = ln(2) ? Study the resulting plots as a
function of the trajectory length. Also, add noise to the data using
addnoise or
makenoise .
You should observe that more than about 2% of noise (in root mean square
sense) will destroy the straight lines with slope log(2.).