iterations : keyword option of the form iterations=val, where val is a posint specifying the number of times the function defined by exprs will be applied. The default value is 10000.

outputimage : keyword option of the form outputimage=img where img is a color image as used by the ImageTools package, with dimensions of height and width. Attractor will fill img with the generated image defined below.

height : keyword option of the form height=val, where val is a positive integer specifying the height of the image. If outputimage is defined the default value is the height of outputimage. Otherwise, the default value is 500.

width : keyword option of the form width=val, where val is a positive integer specifying the width of the image. If outputimage is defined the default value is the width of outputimage. Otherwise, the default value is 500.

xmin : keyword option of the form xmin=val, where val is a real number specifying the minimum value displayed on the vertical axis. The default value is ${\mathrm{init}}_1-1$

xmax : keyword option of the form xmax=val, where val is a real number specifying the maximum value displayed on the vertical axis. The default value is ${\mathrm{init}}_1+1$

rvariable : keyword option of the form rvariable=param, where param is a name specifying the parameter in exprs that will be varied along the horizontal axis. The default value is $r$

loopvariable : keyword option of the form loopvariable=var, where var is a name specifying the variable representing the current iteration index. The default value is $i$

redvariable : keyword option of the form redvariable=var, where var is a name specifying the red color variable at the current position in the image. The default value is $R$

greenvariable : keyword option of the form greenvariable=var, where var is a name specifying the green color variable at the current position in the image. The default value is $G$

bluevariable : keyword option of the form bluevariable=var, where var is a name specifying the blue color variable at the current position in the image. The default value is $B$

redexpression : keyword option of the form redexpression=expr, where expr is an expression defining the coloring procedure of the red color value at the current position the image. If redexpression=$\mathrm{null}$ then no procedure is applied to the red value at the current point. The default value is $R+1$

greenexpression : keyword option of the form greenexpression=expr, where expr is an expression defining the coloring procedure of the green color value at the current position the image. If greenexpression=$\mathrm{null}$ then no procedure is applied to the green value at the current point. The default value is $\mathrm{null}$

blueexpression : keyword option of the form blueexpression=expr, where expr is an expression defining the coloring procedure of the blue color value at the current position the image. If blueexpression=$\mathrm{null}$ then no procedure is applied to the blue value at the current point.The default value is $\mathrm{null}$

The Bifurcation module generates an Array image for an iterative function (rvariable against the first variable in vars) being applied iter times, to a domain of equally spaced rvariable values.

Within the domain [rmin, rmax], rvariable values will be selected at equally spaced intervals.

For each r value, starting at the point $X_0$=init in ${ℝ}^N$, where N is the number of variables in vars, the sequence $X_n$=$F\left(X_{n-1}\right)$, 0 $\le n\le$iterations, is generated.

The function $F:{ℝ}^n\to {ℝ}^n$, is defined by the following algorithm:

Start at the point $X_n=\left(X_{n,1}\,\mathrm{..}\,X_{n,N}\right)$, $j=1$

Apply the function $F_j$ to the current point: $y=F_j\left(X_n\right)$

Replace the jth component of $X_n$ with $y$: $X_n=\left(X_{n,1}\,\mathrm{..}\,X_{n,j-1}\,y\,\mathrm{..}\,X_{n,N}\right)$

Increment j by 1, $j=j+1$ and repeat the process with the new $X_n$. Stop when $j=N+1$, where $N$ is the number of variables in vars.

Each $F_j$ is defined by evaluating ${\mathrm{exprlist}}_j$ at the current values of the variables in vars.

If given an exprlistlist, then one of the functions in exprlistlist will be randomly selected to be applied to $X_n$, for each 0 $\le n\le$iterations-1. The probability that the ith function defined in exprlistlist is applied to $X_n$ is equal to $\frac{{\mathrm{probabilities}}_i}{\sum _{i=1}^N{\mathrm{probabilities}}_i}$.

Whenever this process yields a point $X_n$, the values rvariable, $X_{n,1}$ are examined to find which pixel in the image they correspond to. Then, the expressions $\mathrm{redexpression}$, $\mathrm{greenexpression}$, $\mathrm{blueexpression}$ are evaluated, substituting the values $X_{n,1},\mathrm{...},X_{n,N}$ for the variables ${\mathrm{vars}}_1,\mathrm{...},{\mathrm{vars}}_N$, and the current red, green, and blue values of that pixel for the variables $\mathrm{redvariable}$, $\mathrm{greenvariable}$, and $\mathrm{bluevariable}$, respectively.

The red, green, and blue values of the pixel are then updated to the newly evaluated values.

If a coloring expression is $\mathrm{null}$ then the color, corresponding to the expression, of the pixel will be unchanged

Note that redexpression, greenexpression, blueexpression are applied to an image as defined by ImageTools:-Image, thus the minimum and maximum color values are 0 and 1 respectively.

