**9.1. The coefficient of Lyapunov**

The standard procedure of determining whether or not a system is chaotic is through the exponents of Lyapunov represented by λ. When considering two nearest points in a stage n, x<sup>n</sup> and xn + d xn, in the next temporal stage, they will diverge, particularly at x n+1 and x n+1 + d x n+1. It is this average ratio of divergence (or convergence) that the exponents of Lyapunov capture. Another way of thinking about the exponents of Lyapunov is as a proportion in which the information about the initial conditions lashes. There are so many exponents of Lyapunov as a dimension of phase space.

The signs of the Lyapunov exponents, λ, provide a qualitative picture of a system's dynamics. One-dimensional maps are characterized by a single Lyapunov exponent.

If the exponent of Lyapunov is positive, λ > 0, then the system is chaotic and unstable [55, 56]. Next points will diverge regardless of how close they are. Although there is no order, the system is still deterministic. The magnitude of Lyapunov exponents is a measure of sensitivity to initial conditions, the primary characteristic of a chaotic system.

If λ < 0, then the system is attracted to a fixed point or stable periodic orbit [55]. The absolute value of the exponents indicates the degree of stability.

If λ = 0, the system is in a marginally stable orbit [55].

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

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

$$\lambda = \lim\_{T \to \infty} \frac{1}{T} \sum\_{n=1}^{L} \ln \left| \frac{d \,\, x\_{n+1}}{d \,\, x\_n} \right| \tag{5}$$

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

play a role at all levels of nature: chemical, ecological, climatological, biological - with the

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

"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 cor-

All processes are irreversible because they are connected entropically making the complexity

Chaotic systems consume considerable energy and information to maintain their level of

The standard procedure of determining whether or not a system is chaotic is through the exponents of Lyapunov represented by λ. When considering two nearest points in a stage n, x<sup>n</sup> and xn + d xn, in the next temporal stage, they will diverge, particularly at x n+1 and x n+1 + d x n+1. It is this average ratio of divergence (or convergence) that the exponents of Lyapunov capture. Another way of thinking about the exponents of Lyapunov is as a proportion in which the information about the initial conditions lashes. There are so many exponents of Lyapunov as

The signs of the Lyapunov exponents, λ, provide a qualitative picture of a system's dynamics.

If the exponent of Lyapunov is positive, λ > 0, then the system is chaotic and unstable [55, 56]. Next points will diverge regardless of how close they are. Although there is no order, the system is still deterministic. The magnitude of Lyapunov exponents is a measure of sensitivity to

If λ < 0, then the system is attracted to a fixed point or stable periodic orbit [55]. The absolute

responds to 70% of that of the world biosphere in 1961, growing up to 120% in 1999".

complexity while being very sensitive to environmental fluctuations [54].

One-dimensional maps are characterized by a single Lyapunov exponent.

initial conditions, the primary characteristic of a chaotic system.

value of the exponents indicates the degree of stability. If λ = 0, the system is in a marginally stable orbit [55].

kind of antientropy: "*the universe of non-equilibrium is a connected universe.*".

formation of biomolecules - and finally cosmological".

increases. Human activity is not exempt from this principle.

**9. Some general mathematical concepts**

**9.1. The coefficient of Lyapunov**

a dimension of phase space.

According to Wackernagel et al. [11]:

56 Behavior Analysis

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. H takes values between 0 and 1:

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

0.5 < H < 1, it is a persistent series, graphically presents a smooth appearance. H ≈ 1 indicates that the degree of persistence or long-term dependence holds.

H < 0.5, corresponds to anti-persistence, contrary to long-range dependency (LRD), indicates a strong negative correlation of the process that fluctuates violently.

0 < H < 0.5 indicates that the time series, in the long term, change high and low values of adjacent pairs of data; this tendency remains to fluctuate for a long time [57–59].

H is an index for the categorization of complexity, quantifies the chaotic dynamics, and is directly related to the fractal dimension, D, where 1 < D < 2, such that D = 2–H.
