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=                          Chaos computing                           =
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                            Introduction
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Chaos computing is the idea of using chaotic systems for computation.
In particular, chaotic systems can be made to produce all types of
logic gates and further allow them to be morphed into each other.


Chaotic systems generate large numbers of patterns of behavior and are
irregular because they switch between these patterns. They exhibit
sensitivity to initial conditions which, in practice, means that
chaotic systems can switch between patterns extremely fast.

Modern digital computers perform computations based upon digital logic
operations implemented at the lowest level as logic gates.  There are
essentially seven basic logic functions implemented as logic gates:
AND, OR, NOT, NAND, NOR, XOR and XNOR.

A chaotic morphing logic gate consists of a generic nonlinear circuit
that exhibits chaotic dynamics producing various patterns. A control
mechanism is used to select patterns that correspond to different
logic gates. The sensitivity to initial conditions is used to switch
between different patterns extremely fast (well under a computer clock
cycle).


                          Chaotic Morphing
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As an example of how chaotic morphing works, consider a generic
chaotic system known as the Logistic map. This nonlinear map is very
well studied for its chaotic behavior and its functional
representation is given by:

:\qquad x_{n+1} = r x_n (1-x_n)

In this case, the value of  is chaotic when  >~ 3.57... and rapidly
switches between different patterns in the value of  as one iterates
the value of . 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  that lies above a
given threshold value, *,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. Since the system is chaotic, we can then
switch between various gates ("patterns") exponentially fast.


                              ChaoGate
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The 'ChaoGate' is an implementation of a chaotic morphing logic gate
developed by the inventor of the technology William Ditto, along with
Sudeshna Sinha and K. Murali.

A Chaotic computer, made up of a lattice of ChaoGates, has been
demonstrated by Chaologix Inc.


                              Research
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Recent research has shown how chaotic computers can be recruited in
Fault Tolerant applications, by introduction of dynamic based fault
detection methods. Also it has been demonstrated that multidimensional
dynamical states available in a single ChaoGate can be exploited to
implement parallel chaos computing,
"Parallel computing with extended dynamical systems" S. Sinha, T.
Munakata and W.L. Ditto
Physical Review E, 65 036214 [1-7](2002) and as an example, this
parallel architecture can lead to constructing an SR like memory
element through one ChaoGate. As another example, it has been proved
that any logic function can be constructed directly from just one
ChaoGate.


                              See also
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* Chua's circuit


                             References
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*"The 10 Coolest Technologies You�ve Never Heard Of - Chaos
Computing," PC Magazine, Vol. 25, No. 13, page p. 66, August 8, 2006.
[https://www.pcmag.com/article2/0,2704,1990288,00.asp]
*"Logic from Chaos," MIT Technology Review, June 15, 2006.
[http://www.technologyreview.com/Biztech/16989/]
*"Exploiting the controlled responses of chaotic elements to design
configurable hardware," W. L. Ditto and S. Sinha, Philosophical
Transactions of the Royal Society London A, 364, pp. 2483-2494 (2006)

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