plot::Iteration – plotting iterated functions

plot::Iteration(f, x0, n, x = $`x_{min}`..`x_{max}`$ ) is a graphical object visualizing the iteration () of the given starting point .

→ Examples

Call:

plot::Iteration(f, x0, <n>, x = $`x_{min}`$ .. $`x_{max}`$ , <a = amin .. amax>, Options)

Parameters:

f:	the iteration function: an arithmetical expression in the independent variable x and the animation parameter a. Alternatively, a MuPAD procedure that accepts 1 input parameter x or 2 input parameters x, a and returns a real numerical value when the input parameters are numerical. f is equivalent to the attribute Function.
x0:	the starting point for the iteration: x0 must be a numerical real value or an expression in the animation parameter a. x0 is equivalent to the attribute StartingPoint.
n:	the number of iterations: n must be a positive integer or an expression in the animation parameter a. n is equivalent to the attribute Iterations.
x:	the independent variable: an identifier or an indexed identifier. x is equivalent to the attribute XName.
$`x_{min}`$ .. $`x_{max}`$ :	the plot range: $`x_{min}`$ , $`x_{max}`$ must be numerical real values or expressions in the animation parameter . $`x_{min}`$ .. $`x_{max}`$ is equivalent to the attributes XRange, XMin, XMax.

See Also:

plot, plot::copy, plot::Lsys

Details:

plot::Iteration(f, x0, n, x = $`x_{min}`..`x_{max}`$ ) visualizes the iteration for starting with the point .

The iteration is visualized by connecting the points and by a vertical line. For any step of the iteration, a horizontal line is drawn from the point (on the graph of ) to the point on the main diagonal. From there, a vertical line is drawn to the next pair of the iteration.
The iteration object neither includes the graph of the function nor the main diagonal . You need to plot them separately if you wish the function and/or the diagonal to be in your picture! See the examples.
The iteration is stopped prematurely when the iterated point leaves the plot range $`x_{min}`..`x_{max}`$ . Cf. example 3.
Despite the fact that the number of iterations n represents an integer, it can be animated! Cf. example 4
The default color used for the iteration plot is RGB::Grey50. It can be modified by setting the attribute Color or LineColor. Cf. example 1.
The default line style is solid. It can be modified by setting the attribute LineStyle.

Example 1

We consider the logistic map for the parameter value , i.e., the parabola for . We iterate the starting point :

f := plot::Function2d(3*x*(1 - x), x = 0..1,
Color = RGB::Blue):
x0 := 0.5:

We plot the iteration (without specifying the number of iterations), the parabola and the diagonal line :

g := plot::Function2d(x, x = 0..1, Color = RGB::Red):
it := plot::Iteration(3*x*(1 - x), x0, x = 0..1):
plot(f, g, it)

MuPAD graphics

We increase the number of iterations to and change the color of the lines to RGB::Black:

it::Iterations := 50:
it::Color := RGB::Black:
plot(f, g, it)

MuPAD graphics

Finally, we animate the number of steps, allowing to follow the course of the iteration:

it := plot::Iteration(3*x*(1 - x), x0, n, x = 0..1,
n = 1..50, Color = RGB::Black):
plot(f, g, it)

MuPAD graphics

delete f, g, it:

Example 2

We consider the logistic map for and the animation parameter running from to :

f := plot::Function2d(a*x*(1 - x), x = 0..1, a = 2..4,
Color = RGB::Black):

We define the iteration of the starting point by and plot it together with the function graph of and the diagonal line :

g := plot::Function2d(x, x = 0..1, Color = RGB::Black):
it1 := plot::Iteration(a*x*(1 - x), 0.2, 30, x = 0..1,
a = 2..4, Color = RGB::Red):
plot(f, g, it1)

MuPAD graphics

We define an additional iteration starting at and add it to the plot:

it2 := plot::Iteration(a*x*(1 - x), 0.21, 30, x = 0..1,
a = 2..4, Color = RGB::Blue):
plot(f, g, it1, it2)

MuPAD graphics

For small values of , the two iterations converge to the same fixed point. When approaches the value , the iterations drift into chaos.

delete f, g, it1, it2:

Example 3

Consider the iteration of the starting point by the logistic map with the plot range :

f := plot::Function2d(x*(x - 1), x = 0..1):
it := plot::Iteration(x*(x - 1), 0.2, x = 0..1):
plot(f, it)

MuPAD graphics

We see that only one step of the iteration is plotted. The reason is that the point is negative and, hence, not contained in the requested plot range x = 0..1. We modifiy the plot range:

f::XRange:= -0.5..1:
it::XRange:= -0.5..1:
plot(f, it)

MuPAD graphics

delete f, it:

Example 4

We animate the parameter that sets the number of iterations. We set the time range for the animation to 40 (seconds). Using Frames, the total number of frames is chosen such that approximately 10 frames are used to visualize the step from to :

f := plot::Function2d(4*x*(1 - x), x = 0..1):
g := plot::Function2d(x, x = 0..1):
it := plot::Iteration(4*x*(1 - x), 0.4, n, x = 0..1,
                      LineStyle = Dashed,
                      n = 0..40, Frames = 411,
                      TimeRange = 0..40):
plot(f, g, it)

MuPAD graphics

delete f, g, it:

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