Fractale de Lyapunov

See also: Lyapunov

In Mathematical, the fractales of Lyapunov (known also under the name of Markus-Lyapunov ) are fractales obtained thanks to an extension of the logistic function defining the evolution of a normally definite population like:

$P\_\; \{i+1\}\; \backslash \; =\; \backslash \; K\; \backslash \; times\; P\_i\; \backslash \; times\; (1\; -\; P\_i)$, where K is the constant of growth of the population.

The population varies between 0 and 1.

Unidimensional study of the behavior of the logistic continuation

It is shown that this continuation converges towards a stable limit for values of K lower than 3, and diverges for values higher than 4 (where the continuation cannot be defined any more for an infinite number of elements because it would exceed its field of value). For the majority of the values of K between 3,57 (approximately) and 4 (excluded terminals) the continuation (which represents the evolution of a population in time) presents a chaotic behavior, independent of the starting value (the other values of K result in a continuation converging towards a constant cycle including/understanding a finished number of values). The chaotic character is what makes it possible to trace a figure fractale.

It is shown that the chaotic criterion of the continuation is obtained when the exhibitor of Lyapunov (which represents the average logarithm of growth of the population) calculated on the continuation is positive but lower than 1. When this exhibitor is negative, the population decrease and converges towards 0. When the exhibitor is higher than 1, the population believes infinitely.

To calculate it, one must define the function $f\_k$ transforming an element of the continuation into the following element:

$P\_\; \{i+1\}\; \backslash \; =\; \backslash \; f\_k\; (P\_i)$, for example for the logistic function $f\_k$:

$f\_k\; (p)\; \backslash \; =\; \backslash \; K\; \backslash \; times\; p\; \backslash \; times\; (1\; -\; p)$

Unidimensional function fractale

A curve fractale (chaotic) unidimensional is then obtained by the evaluation of the exhibitor of Lyapunov according to K . The exhibitor of Lyapunov of the continuation is easily calculable if the function $f\_k$ is derivable on the field of values of the continuation P , as follows:

$\backslash \; lambda\; (P\_0)\; \backslash \; =\; \backslash \; \backslash \; lim\_\; \{N\; \backslash \; rarr\; +\; \backslash \; infin\}\; \backslash \; frac\; \{1\}\; \{N\}\; \backslash \; sum\_\; \{I\; =\; 1\}\; ^\; \{N\}\; \{\backslash \; ln\; \backslash \; left|\; \{f\_k\}\; ^\; \backslash \; premium\; (P\_\; \{I\; -\; 1\})\; \backslash \; right|\}$, here

$\{f\_k\}\; ^\; \backslash \; premium\; (p)\; \backslash \; =\; \backslash \; K\; \backslash \; times\; (1\; -\; 2p)$

It should be noted however that the sum above is not always convergent and tends towards less the infinite one for fixed values of K , but it is continuous between these values. These points of discontinuity more and are dispersed chaotiquement in the zone of chaos of the continuation P where the exhibitor generally takes positive values. That means that there exists an infinity of intervals in this zone where the continuation P chaotic, is separated by an infinity from very small intervals for K where the exhibitor of Lyapunov takes negative values and where the continuation P tends towards a cycle (all the more quickly as this exhibitor is strongly negative).

To accelerate calculations of the exhibitor of Lyapunov on great values of N (and to increase the precision of the result), one will be able to group the elements of the sum by groups of finished size and by carrying out only their product, as long as this product does not exceed the limiting terminals of precision of the result, and by summoning only the logarithms of the product of each group. (Within each product it is not necessary to evaluate the absolute value of each term, the absolute value being able to be deferred right before the evaluation of the logarithm).

Obtaining a two-dimensional fractale

The chaotic behavior can be calculated as simply by applying the logistic function successively but by recurringly alternating the values of the degree K of the growth of the population, in a not using cycle as two values has and B , this periodicity names root. For example, for the sequence root ( has , B ) of periodicity 2, one obtains the continuation:

$P\_\; \{2n+1\}\; \backslash \; =\; \backslash \; has\; \backslash \; times\; P\_\; \{2n\}\; \backslash \; times\; (1\; -\; P\_\; \{2n\})$

$P\_\; \{2n+2\}\; \backslash \; =\; \backslash \; B\; \backslash \; times\; P\_\; \{2n+1\}\; \backslash \; times\; (1\; -\; P\_\; \{2n+1\})$

by fixing the initial values arbitrarily $P\_0$ and $P\_1$ in the field of value] 0,1 where the continuation is not constant and null as of the first elements.

If one considers only the under-continuation of the elements of even index (i.e. multiple of the period of the cycle of constant), it can be expressed simply using a function $f\_\; \{has,\; B\}$ single:

$f\_\; \{has,\; B\}\; (X)\; \backslash \; =\; \backslash \; f\_b\; (f\_a\; (X))$, and $P\_\; \{2n+2\}\; \backslash \; =\; \backslash \; f\_\; \{has,\; B\}\; (P\_\; \{2n\})$

And the chaotic character or not of the new continuation is calculated in the same way with the exhibitor of Lyapunov applied this time to the function $f\_\; \{has,\; B\}$ characteristic of the under-continuation (and of the very whole continuation $P$…). It is enough that $f\_\; \{has,\; B\}$ is derivable, which is the case because the functions $f\_a$ (or $f\_b$) are derivable for any value of the constants $a$ or $b$.

To draw the fractale, one defines a rectangle whose horizontal axis corresponds to the parameter has and the vertical axis in the parameter B . For each point ( has , B ), one calculates the exhibitor of Lyapunov. One allots a color to this point according to the result (and a color distinct in the case or the evaluation from the exhibitor of Lyapunov does not converge towards a finished value). One obtains a figure similar to the first image fractale above (where the colors of the points on the principal diagonal correspond to the unidimensional development fractal of the exhibitor of Lyapunov on the logistic continuation presented previously).

It will be noted that the figure has a relative symmetry on both sides diagonal, but this symmetry is not perfect because the order of use of the constants has and B in the continuation P is shifted of a position, which affects the total sum of the exhibitor of Lyapunov (approximate on a number necessarily finished of elements), the continuations differing mainly by their first term considered.

On the other hand, the images obtained differ little in their structure according to the value defined for the first element $P\_0$ of the continuation; in practice, one will often choose $P\_0\; =\; 0,5$.

Others fractales similar two-dimensional can be obtained by modifying the reason of the cycle root, or at higher intervals (but one increases the degree of the polynomial F , and thus the computing time of the image). For example with the cycle of period 5 ( has , has , B , has , B ) as in the second image fractale above.

One can also obtain fractales of higher size by increasing the number of parameters, for example with the cycle ( has , B , C ), and which one can use a projection in an unspecified plan to obtain a two-dimensional image (for example by fixing C , which allows an infinity of images fractales according to the value of C ).

These types of fractales model well for example the development of pluricellular organizations or the crystal growth, subjected to concentrations of nutrients or thermal or luminous exposures graduated, or variable fields of force according to space, and can explain the formation of geometrical figures regular and fractales .

External bonds

• complete Site on the fractales of Lyapunov
• Z-Lyapunov, excels program making it possible to calculate the fractales of Lyapunov
• Second program, perhaps slightly faster than the precedent, and with the advantage of placing at the disposal its sources, in Delphi 4

Random links:

Jacques II of England | Tournon-saint-stone | (217) Eudora | Paradorn Srichaphan | Daniel Lesueur | Sergio Barroso | Cormorant

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org