Phase One Findings

According to Cambel (1993) a system must meet six criteria to be considered technically chaotic. Those criteria include:

1. The system must be nonlinear and its time series should be irregular.
2. Random components must exist.
3. The behavior of the system must be sensitive to initial conditions.
4. The system should have a strange attractor, which generally means a fractal dimension.
5. The Kolmogorov entropy should be positive.
6. There must be at least one positive Lyapunov coefficient.

Figure 1 shows the irregular pattern of the time series. However, the series also contains obvious periodicities. The power spectrum in Figure 2 shows the apparent dichotomy between strong peaks representing frequencies of periods 2.3, 3.5 and 7 days and the "floor" of the spectrum usually interpreted as uncorrelated noise. Therefore, criteria one and two appear to be met.

Figure 2. Spectral Density for Teen Births 1964 through 1998.

Criteria three and six overlap and were determined to be met by a positive Lyapunov coefficient of .934+/-.013. However, strong caution is in order here. Rapp (1994) asserts if the number of data points needed to calculate the correlation dimension is N, the number of data points needed to calculate the Lyapunov exponent is N². The data set under investigation consists of 12,784 data points. Using the Chaos Data Analyzer, Professional Version, (Sprott, 1995) the team obtained a correlation dimension of 4.61+/- .083. It is troubling to note, however, that a dimension of 5 would be the maximum valid dimension possible to obtain with a data set of the magnitude of the one we were analyzing. The data set of interest is of such a size to make a valid correlation dimension higher than five and an accurate calculation of its Lyapunov exponent impossible. This size limitation is ubiquitous in sociodemographic data sets of interest and often makes the chaotic classification of social processes a fanciful endeavor.

Figure 3. Three-dimensional attractor for birth data.

The attractor obtained from the data is shown in Figure 3. This three-dimensional structure is based on a lag of 7 days. Due to problems of sample size discussed above, it was felt that no stable fractal dimension could be obtained using extant procedures requiring extremely large data sets. Therefore, Cambel’s criterion 4 was not met.

Entropy of the process was not explicitly examined in Phase One of our investigations. However, this aspect of the process is currently receiving focused attention (Scafetta, Grigolini, & Hamilton, 2000).

In summary, Phase One was inconclusive in determining whether the social process under investigation, births to teens, could accurately be classified chaotic. The spectrum of the time series shows both randomness and irregularity in addition to strong periodicities. Data limitations prevented a confident conclusion about the fractal dimension, the attractor and the Lyapunov exponent. Quoting Rapp (1994), "Used naively, these methods routinely produce erroneous results. The magnitude of this potential for computational mischief should not be underestimated." (p. 311). Misgivings arising out of Phase One motivated the team to enter Phase Two in which two methods of analysis of nonlinear dynamics could be evaluated for their feasibility in application to demographic data sets of the size often encountered in "real world" situations.