**8. Irreversibility and the sustainability of learning**

The position of Ilya Prigogine [52, 53] on irreversibility and entropy varies that of traditional physics. In his lecture The Birth of Time (Rome, 1987), Prigogine argued:

"Entropy always contains two dialectical elements: a creator element of disorder, but also a creative element of order. (…) We see, then, that instability, fluctuations, and irreversibility play a role at all levels of nature: chemical, ecological, climatological, biological - with the formation of biomolecules - and finally cosmological".

In a three-dimensional continuous dissipative dynamical system, the only possible spectra, and the attractors they describe, are as follows: (+,0,−), a strange attractor; (0,0,−), a two-torus;

> \_\_1 *<sup>T</sup>* ∑ *n*=1 .*L*

H is used as a measure of long-term memory of time series. It refers to the autocorrelations of the time series, and the speed at which they decrease as the gap between pairs of values increases. The inverse of the Hurst exponent is equal to the fractal dimension of a time series.

0.5 < H < 1, it is a persistent series, graphically presents a smooth appearance. H ≈ 1 indicates

H < 0.5, corresponds to anti-persistence, contrary to long-range dependency (LRD), indicates

0 < H < 0.5 indicates that the time series, in the long term, change high and low values of adja-

H is an index for the categorization of complexity, quantifies the chaotic dynamics, and is

The Embedding Theorem serves to remake from the observed or measured time series, the evolution of the states in the phase space, where the exponents of Lyapunov and the fractal dimension can be calculated (for example). It uses the method of delayed coordinates (recon-

If the system is random, the fractal dimension grows, as the dimension of the embedding

If the system is periodic, the fractal dimension grows to a value k and then remains constant

If the system is chaotic, the fractal dimension stabilizes for a certain embedding dimension p.

, x2 , x3 , x4

. The dynamics of the empirical system represented by the "minimum" dynamics (in

ln|*<sup>d</sup> <sup>x</sup>* \_\_\_\_\_\_\_\_\_\_ *<sup>n</sup>*+<sup>1</sup>

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

*<sup>d</sup> xn* <sup>|</sup> (5)

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

57

… xn, we can form the set of points

…, xp+i). These points determine a trajectory in the

The equation that allows calculating the coefficient of Lyapunov is given by:

(0,−,−), a limit cycle; and (−,−,−), a fixed point [55].

H ≈ 0.5 indicates the absence of long-term dependence [57].

that the degree of persistence or long-term dependence holds.

a strong negative correlation of the process that fluctuates violently.

cent pairs of data; this tendency remains to fluctuate for a long time [57–59].

directly related to the fractal dimension, D, where 1 < D < 2, such that D = 2–H.

, x<sup>i</sup> <sup>+</sup> <sup>1</sup>

*λ* = lim*<sup>T</sup>*→<sup>∞</sup>

**9.2. The exponent of Hurst, H**

H takes values between 0 and 1:

**9.3. Embedding dimension**

…, xp), (x<sup>2</sup>

space increases, that is, p.

and whole (it is not fractal).

, x3

(x1 , x2

space R<sup>P</sup>

struction with delays). If we have the data series x<sup>1</sup>

) …, and (x<sup>i</sup>

Also, at least one exponent of Lyapunov will be positive.

…, xp <sup>+</sup> <sup>1</sup>

a dimensional sense) of this set of points:

In this way, it was observed that the phenomenon of irreversibility for Prigogine is constructive, highlighting the "creative role of time," which, at least at a macroscopic level, supposes a kind of antientropy: "*the universe of non-equilibrium is a connected universe.*".

According to Wackernagel et al. [11]:

"Sustainability requires that life is within the regeneration capacity of the biosphere. In an attempt to measure the degree to which humanity satisfies this requirement, existing data have been used to translate human demand on the environment in the area required for the production of food and other goods, as well as in the absorption of waste. Numerical estimates indicate that human demand may well have outgrown the regenerative capacities of the biosphere since the 1980s. According to this preliminary and exploratory evaluation, the carrying capacity of humanity corresponds to 70% of that of the world biosphere in 1961, growing up to 120% in 1999".

All processes are irreversible because they are connected entropically making the complexity increases. Human activity is not exempt from this principle.

Chaotic systems consume considerable energy and information to maintain their level of complexity while being very sensitive to environmental fluctuations [54].
