Lyapunov diagram

Lyapunov fractal with the sequence ba, known as Lyapunov Space

Lyapunov fractal with the sequence bbbbbbaaaaaa, known as Zircon Zity

Lyapunov diagrams (after Alexander Michailowitsch Lyapunow ; also known as Lyapunov fractals or Markus Lyapunov fractals ) are fractals that result from a modification of the logistic equation . In contrast to the logistic equation, the growth rate of the population - - is not kept constant for each point, but is switched in periodic sequences (e.g. sequence "ABAAB") between two values and , with . ${\ displaystyle r}$ ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle 0 \ leq a, b \ leq 4}$

The logistic equation is

{\ displaystyle x_ {n + 1} = r_ {n} x_ {n} (1-x_ {n})}

with the usual starting value . In this example (sequence "ABAAB" with length 5) would ${\ displaystyle x_ {0} = 0 {,} 5}$ ${\ displaystyle r}$

{\ displaystyle r_ {0} = r_ {5} = \ ldots = r_ {5k + 0} = a}

,

{\ displaystyle r_ {1} = r_ {6} = \ ldots = r_ {5k + 1} = b}

,

{\ displaystyle r_ {2} = r_ {7} = \ ldots = r_ {5k + 2} = a}

,

{\ displaystyle r_ {3} = r_ {8} = \ ldots = r_ {5k + 3} = a}

,

{\ displaystyle r_ {4} = r_ {9} = \ ldots = r_ {5k + 4} = b}

to get voted.

This results in the following mathematical and design differences to the logistic equation:

Instead of one number, you have two numbers and choose. This results in a two-dimensional function instead of a one- dimensional function . ${\ displaystyle r}$ ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle f (r)}$ ${\ displaystyle f (a, b)}$
Therefore it is not the values of the series , , as a function , but rather just like the Mandelbrot set the convergence of the series as a map . ${\ displaystyle x_ {0}}$ ${\ displaystyle x_ {1}}$ ${\ displaystyle \ ldots}$ ${\ displaystyle r _ {\ mathrm {min}} \ leq r \ leq r _ {\ mathrm {max}}}$ ${\ displaystyle (a _ {\ mathrm {min}} \ leq a \ leq a _ {\ mathrm {max}}) \ times (b _ {\ mathrm {min}} \ leq b \ leq b _ {\ mathrm {max}} )}$
You have the sequence as a further design factor.

Then the iteration values of the logistic equation are calculated and the Lyapunov exponent is calculated for values (a, b) from intervals, which - in order to get interesting figures - are usually selected in the range and : ${\ displaystyle 0 \ leq a \ leq 4}$ ${\ displaystyle 0 \ leq b \ leq 4}$

{\ displaystyle \ 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}) |}

If the value is , one chooses for the point with the coordinates (a, b) z. B. yellow as the color, if it is greater than zero (which leads to exponential growth, chaos ), one chooses z. B. blue as color. Correspondingly, the color values can be graded depending on the size of . The result is the Lyapunov diagram, which is often fractal in nature. An example is the Zircon Zity diagram , formed with and and the sequence (bbbbbbaaaaaa). ${\ displaystyle \ lambda <0}$ ${\ displaystyle \ lambda}$ ${\ displaystyle 3 {,} 4 \ leq a \ leq 4 {,} 0}$ ${\ displaystyle 2 {,} 5 \ leq b \ leq 3 {,} 4}$