Rainer Hegger |
Holger Kantz |
Thomas Schreiber |

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

x_{n+1}=1-
ax_{n}^{2}+by_{n} |

y_{n+1}=x_{n} |

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.

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 x

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.

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-x
^{2})). - 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 .

When performing numerical simulations, Lyapunov exponent(s) should

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.).