The logistic map¶
Introduced by the biologist Robert May in 1976, the logistic map turned out to be a lot more complex that meets the eye.
Definitions¶
The logistic map is a recursive sequence defined as:
where \(r\in [1,4]\) is the growth ratio and with \(x_0\in[0,1]\). The goal of this module is to be able tu visualize the logistic map, but also to create its bifurcation diagram and link it to the Mandelbrot set.
The bifurcation diagram is a plot of where the logistic map tends to. It shows us that as the growth ratio approaches \(4\), the chaotic behaviour appears.
Summary of the outputs¶
| Methods | Output | ||
|---|---|---|---|
| float | video | matplotlib plot | |
| animate_logistic | X | ||
| connections | X | ||
| logistic | X | ||
| logistic_draw | X | ||
| bifurcation | X | X | |
| logi_branch_points | X | ||
| plot_logi_interact | X | ||
-
animate_logistic(save=False)¶ Represents the evolution of the logistic map.
We can see the creation of the sequence step by step. The graph of the function is animated. It shows the convergence of the sequence when r is small enough, then, the chaotic representation.
- Parameters
save (boolean) – the ability to save the video in a file
- Returns
the matplotlib animation. If save==True, it saves the animation in ‘temp’ and in a .mp4 format
- Return type
plot the animated graph
-
bifurcation(show=True)¶ Builds the bifurcation diagramm of the logistic map.
The x-axis corresponds to the common ratio, r (in [0,4]), and the y-axis corresponds to the x points of the logistic function. This graph summarizes the different possibilities of the logistic draw according to r. The final graph represents bifurcations. When r is equal to 3, we see the first bifurcation.
- Parameters
show – True. Otherwise, the tuples of points coordinates.
- Returns
the bifurcation graph or the coordinates
- Return type
matplotlib graph or list of float tuples
-
connections()¶ Shows the link between Mandlebrot, logistic map and bifurcation.
This function shows the connections with the two Mandlebrot objects of this package: the Mandlebrot set and the logistic map. The moving red line shows the place that we are on these three graphs, at the same time. It saves the video in .avi in the ‘temp’ directory.
- Return type
Matplotlib animation and a save in a .avi format
-
logi_branch_points(x0, mu, nb_iter=100)¶ It creates of the logistic sequence points.
This function makes all coordinates of the sequence’s points and puts them in a list. It creates stairs too.
- Parameters
x0 (float) – the starting point included in [0,1]
mu (float) – the common ratio
- Nb_iter
the number of iterations of the sequence
- Returns
list of sequence points
- Return type
list of tuples
-
logistic(r, x)¶ Calculates values of the logisitc function.
- Parameters
r (float) – the common ratio
x (float) – the value of the research point
- Returns
the logistic value of x
- Return type
float
-
logistic_draw(x0, r, iteration, points)¶ First way to be able to draw the logistic sequence.
It draws the logistic function, the (y=x) line. With these lines, the logistic sequence can be drawn.
- Parameters
x0 (float) – the starting point included in [0,1]
r (float) – the common ratio
iteration (integer) – the number of iteration to draw these lines
points (integer) – the number of points between 0 and 1 on the x-axis
- Returns
the curves (y=x) and logistic function with the creation of the logistic sequence
- Return type
plot the graph
-
plot_logi_interact(x0, r, nb_iter=100, linsdim=100, colors='red')¶ Another way to plot the logistic function which is faster so more pleasant to use with the interaction.
This function creates the logistic sequence. Meaning, we start with x0, then we know the second point drawing the ‘stair’ with (y=x) and the logistic function. And we do it again, up to nb_iter. But here, we know all points thanks to the logi_branch_points function.
- Parameters
x0 (float) – the starting point included in [0,1]
r (float) – the growth ratio
nb_iter (integer) – the number of iterations of the sequence
linsdim (integer) – number of points between 0 and 1 on the x-axis
colors (string) – color for the cobweb
- Returns
graph representing logistic sequence
- Return type
matplotlib plot