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[[Image:Lyapunov-fractal-AB.png|thumb|Standard Lyapunov logistic fractal with iteration sequence AB, in the region [2, 4] × [2, 4].]]
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[[Image:Lyapunov-fractal-AABAB.png|thumb|Generalized Lyapunov logistic fractal with iteration sequence AABAB, in the region [2, 4] × [2, 4].]]
 
[[Image:lyapunov-fractal.png|thumb|Generalized Lyapunov logistic fractal with iteration sequence BBBBBBAAAAAA, in the growth parameter region (''A'',''B'') in [3.4, 4.0] × [2.5, 3.4], known as ''Zircon Zity''.]]
 
In [[mathematics]], '''Lyapunov fractals''' (also known as '''Markus–Lyapunov fractals''') are [[bifurcation theory|bifurcational]] [[fractal]]s derived from an extension of the [[logistic map]] in which the degree of the growth of the population, ''r'', periodically switches between two values ''A'' and ''B''.
 
A [[Aleksandr Lyapunov|Lyapunov]] fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the [[Lyapunov exponent]] <math>\lambda</math>) in the ''a''&minus;''b'' plane for given periodic sequences of ''a'' and ''b''. In the images, yellow corresponds to <math>\lambda < 0</math> (stability), and blue corresponds to <math>\lambda > 0</math> (chaos).
 
==Properties==
Lyapunov fractals are generally drawn for values of ''A'' and ''B'' in the interval <math>[0,4]</math>. For larger values, the interval [0,1] is no longer stable, and the sequence is likely to be attracted by infinity, although convergent cycles of finite values continue to exist for some parameters. For all iteration sequences, the diagonal ''a = b'' is always the same as for the standard one parameter logistic function.
 
The sequence is usually started at the value 0.5, which is a [[critical point (mathematics)|critical point]] of the iterative function. The other (even complex valued) critical points of the iterative function during one entire round are those that pass through the value 0.5 in the first round. A convergent cycle must attract at least one critical point{{Citation needed|date=April 2007}}; therefore all convergent cycles can be obtained by just shifting the iteration sequence, and keeping the starting value 0.5. In practice, shifting this sequence leads to changes in the fractal, as some branches get covered by others; notice for instance how the Lyapunov fractal for the iteration sequence AB is not perfectly symmetric with respect to ''a'' and ''b''.
 
==Algorithm for generating Lyapunov fractals==
An [[algorithm]], for computing the fractal is summarized as follows.
 
# Choose a string of As and Bs of any nontrivial length (e.g., AABAB).
# Construct the sequence <math>S</math> formed by successive terms in the string, repeated as many times as necessary.
# Choose a point <math>(a,b) \in [0,4] \times [0,4]</math>.
# Define the function <math>r_n = a</math> if <math>S_n = A</math>, and <math>r_n = b</math> if <math>S_n = B</math>.
# Let <math>x_0 = 0.5</math>, and compute the iterates <math>x_{n+1} = r_n x_n (1 - x_n)</math>.
# Compute the Lyapunov exponent:<br><math>\lambda = \lim_{N \rightarrow \infty} {1 \over N} \sum_{n = 1}^N \log \left|{dx_{n+1} \over dx_n}\right| = \lim_{N \rightarrow \infty} {1 \over N} \sum_{n = 1}^N \log |r_n (1 - 2x_n)|</math><br>In practice, <math>\lambda</math> is approximated by choosing a suitably large <math>N</math>.
# Color the point <math>(a,b)</math> according to the value of <math>\lambda</math> obtained.
# Repeat steps (3–7) for each point in the image plane.
 
==External links==
*[http://www.efg2.com/Lab/FractalsAndChaos/Lyapunov.htm EFG's Fractals and Chaos – Lyapunov Exponents]
*{{cite web |
last=Elert |
first=Glenn |
title= Lyapunov Space |
work=The Chaos Hypertextbook |
url=http://hypertextbook.com/chaos/44.shtml }}
 
{{Fractals}}
 
[[Category:Fractals]]

Revision as of 09:37, 27 February 2014

Weight problem is just bad for health it may have a bad impact on earning force. Simply 10% increase in body mass index , a measure of fat relative to height, can cut a man's real earnings by 3.3% and a woman's by 1.8%, Economists mentioned on wednesday.

Doctors and dietitians use body mass index (BMI) plus measurement of the waist to assess a person's risk of developing certain diseases, for example, diabetes plus heart disease. If your BMI is greater than 25, you're overweight. BMI equal to or greater than 30 indicates obesity.

Total body fat percentage consists of fat essential to the metabolism and also stored fat. Essential fat is indispensable inside terms of preserving life plus reproductive functions of the body. The important percentage is better for females than which for men because their bodies have to deal with all the demands of childbearing and additional hormonal functions. According to the accepted norms, imperative fat is 2-5% for guys and 10-13% for ladies. The minimum total body fat percentage recommended by medical professionals is a bit more than this necessary fat percentage. Storage fat refers to the fat accumulation in adipose tissue. Some of the adipose tissue is important for safeguarding internal organs in the chest plus abdomen plus it happens to be slightly high for ladies than for men.

There are a few drawbacks to the bmi calculator you need to be aware of. Older persons plus others that have lost muscle mass will get a false reading, considering average muscle is factored into the equation. Athletes will get the mistaken idea that they are overweight, when in actual fact they have a greater amount of lean tissue as opposed to fatty tissue.

In order to know where we fall found on the body fat charts, you need to calculate a body fat. There are several ways to do this, from a do it oneself skin fold test, to online calculators. The skin fold test is not (omit) because correct because alternative methods. Basically, we are pinching some skin at different body places, which is measured. This can show the changes inside body composition over time, whether or not it's not a totally correct method of calculating body fat.

To watch both videos under takes about 90 minutes. Here are the highlights of it. It shows 6 individuals in their house towns with diabetes going to Dr. Cousen's center. One older guy had his blood glucose brought down to usual nevertheless left after 2 week because the diet was too difficult to follow. We can eat as much as you want yet it is nevertheless difficult for individuals to follow.

Well, this would need time administration. There are a great deal of exercise programs available even online for you to do plus accomplish. It's all up to we. If inside case because of when you never have the time to do it, then perhaps doing a small adjustment in the daily escapades would be a wise begin.

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