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[[File:Chua's circuit.svg|thumb|350px|A version of Chua's circuit without [[Chua's Diode]]]]
'''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>
 
==Chaotic criteria==
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:
# 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 {{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.
 
==Model==
[[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]]
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:
 
:<math>\frac{dx}{dt}=\alpha [y-x-f(x)]</math>
 
:<math>\frac{dy}{dt}=x-y+z</math>
 
:<math>\frac{dz}{dt}=-\beta y</math>
 
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 [[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>
 
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, 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 |
author = Leonov G.A., Vagaitsev V.I., Kuznetsov N.V. |
year = 2011 |
title = Localization of hidden Chua's attractors |
journal = Physics Letters A |
volume = 375 |
issue = 23 |
pages = 2230&ndash;2233 |
url = http://www.math.spbu.ru/user/nk/PDF/2011-PhysLetA-Hidden-Attractor-Chua.pdf |
doi = 10.1016/j.physleta.2011.04.037}}
</ref><ref name=2011-IJBC-Hidden-attractors>{{cite journal |
author = Leonov G.A., Kuznetsov N.V. |
year = 2013 |
title = Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits |
journal = International Journal of Bifurcation and Chaos |
volume = 23 |
issue = 1 |
pages = art. no. 1330002|
url = http://www.worldscientific.com/doi/pdf/10.1142/S0218127413300024|
doi = 10.1142/S0218127413300024}}
</ref>
and later smooth and discontinous variants were identified.
<ref name=2012-Physica-D-Hidden-attractor-Chua-circuit-smooth>{{cite journal |
author = Leonov G.A., Vagaitsev V.I., Kuznetsov N.V. |
year = 2012 |
title = Hidden attractor in smooth Chua systems |
journal = Physica D |
volume = 241 |
issue = 18 |
pages = 1482&ndash;1486 |
url = http://www.math.spbu.ru/user/nk/PDF/2012-Physica-D-Hidden-attractor-Chua-circuit-smooth.pdf |
doi = 10.1016/j.physd.2012.05.016}}
</ref><ref name=2013-LNEE-Hidden-attractor-electrical-Chua-circuit>{{cite journal |
author = Kuznetsov N., Kuznetsova O., Leonov G., Vagaitsev V.  |
year = 2013 |
title = Analytical-numerical localization of hidden attractor in electrical Chua’s circuit |
journal = Lecture Notes in Electrical Engineering|
volume = 174 LNEE |
pages = 149&ndash;158 |
url = http://www.math.spbu.ru/user/nk/PDF/2013-LNEE-Hidden-attractor-electrical-Chua-circuit.pdf |
doi = 10.1007/978-3-642-31353-0_11}}
</ref>
 
== See also ==
{{Portal|Electronics}}
*[http://www.chuacircuits.com/sim.php  Interactive Chua's circuit 3D simulation] Double Scroll Example
*[[Memristor]]
 
==Notes==
{{reflist}}
 
==References==
*''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, ISBN 978-981-283-924-4
 
==External links==
*[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]
*[http://www.eecs.berkeley.edu/~chua/circuitrefs.html References]
*[http://blog.wired.com/gadgets/2008/04/scientists-prov.html Chua and Memristors]
* [http://www.math.spbu.ru/user/nk/PDF/Hidden_Chua_Attractor_Localization.pdf Hidden attractor in Chua's system]
* http://www.eecs.berkeley.edu/~chua/papers/Arena95.pdf
 
{{Chaos theory}}
 
{{DEFAULTSORT:Chua's Circuit}}
[[Category:Chaotic maps]]
[[Category:Oscillators]]

Revision as of 19:50, 23 April 2013

A version of Chua's circuit without Chua's Diode

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.[1] 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."[2]

Chaotic criteria

An autonomous circuit made from standard components (resistors, capacitors, inductors) must satisfy three criteria before it can display chaotic behaviour.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. It must contain:

  1. one or more nonlinear elements
  2. one or more locally active resistors
  3. three or more energy-storage elements.

Chua's circuit is the simplest electronic circuit meeting these criteria Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.. As shown in the figure, the energy storage elements are two capacitors (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 diodes. 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.

Model

Output of MATLAB simulation of Chua's circuit after 100 seconds, showing chaotic "double scroll" attractor pattern

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 equations 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:

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 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.[3]

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, а large number of various types of chaotic attractors in Chua's system have been discovered,.[4] 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,[5][6] and later smooth and discontinous variants were identified. [7][8]

See also

Sportspersons Hyslop from Nicolet, usually spends time with pastimes for example martial arts, property developers condominium in singapore singapore and hot rods. Maintains a trip site and has lots to write about after touring Gulf of Porto: Calanche of Piana.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

  • 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, ISBN 978-981-283-924-4

External links

Template:Chaos theory

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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  7. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  8. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang