Editing Logistic map (section)

=== Extension to complex numbers ===
[[File:Verhulst-Mandelbrot-Bifurcation.jpg|thumb|Correspondence between the orbit diagram of a variation of the logistic map (top) and the Mandelbrot set (bottom)]]

Dynamical systems defined by complex analytic functions are also of interest.<!--[ 332 ]-->LP An example is the dynamical system defined by the quadratic function: <!--[ 333 ]-->

{{NumBlk|:|<math>{\displaystyle z_{n+1}=z_{n}^{2}+c}</math>|{{EquationRef|6-3}}}}

where the parameter c and the variable z are complex numbers. <!--[ 333 ]--> This map is essentially the same as the logistic map (1–2). <!--[ 334 ]--> As mentioned above, the map (6–3) is topologically conjugate to the logistic map (1–2) through a linear function. <!--[ 335 ]-->

When the iteration of the map (6–3) is calculated with a fixed parameter c and varying the initial value <math>z_0</math>, a set of <math>z_0</math> such that <math>z_n</math> does not diverge to infinity as n → ∞ is called a filled Julia set.<!--[ 336 ]--> Furthermore, the boundary of a filled Julia set is called a Julia set. <!--[ 336 ]--> When the iteration of the map (6–3) is calculated with a fixed initial value <math>z_0 = 0</math> and varying the parameter {{mvar|c}}, a set of {{mvar|c}} such that {{mvar|z}} does not diverge to infinity is called a Mandelbrot set. <!--[ 337 ]--> The Julia sets and Mandelbrot sets of the map (6–3) generate fractal figures that are described as "mystical looking" and "extremely mysterious".{{attribution needed|date=May 2025}} <!--[ 338 ]--> 

In particular, in the Mandelbrot set, each disk in the diagram corresponds to a region of asymptotically stable periodic orbits of a certain period. <!--[ 339 ]--> By juxtaposing the logistic map orbit diagram with the Mandelbrot set diagram, it is possible to see that the asymptotically stable fixed points, period doubling bifurcations, and period-three windows of the logistic map orbit diagram correspond on the real axis to the Mandelbrot set diagram. <!--[ 340 ]-->