Rainer Hegger |
Holger Kantz |
Thomas Schreiber |

Part IV: Dimensions, Lyapunov exponents and entropies

Use again the data set

- Recall reasonable embedding parameters from the results of the
last exercises and their Theiler-windows.

- Compute the correlation sums of the sets using
d2 .

d2 amplitude.dat -d8 -t100 -o

Study carefully the output files amplitude.dat.c2 and amplitude.dat.d2.

Can you give reasonable dimension estimates?

The best result will be obtained by fitting by eye a power law to the amplitude.dat.c2-curves:

set logscale

plot 'amplitude.dat.c2',.001*x**2.1

Errors of the ampitude.dat.d2-curves are anti-correlated! Use av-d2 with the option -a# to smoothen them. Observe the edge effects and folding effects in the m=1-curve, on larger scales also the m>1-curves cannot show the right dimension. Anyway, although the data represent a chaotic flow where theory tells us that the dimension must be larger than 2, a value of D_{2}>2 can hardly be derived from these curves!

Give an estimate of the noise level.

- Entropies: plot the file amplitude.dat.h2 to
estimate the entropies (on a log-linear scale). Write dimension and entropy estimates and the
noise level into a table (on paper). Do not forget to divide the
h
_{2}-estimate by the time lag.

- Redo the computation with a smaller and a larger time lag, e.g., -d1 and -d24. You should
be able to confirm the values found above, but the scaling ranges
either shrink or larger m-values are needed
for convergence. For smaller lag you find more edge effects, for
larger lags more folding effects. Entropy estimates might somehow
profit from larger lags.

- Maximal Lyapunov exponent: Use
lyap_k and/or lyap_r
, e.g.,.

lyap_k amplitude.dat -M6 -m3 -d8 -t100 -s500 -r.1 -o

If necessary, use the option -n1000 to keep the numerical effort limited. Try to understand on which time interval you expect the exponential increase. Admittedly, flow data are not really suited for such kinds of algorithms.

- Compute the Lyapunov exponent for
the Poincaré-map data of this flow which you produced in the
last exercise, e.g. with the following parameters:

lyap_k max.dat -M5 -m2 -s25 -o

Recall that for map data, the optimal settings are -d1 -t0 which are the defaults, and that there is no use to consider time horizons which are larger than the data set (hence, -s25).

The estimated exponent (slope of the straight line) carries the units of 1/time and thus has to be divided by the average time in between two intersections of the flow with the Poincaré surface of section in order to be compared to the value obtained from the flow data. This time is computed by (length of the flow data set)/(#number of intersections).

- Use the Poincaré-map data max.dat or
amplitude.dat.poin generated
with -C1 as a data base for nstep to create a much longer
synthetic trajectory of length 5000:

nstep -k10 -L5000 max.dat -o

Since the data of max.dat form an almost 1-dimensional graph of a map, -m1,2, i.e., the default values, are fine. Due to the low number of data points, -k10 is a reasonable number of neighbours; to require more would extend the neighbourhoods to outside the linear regime. Verify the validity of the forecasts by a comparison of the attractors in the delay embedding space. Evidently, there is some fake additional structure whose existence cannot be supported by the original data. Other choices of parameters/Poincaré data of the flow might yield better agreements. In any case, the data base (152 points) is rather poor.

- Compute the correlation dimension for these numerical data.

Result: The file max.dat.cast.c2 supports a dimension of D=1.2, but under consideration of what we said before about the fine structure, this number cannot be trusted. When comparing the curves to max.dat.c2, the dimension estimates on max.dat itself, there is some compatibility, but in this case everything is speculation.

**This illustrates our standard warning: One can apply the routines to and exctract numbers from many data sets, but whether the results are meaningful or not remains in the interpretation of the applicant!**

- Compute the maximal Lyapunov exponent for the synthetic Poicare-map data.
Are the results compatible with what you found for max.dat ?

- Can you expect more than one positive exponent in the map data?

Use lyap_spec to compute the full Lyapunov spectrum of the synthetic data.

The resulting set of exponents contains spurious ones. For their identification, run the algorithm with modified neighbourhood size, larger embedding dimension, or a different time lag. How many positive exponents do you find?

Our result is 0.53 for the positive and -1. for the negative exponent. In a 2-dimensional embedding, there are no spurious exponents! Enter the maximal exponent in your table, the sum of all positive as the estimate of the KS-entropy, and compute the Kaplan-Yorke-dimension from the non-spurious exponents.

- How good is the mutual agreement of the different methods?

This is our result:

**data type****dimension****max. Lyapunov****entropy****noise level**flow data D _{2}=2.1±0.05lyap_k=0.015±0.002 h _{2}=0.02±0.0050.5 units map data max.dat D _{2}=1 + 1.2±0.2lyap_k=0.015 h _{2}=0.45/32.9=0.01350.5 units synthetic data max.dat.cast D _{2}=1+ 1.2±.1lyap_k: 0.5/32.9 h _{2}=0.4/32.9synthetic data max.dat.cast D _{KY}=1+1.5lyap_spec: 0.53/32.9= 0.016

**Comments**

The**flow data**are numerically generated data with additive noise, and in so far optimal (no drifts, no interactive noise, no artifacts). However, they represent only about 150 revolutioins of the trajectory on its attractor, and thus are rather few data.

The numerically exact Lyapunov exponent (obtained during the integration of the system) is 0.0173, not so far from the time series value.

The entropy should be bounded by this exponent, but the times series value is also close enough. In fact, the convergence of the entropy estimates as a function of the embedding dimension is from above, i.e., higher embeddings might yield better estimates. However, this fact yields a mismatch of the entropy estimate and the maximal Lypunov exponent obtained from the time series data.

The dimension estimate is reasonable.

The**Poincaré map data**are very few, but since the Poincaré map is essentially one-dimensional, they contain still meaningful information. Nonetheless, the agreement between the results on the map data and the flow data is not excellent.

The invariant characteristics of the**synthetic map data**are in agreement with the Poincaré map data, but as we have seen, the attractor contains structure which cannot be supported by the observations.

The computation of the Lyapunov spectrum yields an overestimation of the dimension by the Kaplan-Yorke formula, and it can be suspected that the negative Lyapunov exponent is underestimated.