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Chua Circuits

This page supports the main stream of the web site by providing very interesting examples of electronic circuits that exhibit amusing behavior confirmed by practical devices. We demonstrate capabilities of MuPad for this topic.

Chua Circuits

1. BRIEF HISTORY OF EVOLUTION Prior to 1983 no uufonomous electronic circuit was known to be chaotic, in spite of numerous attempts by researchers to uncover such examples. In particular, Matsumoto and his students had struggled for years to build an electronic circuit analogue of the Lorenz equation. The history of how Matsumoto’s heart-breaking failure had spurred the author to design a chaotic circuit from first principles was described vividly in Reference 1. Here we only outline the chronological events, which began in the fall of 1983, when this chaotic circuit was designed by the author using a systematic non-linear circuit synthesis technique. After describing his design to Matsumoto and instructing him on how to choose the circuit parameters for a possible chaotic regime, the author’s involvement in this circuit was abruptly interrupted for over a year owing to illness.

Having no prior experimental background, Matsumoto used computer simulation to verify that the author’s circuit, which he had named Chua’s circuit,’ is indeed chaotic. Meanwhile, Matsumoto and his students had followed the author’s suggestion to modify Rosenthal’s circuit in order to obtain an active two terminal non-linear resistor with the desired piecewise-linear characteristic..$ Handicapped by

In 1983, Leon O. Chua from the University of California, Berkely, proposed an electric circuit that exhibits chaotic behavior in the sense of Shilnikov. Later, many other simple circuits were discovered with similar properties, and they all are called today the Chua's circuit family. Such chaotic circuits admit nonperiodic oscillations. So even an extremely small changes of the initial conditions eventualy give rise to entirely different trajectories. All Chua's circuits contain an active piecewise linear active resistor. This active resistor is responsible for the circuit' chaotic behavior because it keeps supplying power to the external circuit. Such circuits are not analog computers, but rather extremely simple real electronic devices that even high school students can build. As a result, all Chua's circuit properties were experimentally observed. Moreover, most of their frequencies are within the audible frequencies, so one can listen to the sound, which is mysterious and amusing. We present some famous examples of Chua's circuits.

The double scroll family Authors (MCK86) Leon O. Chua, Motomasa Komuro, and Takashi Matsumoto, International Journal of Circuit Theory and Applications, Volume 33, Issue ? 1986 pages 1073--1118 .

The double scroll Authors (MCK85) Takashi Matsumoto, L. O. Chua, M. Komuro, International Journal of Circuit Theory and Applications, Volume 32, Issue ? 1985 pages 797--818 .

The double scroll bifurcations Authors (MCK86) T. Matsumoto, L. O. Chua, M. Komuro Published Date April 1986

Bifurcation analysis of a cusp-constrained piecewise-linear circuit Authors Ryuji Tokunaga, Leon O. Chua, Takashi Matsumoto Published Date July 1989 Boundary surfaces in sequential circuits Authors V. Špány, L. Plvka Published Date July 1990 Chua's circuit 10 years later Authors Leon O. Chua Published Date July 1994 A one-dimensional model of dynamics for a class of third-order systems Authors Maciej J. Ogorzalek Published Date November 1990 , International Journal of Circuit Theory and Applications, Volume 18, Issue 6 November/December 1990 Pages 595–624 Transfer maps and return maps for piecewise-linear three-region dynamical systems Authors Claus Kahlert, Leon O. Chua Published Date January 1987

The Double Scroll

The double scroll attractor (see Takashi Matsumoto, Leon O. Chua, Motomasa Komuro, The double scroll, IEEE Transactions on Circuits and Systems, vol. 32, No , 797--818, 1985) has been observed in the circuit on Figure 1, whose only nonlinear element is a three-segment piecewise-linear resistor, with v-i characteristic as shown in Figure 2. The dynamics of Chua's circuit are descrbed by

\begin{align*} C_1 \frac{{\text d}v_{C1}}{{\text d}t} &= G \left( v_{C2} - v_{C1} \right) - g(v_{C1}) , \\ C_2 \frac{{\text d}v_{C2}}{{\text d}t} &= G \left( v_{C1} - v_{C2} \right) + i_L , \\ L\, \frac{{\text d}i_{L}}{{\text d}t} &= - v_{C2} , \qquad (1.1) \end{align*}
where g( . ) is the piecewise linear function:
\[ g(v) = m_0 v + \frac{1}{2} \left( m_1 - m_0 \right) \left[ v+ B_P \right] + \frac{1}{2} \left( m_0 - m_1 \right) \left[ v- B_P \right] . \qquad (1.2) \]
Figure 3 from MCK85 shows trajectories projected onto the (iL, vC1)-plane, (iL, vC2)-plane, and (vC1, vC2)-plane, respectively under the following parameter values:
 C1 = 0.0053 μF  C2 = 0.047 μF  L=6.8 mH
 R = 1.21 kΩ  RB = 56 kΩ  R1 = 1 kΩ
 R2 = 3.3 kΩ  R3 = 88 kΩ  R4 = 39 kΩ
Also Vcc = 29 kΩ. Of course, they are the nominal values; the exact values could fall within 10 percent of these due to component tolerances.

