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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

An autonomous circuit made from standard components (resistors, capacitors, inductors) must satisfy three criteria before it can display chaotic behaviour. 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. 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).

Dynamics

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:

${\displaystyle {\frac {dx}{dt}}=\alpha ,}$

${\displaystyle RC_{2}{\frac {dy}{dt}}=x-y+Rz,}$

${\displaystyle {\frac {dz}{dt}}=-\beta y.}$

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 self-excited 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.

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.

Self-excited and hidden Chua attractors

The classical implementation of Chua circuit is switched on at the zero initial data, thus a conjecture was that the chaotic behavior is possible only in the case of unstable zero equilibrium. In this case a chaotic attractor in mathematical model can be obtained numerically, with relative ease, by standard computational procedure where after transient process a trajectory, started from a point of unstable manifold in a small neighborhood of unstable zero equilibrium, reaches and computes a self-excited attractor. To date, a large number of various types of self-excited chaotic attractors in Chua's system have been discovered. However, in 2009, N. Kuznetsov discovered hidden Chua's attractors coexisting with stable zero equilibrium, and since then various scenarios of the birth of hidden attractors have been described.

Experimental confirmation

First experimental confirmation of self-excited chaos from Chua's circuit was reported in 1985 at the Electronics Research Lab at U.C. Berkeley. First confirmation of hidden chaos was reported in 2022 at the Theoretical Nonlinear Dynamics Lab at the Institute of Radio-engineering and Electronics of the Russian Academy of Sciences.

See also

• Chaos computing
• Multiscroll attractor
• Leon Chua

Notes

1. Matsumoto, Takashi (December 1984). "A Chaotic Attractor from Chua's Circuit" (PDF). IEEE Transactions on Circuits and Systems. CAS-31 (12). IEEE: 1055–1058. doi:10.1109/TCS.1984.1085459. Retrieved 2008-05-01.
2. Madan, Rabinder N. (1993). Chua's circuit: a paradigm for chaos. River Edge, N.J.: World Scientific Publishing Company. Bibcode:1993ccpc.book.....M. ISBN 981-02-1366-2.
3. ^ Kennedy, Michael Peter (October 1993). "Three steps to chaos – Part 1: Evolution" (PDF). IEEE Transactions on Circuits and Systems. 40 (10). Institute of Electrical and Electronics Engineers: 640. doi:10.1109/81.246140. Retrieved February 6, 2014.
4. Kennedy, Michael Peter (October 1993). "Three steps to chaos – Part 2: A Chua's circuit primer" (PDF). IEEE Transactions on Circuits and Systems. 40 (10). Institute of Electrical and Electronics Engineers: 658. doi:10.1109/81.246141. Retrieved February 6, 2014.
5. ^ Kuznetsov N. V.; Mokaev T. N.; Ponomarenko V. I.; Seleznev E. P.; Stankevich N. V.; Chua L. (2023). "Hidden attractors in Chua circuit: mathematical theory meets physical experiments" (PDF). Nonlinear Dynamics. 111 (6): 5859–5887. doi:10.1007/s11071-022-08078-y. S2CID 255049732.
6. Z. Galias, "Positive topological entropy of Chua's circuit: a computer-assisted proof", Int. J. Bifurcations and Chaos, 7 (1997), pp. 331–349.
7. Chua, Leon O.; Matsumoto, T.; Komuro, M. (August 1985). "The Double Scroll". IEEE Transactions on Circuits and Systems. CAS-32 (8). IEEE: 798–818. doi:10.1109/TCS.1985.1085791.
8. Bharathwaj Muthuswamy, "Implementing memristor based chaotic circuits", International Journal of Bifurcation and Chaos, Vol. 20, No. 5 (2010) 1335–1350, World Scientific Publishing Company, doi:10.1142/S0218127410026514.
9. ^ Stankevich N. V.; Kuznetsov N. V.; Leonov G. A.; Chua L. (2017). "Scenario of the birth of hidden attractors in the Chua circuit". International Journal of Bifurcation and Chaos. 27 (12): 1730038–188. arXiv:1710.02677. Bibcode:2017IJBC...2730038S. doi:10.1142/S0218127417300385. S2CID 45604334.
10. Bilotta, E.; Pantano, P. (2008). Gallery of Chua Attractors. World Scientific. ISBN 978-981-279-062-0.
11. Leonov G. A.; Vagaitsev V. I.; Kuznetsov N. V. (2011). "Localization of hidden Chua's attractors" (PDF). Physics Letters A. 375 (23): 2230–2233. Bibcode:2011PhLA..375.2230L. doi:10.1016/j.physleta.2011.04.037.
12. Leonov G. A.; Kuznetsov N. V. (2013). "Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits". International Journal of Bifurcation and Chaos. 23 (1): 1330002–219. Bibcode:2013IJBC...2330002L. doi:10.1142/S0218127413300024.
13. Zhong, G.-Q.; Ayrom, F. (January 1985). "Experimental confirmation of chaos from Chua's circuit". International Journal of Circuit Theory and Applications. 13 (1): 93–98. doi:10.1002/cta.4490130109.
14. University, St Petersburg State. "Mathematicians prove the existence of hidden attractors in an electrical circuit". phys.org. Retrieved 2023-01-25.

