Chua circuit
The Chua Circuit is the simplest electronic circuit exhibiting chaos, and many wellknown bifurcation phenomena, as verified from numerous laboratory experiments, computer simulations, and rigorous mathematical analysis.
Historical Background
The Chua Circuit was invented in the fall of 1983 (Chua, 1992) in response to two unfulfilled quests among many researchers on chaos concerning two wanting aspects of the Lorenz Equations (Lorenz, 1963). The first quest was to devise a laboratory system which can be realistically modeled by the Lorenz Equations in order to demonstrate chaos is a robust physical phenomenon, and not merely an artifact of computer roundoff errors. The second quest was to prove that the Lorenz attractor, which was obtained by computer simulation, is indeed chaotic in a rigorous mathematical sense. The existence of chaotic attractors from the Chua circuit had been confirmed numerically by Matsumoto (1984), observed experimentally by Zhong and Ayrom (1985), and proved rigorously in (Chua, et al, 1986). The basic approach of the proof is illustrated in a guided exercise on Chua’s circuit in the wellknown textbook by Hirsch, Smale and Devaney (2003).
Circuit Diagram and Realization
The circuit diagram of the Chua Circuit is shown in Figure 1. It contains 5 circuit elements. The first four elements on the left are standard offtheshelf linear passive electrical components; namely, inductance L > 0, resistance R > 0, and two capacitances C_{1} > 0 and C_{2} > 0. They are called passive elements because they do not need a power supply (e.g., battery). Interconnection of passive elements always leads to trivial dynamics, with all element voltages and currents tending to zero (Chua, 1969).
Local Activity is Necessary for Chaos
The simplest circuit that could give rise to oscillatory or chaotic waveforms must include at least one locally active (Chua, 1998), (Chua, 2005) nonlinear element, powered by a battery, such as the Chua diode shown in Figure 1, characterized by a current vs. voltage nonlinear function \(i_R = g(v_R)\ ,\) whose slope must be negative somewhere on the curve. Such an element is called a locally active resistor. Although the function \( g(\bullet)\) may assume many shapes,the original Chua circuit specifies the 3segment piecewiselinear oddsymmetric characteristic shown in the right hand side of Figure 1, where m_{0} denotes the slope of the middle segment and m_{1} denotes the slope of the two outer segments ; namely,
\[ g(v_R) = \begin{cases} m_1 v_R + m_1  m_0 & , if \quad v_R \le 1 \\ m_0 v_R & , if \quad 1 \le v_R \le 1 \\ m_1 v_R + m_0  m_1 & , if \quad 1 \le v_R \end{cases} \] where the coordinate of the two symmetric breakpoints are normalized, without loss of generality, to \(v_R\) = \( \pm 1 \ .\)
The Chua Diode is Locally Active
The Chua diode is not an offtheshelf component. However, there are many ways to synthesize such an element using offtheshelf components and a power supply, such as batteries. The circuit for realizing the Chua diode need not concern us since the dynamical behavior of the Chua Circuit depends only on the 4 parameter values L, R, C_{1}, C_{2} and the nonlinear characteristic function \( g(\bullet)\ .\)
Any locally active device requires a power supply for the same reason a mobile phone can not function without batteries (Chua, 1969). A physical circuit for realizing the Chua Circuit in Figure 1 is shown in Figure 2.
Observe the onetoone correspondence between each linear circuit element in Figure 1 and its corresponding physical component in Figure 2 (Gandhi et al, 2007). The Chua diode in Figure 1 corresponds to the small black box with two external wires soldered across capacitance C_{1}. Two batteries are used to supply power for the Chua diode. The parameter values for L, R, C_{1}, and C_{2}, as well as instructions for building the Chua diode in Figure 1 are given in (Kennedy, 1992).
Figure 3 shows the complete Chua Circuit, including the circuit schematic diagram (enclosed inside the box N_{R}) for realizing the Chua diode, using 2 standard Operational Amplifiers (Op Amps) and 6 linear resistors (Gandhi et al. 2007).
The two vertical terminals emanating from each Op Amp (labeled \(V^+\) and \(V^\ ,\) respectively) in Figure 3 must be connected to the plus and minus terminals of a 9 volt battery, respectively.
