Editing Logistic map (section)

==== Exact solutions for special cases ====
For a logistic map with a specific parameter a, an exact solution that explicitly includes the time n and the initial value x 0 has been obtained as follows.

When r = 4<!--[ 247 ]-->

{{NumBlk|:|<math>{\displaystyle x_{n}={\frac {1-\cos \left[2^{n}\arccos(1-2x_{0})\right]}{2}}}</math>|{{EquationRef|3-19}}}}

When r = 2<!--[ 248 ]-->

{{NumBlk|:|<math>{\displaystyle x_{n}={\frac {1-\exp \left[2^{n}\log(1-2x_{0})\right]}{2}}}</math>|{{EquationRef|3-20}}}}

When r = −2<!--[ 249 ]-->

{{NumBlk|:|<math>{\displaystyle x_{n}={\frac {1}{2}}-\cos \left\{{\frac {1}{3}}\left[\pi -(-2)^{n}\left(\pi -3\arccos({\frac {1}{2}}-x_{0})\right)\right]\right\}}		</math>|{{EquationRef|3-21}}}}

Considering the three exact solutions above, all of them are

{{NumBlk|:|<math>{\displaystyle x_{n}={\frac {1}{2}}\left\{1-f\left[a^{n}f^{-1}(1-2x_{0})\right] \right\}}</math>|{{EquationRef|3-22}}}}