**10. Experimental results: Application of the chaos data analyzer software (CDA) to the experimental time series**

In chaos theory, the calculation of the Lyapunov coefficients is fundamental because it allows studying the effect of the initial condition, the irreversibility of the processes, the entropy, the time of predictability, the complexity, and, based on these parameters, to characterize the sustainability of a learning process.

In the cases of weak attractor dynamics and middle attractor dynamics, chaos theory does not apply, since they are predictable or deterministic systems.

#### **10.1. Chaotic attractor dynamics**

For the analysis of the time series, the CDA Software, Chaos Data Analyzer Programs [37, 60], and the Golden Surfer Software were used to fill incomplete time series [65].

Notation:

λ: Exponent of Lyapunov (bits/units of time) [37, 47, 66].

D: Encrusting dimension [37].

n: is the number of sample intervals over which each pair of points followed before a new pair is selected.

> The values of C(t) for time series give results greater than zero for the three time series (X(t), Y (t), and Z (t)), which would characterize a complex system and is coincident with the result

Initial Condition and Behavior Patterns in Learning Dynamics: Study of Complexity and…

http://dx.doi.org/10.5772/intechopen.74140

61

In the learning process studied, applying Chaos Theory to time series arising from characteristic variables that are common to all learning processes, three particularities revealed entropic connectivity, irreversibility, and complexity. In the same way, the sustainability of the process is due to the quotient of the POS/NEG emotions, when positive influence exists (POS) in the case of the Chaotic Attractor Dynamics, which is the one that presents the highest achievement in meaningful learning (rudely indicated as performance). These learnings lead to the consequent formation of patterns of "desired" behavior. Increasing connectivity is increasing entropy, which to maintain the "beloved" order, entropy (negentropy) must be transferred to the environment (to the planet), is quantifiable evidence of such process, the increase of garbage and pollution. This corollary demonstrates the irreversibility of the process in the current narrative. Is it possible to modify this plot? Within the current narrative, it seems unlikely. A new form of relationship between ourselves and with nature must build. (Chaos theory does not admit complexity for the weak and middle attractor, i.e., it does not apply).

The considerable cognitive requirements of life in complex societies have resulted in many primate species having larger and more expensive brains [70], with all that this implies in connectivity. The human immersed in evolution has historically transferred the cost of learning

In the interior of ancient Mesopotamia, agriculture and livestock farming were imposed as the primary economic activity between 6000 and 5000 BC. Due to unfavorable natural conditions

**11. The future in the past? Ancient civilizations**

the complexity of nature, and there is ample evidence.

**11.1. Mesopotamia**

of the matrix of correlation.

**Graph 4.** Eigenvalues of the matrix versus number eigenvalues.

A: is the relative accuracy of the data before the expected noise begins to dominate.

H: Exponent of Hurst is related to the smoothness of the curve and the dimension fractal, according to Mandelbrot [67–69], 0 ≤ *H* ≤ 1. To 0.5 ≤ *H* ≤ 1.0 indicates persistence (the past tends to persist in the future).

S = Correlation entropy [37–60] for each variable (measured in bits/units of time) (**Table 3**):


**Table 3.** The correlation entropy, SK, is the entropy of Kolmogorov, and its reciprocal delivers the time for which the prediction is significant.

Applying the calculation of the correlation matrix to the time series X (t) of the chaotic attractor, the number of eigenvalues, of the order of the correlation dimension, is a measure of the complexity of the system [37]. In this case, two significant eigenvalues were determined (not zero) (**Graph 4**):

The time series of Y (t) and Z (t) treated in a similar way present the same number of eigenvalues. So, we conclude that given the number of eigenvalues, the series represent a complex system.

According to the López-Corona approach [50] for complexity, with normalized *Sk* (**Table 3**) → *SKobs* :

$$C\_{\uparrow}(t) = aS\_{\chi\_{\leftrightarrow}}(1 - S\_{\chi\_{\leftrightarrow}}) > 0, \quad i = X, Y, Z \tag{8}$$

**Graph 4.** Eigenvalues of the matrix versus number eigenvalues.

In the cases of weak attractor dynamics and middle attractor dynamics, chaos theory does not

For the analysis of the time series, the CDA Software, Chaos Data Analyzer Programs [37, 60],

n: is the number of sample intervals over which each pair of points followed before a new

H: Exponent of Hurst is related to the smoothness of the curve and the dimension fractal, according to Mandelbrot [67–69], 0 ≤ *H* ≤ 1. To 0.5 ≤ *H* ≤ 1.0 indicates persistence (the past

S = Correlation entropy [37–60] for each variable (measured in bits/units of time) (**Table 3**):

**Variable Λ D N A H Correlation dimension SK** X 1.150 ± 0.099 1 2 0.0001 0.925 0.719 ± 0.247 1.027 Y 0.723 ± 0.084 1 2 0.0001 0.919 0.728 ± 0.245 0.477 Z 1.469 ± 0.105 1 2 0.0001 0.89 0.737 ± 0.251 0.728

Applying the calculation of the correlation matrix to the time series X (t) of the chaotic attractor, the number of eigenvalues, of the order of the correlation dimension, is a measure of the complexity of the system [37]. In this case, two significant eigenvalues were determined (not

**Table 3.** The correlation entropy, SK, is the entropy of Kolmogorov, and its reciprocal delivers the time for which the

The time series of Y (t) and Z (t) treated in a similar way present the same number of eigenvalues. So, we conclude that given the number of eigenvalues, the series represent a complex

(*t*) <sup>=</sup> *aSKobs*,*<sup>i</sup>*(<sup>1</sup> <sup>−</sup> *SKobs*,*<sup>i</sup>*) <sup>&</sup>gt; 0, *<sup>i</sup>* <sup>=</sup> *<sup>X</sup>*, *<sup>Y</sup>*, *<sup>Z</sup>* (8)

According to the López-Corona approach [50] for complexity, with normalized *Sk*

A: is the relative accuracy of the data before the expected noise begins to dominate.

and the Golden Surfer Software were used to fill incomplete time series [65].

apply, since they are predictable or deterministic systems.

λ: Exponent of Lyapunov (bits/units of time) [37, 47, 66].

**10.1. Chaotic attractor dynamics**

D: Encrusting dimension [37].

tends to persist in the future).

Notation:

60 Behavior Analysis

pair is selected.

zero) (**Graph 4**):

prediction is significant.

*Ci*

system.

→ *SKobs* : The values of C(t) for time series give results greater than zero for the three time series (X(t), Y (t), and Z (t)), which would characterize a complex system and is coincident with the result of the matrix of correlation.

In the learning process studied, applying Chaos Theory to time series arising from characteristic variables that are common to all learning processes, three particularities revealed entropic connectivity, irreversibility, and complexity. In the same way, the sustainability of the process is due to the quotient of the POS/NEG emotions, when positive influence exists (POS) in the case of the Chaotic Attractor Dynamics, which is the one that presents the highest achievement in meaningful learning (rudely indicated as performance). These learnings lead to the consequent formation of patterns of "desired" behavior. Increasing connectivity is increasing entropy, which to maintain the "beloved" order, entropy (negentropy) must be transferred to the environment (to the planet), is quantifiable evidence of such process, the increase of garbage and pollution. This corollary demonstrates the irreversibility of the process in the current narrative. Is it possible to modify this plot? Within the current narrative, it seems unlikely. A new form of relationship between ourselves and with nature must build. (Chaos theory does not admit complexity for the weak and middle attractor, i.e., it does not apply).
