Editing Logistic map (section)

=== A class of mappings that exhibit homogeneous behavior ===
[[File:正弦関数による力学系のグラフ.png|class=skin-invert-image|thumb|Graph of the sine map ( 4-1 )]]
[[File:正弦関数による力学系の軌道図.png|class=skin-invert-image|thumb|Orbit diagram of the sine map ( 4-1 )]]

The bifurcation pattern shown above for the logistic map is not limited to the logistic map <!--[ 239 ]-->. It appears in a number of maps that satisfy certain conditions . The following dynamical system using sine functions is one example <!--[ 251 ]-->:

{{NumBlk|:|<math>{\displaystyle x_{n+1}=b\sin \pi x_{n}}</math>|{{EquationRef|4-1}}}}

Here, the domain is 0 ≤ b ≤ 1 and 0 ≤ x ≤ 1 <!--[ 251 ]-->. The sine map ( 4-1 ) exhibits qualitatively identical behavior to the logistic map ( 1-2 ) <!--[ 251 ]-->: like the logistic map, it also becomes chaotic via a period doubling route as the parameter b increases, and moreover, like the logistic map, it also exhibits a window in the chaotic region <!--[ 251 ]-->.

Both the logistic map and the sine map are one-dimensional maps that map the interval [0, 1] to [0, 1] and satisfy the following property, called unimodal <!--[ 252 ]-->.

<math>f(0)=f(1)= 0</math>.
The map is differentiable and there exists a unique critical point c in [0, 1] such that <math>f'( c ) = 0</math>.
In general, if a one-dimensional map with one parameter and one variable is unimodal and the vertex can be approximated by a second-order polynomial, then, regardless of the specific form of the map, an infinite period-doubling cascade of bifurcations will occur for the parameter range 3 ≤ r ≤ 3.56994... , and the ratio δ defined by equation ( 3-13 ) is equal to the Feigenbaum constant, 4.669... <!--[ 253 ]-->.

The pattern of stable periodic orbits that emerge from the logistic map is also universal <!--[ 254 ]--> . For a unimodal map, <math>x_{n +1} = cf ( x_n )</math> , with parameter c, stable periodic orbits with various periods continue to emerge in a parameter interval where the two fixed points are unstable, and the pattern of their emergence (the number of stable periodic orbits with a certain period and the order of their appearance) is known to be common <!--[ 255 ]--><!--[ 256 ]-->. In other words, for this type of map, the sequence of stable periodic orbits is the same regardless of the specific form of the map <!--[ 257 ]--> . For the logistic map, the parameter interval is 3 < a < 4, but for the sine map ( 4-1 ), the parameter interval for the common sequence of stable periodic orbits is 0.71... < b < 1 <!--[ 256 ]-->. This universal sequence of stable periodic orbits is called the U sequence <!--[ 254 ]-->.

In addition, the logistic map has the property that its Schwarzian derivative is always negative on the interval [0, 1] . The Schwarzian derivative of a map f (of class C3 ) is 

{{NumBlk|:|<math>{\displaystyle Sf(x)={\frac {f'''(x)}{f'(x)}}-{\frac {3}{2}}\left({\frac {f''(x)}{f'(x)}}\right)^{2}}	</math>|{{EquationRef|4-2}}}}

<!--[ 258 ]--> 
In fact, when we calculate the Schwarzian derivative of the logistic map, we get

{{NumBlk|:|<math>{\displaystyle S(ax(1-x))={\frac {-6}{(1-2x)^{2}}}<0}</math>|{{EquationRef|4-3}}}}

where the Schwarzian derivative is negative regardless of the values of a and x . <!--[ 259 ]--> It is known that if a one-dimensional mapping from [0, 1] to [0, 1] is unimodal and has a negative Schwarzian derivative, then there is at most one stable periodic orbit . <!--[ 260 ]-->