Iterated function: Difference between revisions

Revision as of 00:39, 27 October 2023

Revision as of 18:23, 27 October 2023

Line 155:

Well-known iterated functions include the [[Mandelbrot set]] and [[iterated function systems]].

[[Ernst Schröder (mathematician)|Ernst Schröder]],<ref name="schr">{{cite journal |last=Schröder |first=Ernst |author-link=Ernst Schröder (mathematician) |year=1870 |title=Ueber iterirte Functionen|journal=Math. Ann. |volume=3 |issue= 2|pages=296–322 | doi=10.1007/BF01443992 |s2cid=116998358 }}</ref> in 1870, worked out special cases of the [[logistic map]], such as the chaotic case {{math|1=''f''(''x'') = 4''x''(1 − ''x'')}}, so that {{math|1=Ψ(''x'') = arcsin</sup>({{radic|''x''}})²}}, hence {{math|1=''f'' ^''n''(''x'') = sin</sup>(2^''n'' arcsin({{radic|''x''}}))²}}.

A nonchaotic case Schröder also illustrated with his method, {{math|1=''f''(''x'') = 2''x''(1 − ''x'')}}, yielded {{math|1=Ψ(''x'') = −{{sfrac|1|2}} ln(1 − 2''x'')}}, and hence {{math|1=''f''^''n''(''x'') = −{{sfrac|1|2}}((1 − 2''x'')^{2^''n''} − 1)}}.