There are many other circuits for realizing the Chua diode. The most compact albeit expensive way is to design an integrated electronic circuit, such as the physical circuit shown in Figure 4, where the black box in Figure 2 had been replaced by a single IC chip (Cruz and Chua, 1993), and powered by only one battery.
Oscilloscope Displays of Chaos
Using the Chua Circuit shown in Figure 4, the voltage waveforms \( v_{C_1} (t)\) and \( v_{C_2} (t)\) across capacitors C_{1} and C_{2}, and the current waveform \( i_L (t)\) through the inductor L in Figure 1, were observed using an oscilloscope and displayed in Figure 5 (a), (b), and (c) (left column), respectively.
The Lissajous figures associated with 3 permutated pairs of waveforms are displayed on the right column Figure 5; namely, in the \(v_{C_1}i_L\) plane in Figure 5(d), the \(v_{C_1}v_{C_2}\) plane in Figure 5(e), and the \(v_{C_2}i_L\) plane Figure 5(f). They are 2dimensional projections of the chaotic attractor, called the double scroll, traced out by the 3 waveforms from the left column in the 3dimensional \( v_{C_1}  v_{C_2}  i_L\) space.
It is important to point out that the Chua Circuit is not an analog computer. Rather it is a physical system where the voltage, current, and power associated with each of the 5 circuit elements in Figure 1 can be measured and observed on an oscilloscope, and where the power flow among the elements makes physical sense. In an analog computer (usually using Op Amps interconnected with other electronic components to mimic some prescribed set of differential equations), the measured voltages have no physical meanings because the corresponding currents and powers can not be identified, let alone measured, from the analog computer.
Chua Equations
By rescaling the circuit variables \( v_{C_1}\ ,\) \( v_{C_2}\ ,\) and \( i_L\) from Figure 1, we obtain the following dimensionless Chua Equations involving 3 dimensionless state variables x, y, z, and only 2 dimensionless parameters \(\alpha\) and β :
Equations 

where \(\alpha\) and β are real numbers, and \( \phi(x)\) is a scalar function of the single variable \( x \ .\) The Chua Equations are simpler than the Lorenz Equations in the sense that it contains only one scalar nonlinearity, whereas the Lorenz Equations contains 3 nonlinear terms, each consisting of a product of two variables (Pivka et al, 1996). In the original version studied indepth in (Chua et al, 1986), \( \phi(x)\) is defined as a piecewiselinear function
\[ \phi(x) \stackrel{\triangle}{=} x + g(x) = m_1 x + \frac{1}{2} (m_0  m_1)[x+1x1] \]
where m_{0} and m_{1} denote the slope of the inner and outer segments of the piecewiselinear function in Figure 1, respectively. Although simpler smooth scalar functions, such as polynomials, could be chosen for \( \phi(x)\) without affecting the qualitative behaviors of the Chua Equations, a continuous (but not differentiable) piecewiselinear function was chosen strategically from the outset in (Chua et al, 1986) in order to devise a rigorous proof showing the experimentally and numerically derived double scroll attractor is indeed chaotic. Unlike the Lorenz attractor (Lorenz, 1963), which had not been proven to be chaotic until 36 years later (Stewart, 2000) by Tucker (1999), it was possible to prove the double scroll attractor from the Chua Circuit is chaotic by virtue of the fact that certain Poincare return maps associated with the attractor can be derived explicitly in analytical form via compositions of eigen vectors within each linear region of the 3dimensional state space (Chua et al, 1986), (Shilnikov, 1994).
Fractal Geometry of the Double Scroll Attractor
Based on an indepth analysis of the phase portrait located in each of the 3 linear regions of the xyz state space, as well as from a detailed numerical analysis of the double scroll attractor shown in Figure 6, the geometrical structure of the double scroll attractor is found to consist of a juxtaposition of infinitely many thin, concentric, oppositelydirected fractallike layers. The local geometry of each cross section appears to be a fractal at all cross sections and scales. This fractal geometry is depicted in the caricature shown in Figure 7. A 3dimensional model of the double scroll attractor, accurate to millimeter scales, has been carefully sculpted using red and blue fiber glass, and displayed in Figure 8.
