Editing Logistic map (section)

==== When r > 4 ====
[[File:A=4.5のロジスティック写像.png|class=skin-invert-image|thumb|For the logistic map with r = 4.5, trajectories starting from almost any point in [0, 1] go towards minus infinity.]]

When the parameter r exceeds 4, the vertex r /4 of the logistic map graph exceeds 1.<!--[ 238 ]--> To the extent that the graph penetrates 1, trajectories can escape [0, 1]<!--[ 238 ]--.. As a result, trajectories that start from almost any point in [0, 1] will at some point escape [0, 1] and eventually diverge to minus infinity <!/--[ 239 ]-->.

The bifurcation at r = 4 is also a type of crisis, specifically a boundary crisis.<!--[ 240 ]--> In this case, the attractor at [0, 1] becomes unstable and collapses, and since there is no attractor outside it, the trajectory diverges to infinity.<!--[ 240 ]-->

On the other hand, there are orbits that remain in [0, 1] even if r > 4.<!--[ 241 ]--> Easy-to-understand examples are fixed points and periodic points in [0, 1], which remain in [0, 1].<!--[ 241 ]--> However, there are also orbits that remain in [0, 1] other than fixed points and periodic points.<!--[ 242 ]-->

Let <math>A_0</math> be the interval of x such that f  (x) > 1. As mentioned above,once a variable <math>x_n</math> enters <math>A_0</math>, it diverges to minus infinity. There is also <math>r_n</math> x in [0, 1] that maps to <math>A_0</math> after one application of the map. This interval of x is divided into two, which are collectively called <math>A_1</math>. Similarly, there are four intervals that map to <math>A_1</math> after one application of the map, which are collectively called <math>A_2</math>. Similarly,there are 2n intervals <math>A_n</math> that reach <math>A_0</math> after n iterations.<!--[ 243 ]--> Therefore, the interval <math>\Lambda</math> obtained by removing <math>A_n</math> from [0, 1] an infinite number of times as follows is a collection of orbits that remain in I.<!--[ 244 ]-->

{{NumBlk|:|<math>{\displaystyle \Lambda =\left[0,\ 1\right]-\bigcup _{n=0}^{\infty }A_{n}}</math>|{{EquationRef|3-18}}}}

The process of removing <math>A_n</math> from [0, 1] is similar to the construction of the Cantor set mentioned above, and in fact Λ exists in [0, 1] as a Cantor set (a closed, completely disconnected, and complete subset of [0, 1]).<!--[ 245 ]--> Furthermore, on <math>\Lambda</math>, the logistic map <math>f_{r >4}</math> is chaotic.<!--[ 246 ]-->