References

• Chaos synchronization in Chua's circuit, Leon O Chua, Berkeley: Electronics Research Laboratory, College of Engineering, University of California, , 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

Further reading

• Recai Kilic (2010). A Practical Guide for Studying Chua's Circuits. World Scientific. Bibcode:2010pgsc.book.....K. ISBN 978-981-4291-14-9.

External links

• Chua's Circuit: Diagram and discussion Archived 2012-01-26 at the Wayback Machine
• NOEL laboratory. Leon O. Chua's laboratory at the University of California, Berkeley
• References
• Chua and Memristors
• Hidden attractor in Chua's system
• https://eecs.berkeley.edu/~chua/papers/Arena95.pdf
• Interactive Chua's circuit 3D simulation
• Chua's circuit 3D numerical interactive experiment, experiences.math.cnrs.fr

v t e Chaos theory
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Theoretical branches	Bifurcation theory Control of chaos Dynamical system Ergodic theory Quantum chaos Stability theory Synchronization of chaos
Chaotic maps (list)	Discrete Arnold's cat map Baker's map Complex quadratic map Coupled map lattice Duffing map Dyadic transformation Dynamical billiards outer Exponential map Gauss map Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Irrational rotation Kaplan–Yorke map Langton's ant Logistic map Standard map Tent map Tinkerbell map Zaslavskii map Continuous Double scroll attractor Duffing equation Lorenz system Lotka–Volterra equations Mackey–Glass equations Rabinovich–Fabrikant equations Rössler attractor Three-body problem Van der Pol oscillator
Physical systems	Chua's circuit Convection Double pendulum Elastic pendulum FPUT problem Hénon–Heiles system Kicked rotator Multiscroll attractor Population dynamics Swinging Atwood's machine Tilt-A-Whirl Weather
Chaos theorists	Michael Berry Rufus Bowen Mary Cartwright Chen Guanrong Leon O. Chua Mitchell Feigenbaum Peter Grassberger Celso Grebogi Martin Gutzwiller Brosl Hasslacher Michel Hénon Svetlana Jitomirskaya Bryna Kra Edward Norton Lorenz Aleksandr Lyapunov Benoît Mandelbrot Hee Oh Edward Ott Henri Poincaré Itamar Procaccia Mary Rees Otto Rössler David Ruelle Caroline Series Yakov Sinai Oleksandr Mykolayovych Sharkovsky Nina Snaith Floris Takens Audrey Terras Mary Tsingou Marcelo Viana Amie Wilkinson James A. Yorke Lai-Sang Young
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