The experimental observations for the double scroll are confirmed by solving the system of differential equations with the following rscaled parameter values:
\[ C_1 = 1/9, \quad C_2 =1, \quad L= 1/7, \quad G=7/10, \quad m_0 = -0.5, \quad m_1 =-0.8, \quad B_P =1. \qquad (1.3) \]
The double scroll system of equations can be transformed into the following one:
\begin{align*} \frac{{\text d}x}{{\text d}\tau} &= \alpha \left( y - f(x) \right) , \\ \frac{{\text d}y}{{\text d}\tau} &= x-y+z , \\ \frac{{\text d}z}{{\text d}\tau} &= -\beta \,y , \qquad (3.3) \end{align*}
where
\[ f(x) = \begin{cases} b\,x +a-b , & \ x\ge 1, \\ a\,x , & \ |x| \le 1, \\ b\,x+b-a , & \ x\le -1 . \end{cases} \]
via
  x = vC1 / BP   y = vC2 / BP   z = i / (BPG)
  τ = t G / C2   a = m1 / G + 1   b = m0 / G + 1
  α = C2 / C1   β = C2 / (L G2)  
To find equilibria, we consider the equations
\[ \begin{cases} f(x) &= 0 , \\ y &= 0 , \\ x+z &= 0 . \end{cases} \]
It follows from the form of f(.) that the system of Chua's differential equations has a unique equilibrium in each of the following three subsets of \( \mathbb{R}^3 : \)
\[ \begin{split} D_1 &= \left\{ (x,y,z) \, : \, x \ge 1 \right\} , \\ D_0 &= \left\{ (x,y,z) \, : \, |x| \le 1 \right\} , \\ D_{-1} &= \left\{ (x,y,z) \, : \, x \le -1 \right\} , \end{split} \]
provided that \( a, b \ne -1 . \) The equlibria are explicitely given by
\[ \begin{split} P^{+} &= (k, 0, -k ) \in D_1 , \\ 0 &= (0, 0, 0) \in D_0 , \\ P^{-} &= (-k, 0, k ) \in D_{-1} , \end{split} \]
where k = (b-a)/(b+1).

In each of the domains, the Chau's system of equations is linear. Setting

Folded Torus

The Double Hook

based on paper:

Philippe Bartissol and Leon Chua, The Double Hook, IEEE Transactions on Circuits and Systems, vol. 35, No 12, 1512--1522, 1988.

https://people.eecs.berkeley.edu/~chua/papers/Bartissol88.pdf

Refernces

[1] Leon O. Chua, Motomasa Komuro, and Takashi Matsumoto, "The double scroll family," International Journal of Circuit Theory and Applications, Volume 33, Issue ? 1986 pages 1073--1118 .

[2] Takashi Matsumoto, Leon O. Chua, and Motomasa Komuro, The double scroll, IEEE Transactions on Circuits and Systems, vol. 32, No , 797--818, 1985.

[3] M. Pyee, "Fonctions et systemes electroneques," Tome 1, Escole Nationale Superieure de l'Aeronautique et de l'Espace, Toulouse, France, 1986.

[4] Leon O. Chua and F. Aurom "Desining non-linear single op-amp circuits: A cook-hook approach," International Journal of Circuit Theory and Applications, Volume 13, Issue ?, 235--268, 1985.

[5] Leon O. Chua and Guo-Qun Zhong, "Negative resistance curve tracer," IEEE Transactions on Circuits and Systems, vol. 32, No 1, ??--??, 1985.

[6] Leon O. Chua and T. Sugawara, "Panoramic view of strange attractors," Proceedings IEEE, August, vol. 75, 1107--1120, 1987.

[7] C.P. Silva and L.O. Chua, "The overdamped double scroll family; Part 1: Piecewise linear geometry and normal form," International Journal of Circuit Theory and Applications, Volume 16, Issue ?, 233--302, 1988.

[8] L.O. Chua, C.A. Desoer, and E.S. Kuh, "Linear and Nonlinear Circuits," New York, McGraw Hill, 1987.

[9] S. Wu, "Chua's circuit family," Proceedings IEEE, August, vol. 75, 1022--1032, 1987.

The double hook attractor is a new strange attractor displayed by the same system of differential equations, but with the following parameter values:

\[ C_1 \approx -0.0647, \quad C_2 \approx 0.3180, \quad L \approx -0.3005, \quad G \approx 0.539, \quad m_0 \approx -0.5013, \quad m_1 \approx -1.3475, \quad B_P =1. \qquad (1.4) \]
The name "double hook" is chosen to call attention to the geometrical structure of the cross section through the origin (see orange countour in Fig. 11(a)), which resembles two opositely pointed "hooks," joined together on one side.

Figure 3 shows the new attractor corresponding to this set of parameter values with an appropriaate time scaling. This simulation has been obtained by computing the dimensionless form of system (1.1)

Note that the most significant difference between the tow parameter sets (1.3) and (1.4) is that the circuit giving the new attractor requires a negative capacitor and a negative inductor. The realization of such negative devices is not easy.

One technique for synthesizing negative capacitance, or negative inductance, is to use a Negative Impedance Converter (NIC) from [3] and a positive-valued capacitance or inductance. One can realize an NIC in two different ways as shown in Figure 4. Upon reschaling, we get the following system of equations:

\begin{align*} \frac{{\text d}x}{{\text d}\tau} &= \alpha \left( -y - f(x) \right) , \\ \frac{{\text d}y}{{\text d}\tau} &= x-y+z , \\ \frac{{\text d}z}{{\text d}\tau} &= \beta \,y , \qquad (3.3) \end{align*}
where
\[ f(x) = \begin{cases} b\,x +a-b , & \ x\ge 1, \\ a\,x , & \ |x| \le 1, \\ b\,x+b-a , & \ x\le -1 . \end{cases} \]

The Canonical piecewise linear Circuit

Driven RL-Diode