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Fractals[EscapeTime]

 Lyapunov
 Lyapunov fractal generator

 Calling Sequence Lyapunov( n, ab, xrng, yrng ) Lyapunov( n, ab, xrng, yrng, opts )

Parameters

 n - : positive integer specifying the dimensions of the square Array output ab - : {list,Vector}(identical(0,1)) data specifying the alternating conditions xrng - : realcons..realcons the range for the horizontal domain yrng - : realcons..realcons the range for the vertical domain opts - : (optional) keyword options of the form opt=value where opt and value are described below

Options

 • output : keyword option of the form output=value where value is a name or list of names denoting the returned Array image(s). The accepted names are color, or raw. The default value is color.
 • numterms : keyword option of the form numterms=value where value is a positive integer specifying the number of terms used in the summation that approximates the Lyapunov exponent at each input point The default value is 1000.
 • container : An n-by-n-by-2 Array with datatype=float[8] and order=Fortran_order used in-place to store the raw data.

Description

 • The Lyapunov command generates Array images which provide a visualization of a bifurcational fractal that maps regions of chaos and stability given a binary-valued sequence. The particular binary pattern determines the value of r in the iterative formula,

${z}_{i+1}={r}_{i}{z}_{i}\left(1-{z}_{i}\right)$

 • The binary pattern dictates whether r[i] is taken as the x or the y value of a given point in the real plane. The values of z and r are used in computing the Lyapunov exponent according to the following definition.

$\mathrm{lambda}=\underset{n→\mathrm{infinity}}{{lim}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{{\sum }_{i=1}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\mathrm{ln}\left(\left|{r}_{i}\left(1-2{z}_{i}\right)\right|\right)}{n}$

 • The above limit is approximated using the finite sum of numterms individual terms. In regions where lambda>0 the behavior is chaotic, while it is stable in regions where lambda<0.
 • The 2-D grayscale Array image returned by supplying the option output=raw contains the computed values of lambda.
 • The 3-D color Array image returned by supplying the option output=color contains data where the three layers corresponding to red, green, and blue have been computed using the grayscale raw results.

Examples

 > $\mathrm{with}\left(\mathrm{Fractals}:-\mathrm{EscapeTime}\right)$
 $\left[{\mathrm{BurningShip}}{,}{\mathrm{Colorize}}{,}{\mathrm{HSVColorize}}{,}{\mathrm{Julia}}{,}{\mathrm{LColorize}}{,}{\mathrm{Lyapunov}}{,}{\mathrm{Mandelbrot}}{,}{\mathrm{Newton}}\right]$ (5.1)
 > $\mathrm{with}\left(\mathrm{ImageTools}\right):$
 > $n≔200:$
 > $L≔\mathrm{Lyapunov}\left(n,\left[1,0,0,1,1,0\right],2.1..3.9,2.1..3.9\right)$
 ${L}{≔}\left[\begin{array}{c}{\mathrm{1..200 x 1..200 x 1..3}}{\mathrm{Array}}\\ {\mathrm{Data Type:}}{{\mathrm{float}}}_{{8}}\\ {\mathrm{Storage:}}{\mathrm{rectangular}}\\ {\mathrm{Order:}}{\mathrm{C_order}}\end{array}\right]$ (5.2)
 > $\mathrm{Embed}\left(L\right)$

 > $L≔\mathrm{Lyapunov}\left(n,\mathrm{Vector}\left(\left[1,1,0,0,0,0,0,1,1,1\right]\right),2.1..3.9,2.1..3.9,\mathrm{output}=\mathrm{raw}\right):$
 > $R≔\mathrm{Array}\left(1..n,1..n,1..3,\mathrm{datatype}=\mathrm{float}[8],\mathrm{order}=\mathrm{C_order}\right):$
 > $\mathrm{LColorize}\left(L,50,500,300,\mathrm{container}=R\right):$
 > $\mathrm{Embed}\left(\left[\mathrm{FitIntensity}\left(L\right),R\right]\right)$

 > $\mathrm{randomize}\left(\right):$
 > $p≔\mathrm{rand}\left(350..450\right):$
 > $Q≔\left\{\left[0,0,0,0,1,1\right],\left[0,0,1,0,0,1\right],\left[0,1,0,0,0,1\right],\left[0,1,1,0,0,1\right]\right\}$
 ${Q}{≔}\left\{\left[{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{1}\right]{,}\left[{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{1}\right]{,}\left[{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{1}\right]{,}\left[{0}{,}{1}{,}{1}{,}{0}{,}{0}{,}{1}\right]\right\}$ (5.3)
 > $L≔\left[\mathrm{seq}\left(\left[\mathrm{seq}\left(\mathrm{LColorize}\left(\mathrm{Complement}\left(\mathrm{Lyapunov}\left(300,{Q}_{i+\left(k-1\right)\cdot 2},2.1..3.9,2.1..3.9,\mathrm{output}=\mathrm{raw}\right)\right),p\left(\right),2000,p\left(\right)\right),i=1..2\right)\right],k=1..2\right)\right]:$
 > $\mathrm{Embed}\left(L\right)$

 > $\mathrm{Embed}\left(\left[\mathrm{seq}\left(\mathrm{Lyapunov}\left(200,\left[1,0,0,1,1,0\right],2.1..3.9,2.1..3.9,\mathrm{numterms}=i+10\right),i=0..600,200\right)\right]\right)$

 > 

Compatibility

 • The Fractals:-EscapeTime:-Lyapunov command was introduced in Maple 18.