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| [[File:Chua's circuit.svg|thumb|350px|A version of Chua's circuit without [[Chua's Diode]]]]
| | Greetings. Let me start by telling you the writer's title - Phebe. Managing individuals has been his day occupation for a whilst. Puerto Rico is where he and his wife reside. Doing ceramics is what her family and her appreciate.<br><br>My webpage ... std home test ([http://www.pornextras.info/user/G49Z additional reading]) |
| '''Chua's circuit''' (also known as a Chua circuit) is a simple [[electronic circuit]] that exhibits classic [[chaos theory]] behavior. It was introduced in 1983 by [[Leon O. Chua]], who was a visitor at [[Waseda University]] in [[Japan]] at that time.<ref>{{cite journal | last = Matsumoto | first = Takashi | title = A Chaotic Attractor from Chua's Circuit | journal = IEEE Transactions on Circuits and Systems | volume = CAS-31 | issue = 12 | pages = 1055–1058 | publisher = [[IEEE]] | date = December 1984 | url = http://www.eecs.berkeley.edu/~chua/papers/Matsumoto84.pdf | accessdate = 2008-05-01}}</ref> 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."<ref>{{cite book | last = Madan | first = Rabinder N. | title = Chua's circuit: a paradigm for chaos | publisher = World Scientific Publishing Company | year = 1993 | location = River Edge, N.J. | isbn = 981-02-1366-2}}</ref>
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| ==Chaotic criteria==
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| An [[autonomous circuit]] made from standard components ([[resistor]]s, [[capacitor]]s, [[inductor]]s) must satisfy three criteria before it can display chaotic behaviour.{{Citation needed|date=May 2010}} It must contain:
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| # one or more nonlinear elements
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| # one or more locally active resistors
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| # '''three''' or more energy-storage elements.
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| Chua's circuit is the simplest electronic circuit meeting these criteria {{citation needed|date=September 2012}}. As shown in the figure, the energy storage elements are two [[capacitor]]s (labeled C1 and C2) and an [[inductor]] (labeled L1). There is an active [[resistor]] (labeled R). There is a nonlinear resistor made of two linear resistors and two [[diode]]s. At the far right is a [[negative impedance converter]] made from three linear resistors and an [[operational amplifier]]. The section to the right simulates [[Chua's diode]], a component that is currently not sold commercially.
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| ==Model==
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| [[File:Double scroll attractor from Matlab simulation.jpg|thumb|Output of MATLAB simulation of Chua's circuit after 100 seconds, showing chaotic "double scroll" attractor pattern]]
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| By means of the application of the laws of [[electromagnetism]], the dynamics of Chua's circuit can be accurately modeled by means of a system of three [[nonlinear]] [[ordinary differential equation]]s in the variables x(t), y(t) and z(t), which give the voltages across the capacitors C1 and C2, and the intensity of the electrical current in the inductor L1, respectively. These equations read:
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| :<math>\frac{dx}{dt}=\alpha [y-x-f(x)]</math>
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| :<math>\frac{dy}{dt}=x-y+z</math>
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| :<math>\frac{dz}{dt}=-\beta y</math>
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| 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.
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| A [[chaotic attractor]], known as "[[Double scroll attractor|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.<ref>{{cite journal | last = Chua | first = Leon O. | authorlink = Leon O. Chua | coauthors = Matsumoto, T., and Komuro, M. | title = The Double Scroll | journal = IEEE Transactions on Circuits and Systems | volume = CAS-32 | issue = 8 | pages = 798–818 | publisher = [[IEEE]] | date = August 1985 | url = http://ieeexplore.ieee.org/iel5/31/23571/01085791.pdf | accessdate = 2008-05-01}}</ref>
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| 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.
