Chaos and Visual Recurrence Analysis (VRA) of TWSTT.

The data we look at here is the measured time difference of two Hydrogen maser clocks whose output are steered towards UTC. This system would be generically classified as a "driven" system since we are not looking at two free running clocks. In time and frequency keeping most of the "signals" are described as "noise". This will be a first look at a good system with an attempt at isolation of any chaotic signal(s) within. We start with figure 1 which is the time series. These are simple one day averages of nonimnally sampled hour TWSTT experiments. The basic statistics of the data are as indicated below.

N 1194 (days)
Mean 0.174 (Nanoseconds)
Minimum -1.94
Maximum 2.49
Variance 0.478
Std. Deviation 0.691
Skewness 0.276
Kurtosis 0.616


Figure 1 The original TWSTT signal.

Next we generate the power spectra using the method given in reference 1. The time and frequency communiity most often plots the power spectra as log/log plots which makes it easy to detection power law components of the "noise". Iniitially the spectra is flat and later shows a slope change. This might indicate a low-order attractor which often show broadband peaks in power spectra, i.e. there are no sharp peaks indicative of linear periodic signals.


Figure 2 The Fourier spectrum of the TWSTT signal.

Next is shown the standard autocorrelation function. The first minimum is located at a lag of 16, but a deeper first minimum is located at a lag of 32.


Figure 3 The autocorrelation function of the TWSTT signal.

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Figure 4 The global average mutual information of the TWSTT signal.


Figure 5 The global false nearest neighbors of the TWSTT signal.


Figure 6 The correlation dimension of the TWSTT signal.


Figure 7 The unfolded attactor consisting of the "quantum noise" of the time transfer and clocks involved.


Figure 8 shows the VRA plot of the TWSTT signal with dimension of 1 and a time lag of 1. The STE is 83%. All VRA plots were produced using the software in reference 1. (gif - 125 kb)


Figure 9 shows the VRA plot of the TWSTT signal with dimension of 2 and a time lag of 8. The STE is 80%. (gif - 131 kb)


Figure 10 shows the VRA plot of the TWSTT signal with dimension of 8 and a time lag of 8. The STE is 80%. (gif - 121 kb)


Figure 11 shows the VRA plot of the TWSTT signal with dimension of 15 and a time lag of 15. The STE is 80%. (gif - 120 kb)


Figure 12 shows the VRA plot of the TWSTT signal with dimension of 30 and a time lag of 30. There is not enough data to compute the STE. (gif - 101 kb)


Figure 13 shows the VRA plot of the TWSTT signal with dimension of 31 and a time lag of 31. There is not enough data to compute the STE. (gif - 97 kb)


Figure 14 shows the VRA plot of the TWSTT signal with dimension of 32 and a time lag of 32. There is not enough data to compute the STE. (gif - 88 kb)


  1. J. A. Barnes, 1993, "A Digital Equivalent of an Analog Spectrum Analyzer," Proceedings of the 1993 IEEE International Frequency Control Symposium, 2-4 June 1993, Salt Lake City, Utah, USA, p. 270-281.
  2. Kononov, Eugene, (October 1, 1999), Visual Recurrence Analyis, Software Version 4.0.
  3. Applied Nonlinear Sciences, 1997, Contemporary Signal Processing for Windows, cspW, Software Version 1.2.
  4. Schroeder, Manfred, 1991, "Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise", W. H. Freeman Company, New York, pp. 429.
  5. Gutzwiller, Martin C., 1990, Chaos in Classical and Quantum Mechanics, (Interdisciplinary Applied Mathematics Volume 1), Springer-Verlag, New York, pp. 432.