Bifurcation returns an image containing the diagram described above. Further coloring procedures, defined in ImageTools, can be applied to the image.

The default $\mathrm{redexpression}$, $\mathrm{greenexpression}$, $\mathrm{blueexpression}$ return an image that is designed to be used with ImageTools:-ColouringProcedures:-HueToRGB.

$\mathrm{with}\left(\mathrm{IterativeMaps}\right)\:$

$\mathrm{with}\left(\mathrm{ImageTools}\right)\:$

$\mathrm{Logistic}≔\mathrm{Bifurcation}\left(\left[x\right]\,\left[rx\left(1-x\right)\right]\,\left[0.5\right]\,2.5\,4\right)\:$

$\mathrm{ArrayTools}:-\mathrm{Dimensions}\left(\mathrm{Logistic}\right)$

$\left[1..500\,1..500\,1..3\right]$

$\mathrm{ColouringProcedures}:-\mathrm{HueToRGB}\left(\mathrm{Logistic}\right)\:$

$\mathrm{Embed}\left(\mathrm{Logistic}\right)$

$\mathrm{LogisticZoom}≔\mathrm{Bifurcation}\left(\left[x\right]\,\left[rx\left(1-x\right)\right]\,\left[0.5\right]\,3.7\,3.8\,\mathrm{xmin}=0.15\,\mathrm{xmax}=0.95\,\mathrm{width}=160\,\mathrm{height}=500\right)\:$

$\mathrm{ArrayTools}:-\mathrm{Dimensions}\left(\mathrm{LogisticZoom}\right)$

$\left[1..500\,1..160\,1..3\right]$

$\mathrm{ColouringProcedures}:-\mathrm{HueToRGB}\left(\mathrm{LogisticZoom}\right)\:$

$\mathrm{Embed}\left(\mathrm{LogisticZoom}\right)$

$\mathrm{LogisticTentMix}≔\mathrm{Bifurcation}\left(\left[x\right]\,\left[\left[rx\left(1-x\right)\right]\,\left[r\min \left(x\,1-x\right)\right]\right]\,\left[0.5\right]\,1\,2\,\left[\frac{1}{2}\,\frac{1}{2}\right]\right)\:$

$\mathrm{ArrayTools}:-\mathrm{Dimensions}\left(\mathrm{LogisticTentMix}\right)$

$\left[1..500\,1..500\,1..3\right]$

$\mathrm{ColouringProcedures}:-\mathrm{HueToRGB}\left(\mathrm{LogisticTentMix}\right)\:$

$\mathrm{Embed}\left(\mathrm{LogisticTentMix}\right)$

$\mathrm{xD}≔\mathrm{Step}\left(-y-z\right)$

$\mathrm{xD}≔\mathrm{Step}\left(-y-z\right)$

$\mathrm{xProc}≔x+\mathrm{xD}$

$\mathrm{xProc}≔x+\mathrm{Step}\left(-y-z\right)$

$\mathrm{yD}≔\mathrm{Step}\left(\mathrm{tx}+0.2y\right)$

$\mathrm{yD}≔\mathrm{Step}\left(\mathrm{tx}+0.2y\right)$

$\mathrm{yProc}≔y+\mathrm{yD}$

$\mathrm{yProc}≔y+\mathrm{Step}\left(\mathrm{tx}+0.2y\right)$

$\mathrm{zD}≔\mathrm{Step}\left(0.2+z\left(\mathrm{tx}-r\right)\right)$

$\mathrm{zD}≔\mathrm{Step}\left(0.2+z\left(\mathrm{tx}-r\right)\right)$

$\mathrm{zProc}≔z+\mathrm{zD}$

$\mathrm{zProc}≔z+\mathrm{Step}\left(0.2+z\left(\mathrm{tx}-r\right)\right)$

$\mathrm{Step}≔0.01$

$\mathrm{Step}≔0.01$

$\mathrm{Rossler}≔\mathrm{Bifurcation}\left(\left['a'\,x\,y\,z\,\mathrm{tx}\,\mathrm{ty}\right]\,\left[\left[x^2+y^2\,\mathrm{xProc}\,\mathrm{yProc}\,\mathrm{zProc}\,x\,y\right]\right]\,\left[0.1\,0.1\,0.1\,0.1\,0.1\,0.1\right]\,0\,8\,\mathrm{xmin}=0\,\mathrm{xmax}=300\,\mathrm{iterations}=30000\right)\:$

$\mathrm{ArrayTools}:-\mathrm{Dimensions}\left(\mathrm{Rossler}\right)$

$\left[1..500\,1..500\,1..3\right]$

$\mathrm{ColouringProcedures}:-\mathrm{HueToRGB}\left(\mathrm{Rossler}\right)\:$

$\mathrm{Embed}\left(\mathrm{Rossler}\right)$

The IterativeMaps:-Bifurcation command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.