Editing Logistic map (section)

=== Period-doubling route to chaos ===

In the logistic map, we have a function <math>f_r (x) = rx(1-x)</math>, and we want to study what happens when we iterate the map many times. The map might fall into a fixed point, a fixed cycle, or chaos. When the map falls into a stable fixed cycle of length <math>n</math>, we would find that the graph of <math>f_r^n</math> and the graph of <math>x\mapsto x</math> intersects at <math>n</math> points, and the slope of the graph of <math>f_r^n</math> is bounded in <math>(-1, +1)</math> at those intersections. 

For example, when <math>r=3.0</math>, we have a single intersection, with slope bounded in <math>(-1, +1)</math>, indicating that it is a stable single fixed point.

As <math>r</math> increases to beyond <math>r=3.0</math>, the intersection point splits to two, which is a period doubling. For example, when <math>r=3.4</math>, there are three intersection points, with the middle one unstable, and the two others stable.

As <math>r</math> approaches <math>r = 3.45</math>, another period-doubling occurs in the same way. The period-doublings occur more and more frequently, until at a certain <math>r\approx 3.56994567</math>, the period doublings become infinite, and the map becomes chaotic. This is the [[Period-doubling bifurcation|period-doubling route to chaos]].
<div class=skin-invert-image>{{multiple image
| align             = center
| direction         = horizontal
| total_width       = 620
| image1            = ロジスティック写像2回反復グラフの周期倍化分岐の様（a = 2.7）.png
| caption1          = Relationship between <math>x_{n+2}</math> and <math>x_{n}</math> when <math>a=2.7</math>. Before the period doubling bifurcation occurs. The orbit converges to the fixed point <math>x_{f2}</math>.
| image2            = ロジスティック写像2回反復グラフの周期倍化分岐の様（a = 3）.png
| caption2          = Relationship between <math>x_{n+2}</math> and <math>x_{n}</math> when <math>a=3</math>. The tangent slope at the fixed point <math>x_{f2}</math>. is exactly 1, and a period doubling bifurcation occurs.
| image3            = ロジスティック写像2回反復グラフの周期倍化分岐の様（a = 3.3）.png
| caption3          = Relationship between <math>x_{n+2}</math> and <math>x_{n}</math> when <math>a=3.3</math>. The fixed point <math>x_{f2}</math> becomes unstable, splitting into a periodic-2 stable cycle.
}}</div>

<div class=skin-invert-image>{{multiple image
| align             = center
| direction         = horizontal
| total_width       = 620
| image1            = Logistic map iterates, r=3.0.svg
| caption1          = When <math>r=3.0</math>, we have a single intersection, with slope exactly <math>+1</math>, indicating that it is about to undergo a period-doubling.
| image2            = Logistic iterates 3.4.svg
| caption2          = When <math>r=3.4</math>, there are three intersection points, with the middle one unstable, and the two others stable.
| image3            = Logistic iterates r=3.45.svg
| caption3          = When <math>r=3.45</math>, there are three intersection points, with the middle one unstable, and the two others having slope exactly <math>+1</math>, indicating that it is about to undergo another period-doubling.
| image4            = Logistic iterates with r=3.56994567.svg
| caption4          = When <math>r\approx 3.56994567</math>, there are infinitely many intersections, and we have arrived at [[Period-doubling bifurcation|chaos via the period-doubling route]].
| perrow            = 2/2
}}</div>