= Chua's circuit =

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= Chua's circuit =
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Introduction
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Chua's circuit (also known as a Chua circuit) is a simple electronic
circuit that exhibits classic chaotic behavior. This means roughly
that it is a "nonperiodic oscillator"; it produces an oscillating
waveform that, unlike an ordinary electronic oscillator, never
"repeats". It was invented in 1983 by Leon O. Chua, who was a visitor
at Waseda University in Japan at that time. The ease of construction
of the circuit has made it a ubiquitous real-world example of a
chaotic system, leading some to declare it "a paradigm for chaos".

Chaotic criteria
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An autonomous circuit made from standard components (resistors,
capacitors, inductors) must satisfy three criteria before it can
display chaotic behaviour. It must contain:
# one or more nonlinear elements,
# one or more locally active resistors,
# three or more energy-storage elements.
Chua's circuit is the simplest electronic circuit meeting these
criteria. As shown in the top figure, the energy storage elements are
two capacitors (labeled C1 and C2) and an inductor (labeled L; L1 in
lower figure). A "locally active resistor" is a device that has
negative resistance and is active (it can amplify), providing the
power to generate the oscillating current. The locally active resistor
and nonlinearity are combined in the device 'N'R, which is called
"Chua's diode". This device is not sold commercially but is
implemented in various ways by active circuits. The circuit diagram
shows one common implementation. The nonlinear resistor is implemented
by two linear resistors and two diodes. At the far right is a negative
impedance converter made from three linear resistors and an
operational amplifier, which implements the locally active resistance
(negative resistance).

Models
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Analyzing the circuit using Kirchhoff's circuit laws, the dynamics of
Chua's circuit can be accurately modeled by means of a system of three
nonlinear ordinary differential equations in the variables 'x'('t'),
'y'('t'), and 'z'('t'), which represent the voltages across the
capacitors C1 and C2 and the electric current in the inductor L1
respectively. These equations are:
:\frac{dx}{dt} = \alpha [y - x - f(x)],
:RC_2 \frac{dy}{dt} = x - y + Rz,
:\frac{dz}{dt} = -\beta y.

The function 'f'('x') describes the electrical response of the
nonlinear resistor, and its shape depends on the particular
configuration of its components. The parameters α and β are determined
by the particular values of the circuit components.

A computer-assisted proof of chaotic behavior (more precisely, of
positive topological entropy) in Chua's circuit was published in 1997.
A chaotic attractor, known as "the double scroll" because of its shape
in the ('x',�'y',�'z') space, was first observed in a circuit
containing a nonlinear element such that 'f'('x') was a 3-segment
piecewise-linear function.

The easy experimental implementation of the circuit, combined with the
existence of a simple and accurate theoretical model, makes Chua's
circuit a useful system to study many fundamental and applied issues
of chaos theory. Because of this, it has been object of much study and
appears widely referenced in the literature.

Further, Chua' s circuit can be easily realized by using a multilayer
CNN (cellular nonlinear network). CNNs were invented by Leon Chua in
1988. To date, a large number of various types of chaotic attractors
in Chua's system have been discovered. These may be obtained
numerically, with relative ease, by standard computational procedure
(after transient process a trajectory, started from a point of
unstable manifold in a small neighborhood of unstable zero
equilibrium, reaches an attractor and computes it). Also, recently, a
hidden Chua's attractor was discovered in the classical Chua circuit,
and scenarios of the birth of hidden attractors were described.

The Chua diode can also be replaced by a memristor; an experimental
setup that implemented Chua's chaotic circuit with a memristor was
demonstrated by Muthuswamy in 2009; the memristor was actually
implemented with active components in this experiment.

See also
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*Chaos computing

References
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*'Chaos synchronization in Chua's circuit', Leon O Chua, Berkeley:
Electronics Research Laboratory, College of Engineering, University of
California, [1992], OCLC: 44107698
*'Chua's Circuit Implementations: Yesterday, Today and Tomorrow', L.
Fortuna, M. Frasca, M. G. Xibilia, World Scientific Series on
Nonlinear Science, Series A - Vol. 65, 2009,

External links
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*[http://www.cmp.caltech.edu/~mcc/chaos_new/Chua.html Chua's Circuit:
Diagram and discussion]
*[http://nonlinear.eecs.berkeley.edu NOEL laboratory. Leon O. Chua's
laboratory at the University of California, Berkeley]
*[https://people.eecs.berkeley.edu/~chua/circuitrefs.html References]
*[https://www.wired.com/2008/04/scientists-prov/ Chua and Memristors]
*[http://www.math.spbu.ru/user/nk/PDF/Hidden_Chua_Attractor_Localization.pdf
Hidden attractor in Chua's system]
*https://eecs.berkeley.edu/~chua/papers/Arena95.pdf
*[http://www.chuacircuits.com/sim.php Interactive Chua's circuit 3D
simulation]
*[http://experiences.math.cnrs.fr/Circuit-chaotique-de-Chua.html
Chua's circuit 3D numerical interactive experiment],
experiences.math.cnrs.fr

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Original Article: http://en.wikipedia.org/wiki/Chua's_circuit