PeriodDoubling Route to Chaos
By fixing the parameters of the Chua Equations at \(\alpha\) = 15.6, m_{0} = 8/7 and m_{1} = 5/7, and varying the parameter β from β = 25 to β = 51, one observes a classic perioddoubling bifurcation route to chaos (Kennedy, 2005). This is depicted in Figure 9, reproduced from page 377 of (Alligood et al, 1997).
Interior Crisis and Boundary Crisis
By fixing the parameters of the Chua Equations at \(\alpha\) = 15.6, m_{0} = 8/7 and m_{1} = 5/7, and varying the parameter β from β = 32 to β = 30, Figure 10 (reproduced from page 421 of Alligood et al (1997)) shows the bifurcation of a pair of coexisting Rösslerlike attractors with separate basins of attraction moving toward one another until they touch at β = 31, whereupon the two twin attractors merge into a single double scroll attractor. A further reduction to β = 30 triggers a boundary crisis, resulting in a periodic orbit.
Generalizations
There exists several generalized versions of the Chua Circuit. One generalization substitutes the continuous piecewiselinear function \( \phi(x)\) by a smooth function, such as a cubic polynomial (Khibnik et al, 1993), (Shilnikov, 1994), (Huang et al, 1996), (Hirsch et al, 2003), (Tsuneda, 2005), (O’Donoghue et al, 2005). For example, Hirsch, Smale and Devaney chose
\[ \phi(x) \stackrel{\triangle}{=} = \frac{1}{16} x^3  \frac{1}{6}x \]
with \(\alpha\) = 10.91865 and β = 14 to obtain a pair of homoclinic orbits, a much coveted precursor of chaos (Shilnikov, 1994).
Another generalization replaces the third equation in the Chua Equations by
\[ \dot{z} = \beta y  \gamma z \]
thereby introducing a third parameter \( \gamma \) (Chua, 1993). This unfolding of the originl vector field gives rise to a surprisingly large number of topologically distinct chaotic attractors. For example, Bilotta had reported almost a thousand attractors (which appears to exhibit different geometrical structures) from the generalized Chua Equations (Bilotta et al, 2007).
Various forms of the Chua Equations can be found in textbooks on nonlinear dynamics (Hirsch et al, 2003), (Alligood et al, 1997) and chaos (van Wyk and Steeb, 1997), (Sprott, 2003), where a more detailed mathematical analysis can be found.
Applications
The Chua Circuit has been built and used in many laboratories as a physical source of pseudo random signals, and in numerous experiments on synchronization studies, such as secure communication systems and simulations of brain dynamics. It has also been used extensively in many numerical simulations, and exploited in avantgarde music compositions (Bilotta et al, 2005), and in the evolution of natural languages (Bilotta and Pantano, 2006).
Arrays of Chua Circuits have been used to generate 2dimensional spiral waves, 3dimensional scroll waves, (Munuzuri et al, 1993) and stationary patterns, such as Turing and other exotic patterns, (Munuzuri and Chua, 1997), (Madan, 1993), as illustrated in Figures 11(a), (b), and (c), respectively. Such highdimensional attractors have been exploited for applications in image processing, neural networks, dynamic associative memories (Itoh and Chua, 2004), complexity (Chua, 1998), emergence (Arena et al, 2005), etc.
External Links
http://sprott.physics.wisc.edu/chaostsa/
http://www.chuacircuit.com
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Internal references
 John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
 Edward Ott (2006) Basin of attraction. Scholarpedia, 1(8):1701.
 John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
 Valentino Braitenberg (2007) Brain. Scholarpedia, 2(11):2918.
 Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.
 James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
 Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHughNagumo model. Scholarpedia, 1(9):1349.
 Mark Aronoff (2007) Language. Scholarpedia, 2(5):3175.
 Kendall E. Atkinson (2007) Numerical analysis. Scholarpedia, 2(8):3163.
 Arkady Pikovsky and Michael Rosenblum (2007) Synchronization. Scholarpedia, 2(12):1459.
 James Murdock (2006) Unfoldings. Scholarpedia, 1(12):1904.