Further developments are presented in the technique for analysing attractor behaviour from small data sets, based on the observation of scaled structures in families of slope curves of correlation integrals. Scaled doublet structures are investigated systematically for short time series obeying the Mackey and Glass delay differential equation. At an attractor correlation dimension close to 5, ranges of values of T (the length of the time sequence) and of f (the recording frequency) are described in which the scaled doublet structures are unambiguously identified and distinguished from structures that can occasionally be found with randomized time sequences. Implications for the characterization of low-dimension attractors, notably from electroencephalographic recordings, are discussed, including in particular the advantage to be gained from moderately oversampling the data.
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Cerf, R., Ben Maati, M.L. Attractor characterization from scaled doublet structures: simulations for small data sets. Biol. Cybern. 72, 357–363 (1995). https://doi.org/10.1007/BF00202791
- Differential Equation
- Time Series
- Time Sequence
- Correlation Dimension
- Attractor Behaviour