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| 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, а large number of various types of chaotic attractors in Chua's system have been discovered,.<ref>{{cite book |author= Bilotta, E., Pantano, P. |title=Gallery of Chua Attractors |publisher=World Scientific |year=2008 |isbn=978-981-279-062-0}}</ref> These may be obtained numerically, with relative ease, by [[Attractor#Numerical_localization_.28visualization.29_of_attractors|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 oscillation|hidden Chua's attractor]] was discovered in the classical Chua circuit,<ref name=2011-PLA-Hidden-Chua-attractor>{{cite journal |
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| author = Leonov G.A., Vagaitsev V.I., Kuznetsov N.V. |
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| year = 2011 |
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| title = Localization of hidden Chua's attractors |
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| journal = Physics Letters A |
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| volume = 375 |
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| issue = 23 |
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| pages = 2230–2233 |
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| url = http://www.math.spbu.ru/user/nk/PDF/2011-PhysLetA-Hidden-Attractor-Chua.pdf |
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| doi = 10.1016/j.physleta.2011.04.037}}
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| </ref><ref name=2011-IJBC-Hidden-attractors>{{cite journal |
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| author = Leonov G.A., Kuznetsov N.V. |
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| year = 2013 |
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| title = Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits |
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| journal = International Journal of Bifurcation and Chaos |
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| volume = 23 |
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| issue = 1 |
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| pages = art. no. 1330002|
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| url = http://www.worldscientific.com/doi/pdf/10.1142/S0218127413300024|
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| doi = 10.1142/S0218127413300024}}
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| </ref>
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| and later smooth and discontinous variants were identified.
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| <ref name=2012-Physica-D-Hidden-attractor-Chua-circuit-smooth>{{cite journal |
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| author = Leonov G.A., Vagaitsev V.I., Kuznetsov N.V. |
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| year = 2012 |
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| title = Hidden attractor in smooth Chua systems |
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| journal = Physica D |
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| volume = 241 |
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| issue = 18 |
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| pages = 1482–1486 |
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| url = http://www.math.spbu.ru/user/nk/PDF/2012-Physica-D-Hidden-attractor-Chua-circuit-smooth.pdf |
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| doi = 10.1016/j.physd.2012.05.016}}
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| </ref><ref name=2013-LNEE-Hidden-attractor-electrical-Chua-circuit>{{cite journal |
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| author = Kuznetsov N., Kuznetsova O., Leonov G., Vagaitsev V. |
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| year = 2013 |
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| title = Analytical-numerical localization of hidden attractor in electrical Chua’s circuit |
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| journal = Lecture Notes in Electrical Engineering|
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| volume = 174 LNEE |
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| pages = 149–158 |
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| url = http://www.math.spbu.ru/user/nk/PDF/2013-LNEE-Hidden-attractor-electrical-Chua-circuit.pdf |
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| doi = 10.1007/978-3-642-31353-0_11}}
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| </ref>
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| == See also ==
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| {{Portal|Electronics}}
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| *[http://www.chuacircuits.com/sim.php Interactive Chua's circuit 3D simulation] Double Scroll Example
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| *[[Memristor]]
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| ==Notes==
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| {{reflist}}
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| ==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
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| *'' 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, ISBN 978-981-283-924-4
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| ==External links==
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| *[http://www.cmp.caltech.edu/~mcc/chaos_new/Chua.html Chua's Circuit: Diagram and discussion]
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| *[http://nonlinear.eecs.berkeley.edu NOEL laboratory. Leon O. Chua's laboratory at the University of California, Berkeley]
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| *[http://www.eecs.berkeley.edu/~chua/circuitrefs.html References]
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| *[http://blog.wired.com/gadgets/2008/04/scientists-prov.html Chua and Memristors]
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| * [http://www.math.spbu.ru/user/nk/PDF/Hidden_Chua_Attractor_Localization.pdf Hidden attractor in Chua's system]
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| * http://www.eecs.berkeley.edu/~chua/papers/Arena95.pdf
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| {{Chaos theory}}
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| {{DEFAULTSORT:Chua's Circuit}}
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| [[Category:Chaotic maps]]
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| [[Category:Oscillators]]
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Greetings. Let me start by telling you the writer's title - Phebe. Managing individuals has been his day occupation for a whilst. Puerto Rico is where he and his wife reside. Doing ceramics is what her family and her appreciate.
My webpage ... std home test (additional reading)