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Miloslav Ciz 2023-12-07 19:58:18 +01:00
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chaos.md
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@ -16,7 +16,7 @@ A simple example of a chaotic equation is also the function *sin(1/x)* for *x* n
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**Logistic map** is often given as the typical example of a chaotic system. It is the series defined as *x[n + 1] = r * (1 - x[n])*, which for some constant *r* (interpreted as speed of population increase) says how a population evolves from some starting value *x[0]*; for low *x[n]* the population will be increasing proportionally by the rate of *r* but once it reaches a higher value, it will start decreasing (as if by starvation), resulting in oscillation. Now if we only start to be interested in changing the value *r* and then seeing at what value the population stabilizes (for a big *n*), we make some interesting discoveries. This is best seen by plotting the stable values (let's say *x[1000]*) depending on *r*. For *r* approximately between 3.57 and 4 we start to see a chaotic behavior, with results greatly depending on the initial population value (*x[0]*). This demonstrates chaotic behavior.
**Logistic map** is often given as the typical example of a chaotic system. It is the series defined as *x[n + 1] = r * x[n] * (1 - x[n])*, which for some constant *r* (interpreted as speed of population increase) says how a population evolves from some starting value *x[0]*; for low *x[n]* the population will be increasing proportionally by the rate of *r* but once it reaches a higher value, it will start decreasing (as if by starvation), resulting in oscillation. Now if we only start to be interested in changing the value *r* and then seeing at what value the population stabilizes (for a big *n*), we make some interesting discoveries. This is best seen by plotting the stable values (let's say *x[1000]*) depending on *r*. For *r* approximately between 3.57 and 4 we start to see a chaotic behavior, with results greatly depending on the initial population value (*x[0]*). This demonstrates chaotic behavior.
The following is a [fixed point](fixed_point.md) [C](c.md) implementation of the above: