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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