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:<math>\qquad x_{n+1} = r x_n (1-x_n) </math>.
In this case, the value of {{math|''x''}} is chaotic when {{math|''r''}} >~ 3.57... and rapidly switches between different patterns in the value of {{math|''x''}} as one iterates the value of {{math|''n''}}. A simple threshold controller can control or direct the chaotic map or system to produce one of many patterns. The controller basically sets a threshold on the map such that if the iteration ("chaotic update") of the map takes on a value of {{math|''x''}} that lies above a given threshold value, {{math|''x''}}*, then the output corresponds to a 1, otherwise it corresponds to a 0. One can then reverse engineer the chaotic map to establish a lookup table of thresholds that robustly produce any of the logic gate operations.<ref>{{cite journal | last1=Sinha | first1=Sudeshna |author2-link=William Ditto | last2=Ditto | first2=William | title=Dynamics Based Computation | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=81 | issue=10 | year=1998 | issn=0031-9007 | doi=10.1103/physrevlett.81.2156 | pages=2156–2159| bibcode=1998PhRvL..81.2156S }}</ref><ref>{{cite journal | last1=Sinha | first1=Sudeshna | last2=Ditto | first2=William L. | title=Computing with distributed chaos | journal=Physical Review E | publisher=American Physical Society (APS) | volume=60 | issue=1 | date=1999-07-01 | issn=1063-651X | doi=10.1103/physreve.60.363 | pages=363–377| pmid=11969770 | bibcode=1999PhRvE..60..363S }}</ref><ref>{{cite journal | last1=Munakata | first1=T. | last2=Sinha | first2=S. | last3=Ditto | first3=W.L. | title=Chaos computing: implementation of fundamental logical gates by chaotic elements | journal=IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=49 | issue=11 | year=2002 | issn=1057-7122 | doi=10.1109/tcsi.2002.804551 | pages=1629–1633}}</ref> Since the system is chaotic, we can then switch between various gates ("patterns") exponentially fast.
== ChaoGate ==
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