CHAOS IN CHUA’S CIRCUITMike AudetDecember 6, 2007Math 53: Chaos!2INTRODUCTION AND BRIEF HISTORYThis project analyzes Chua’s circuit, a simple electronic circuit that demonstrates chaoticbehavior. The study consists of two main components: numerically exploring (in Matlab) theequations derived from the circuit, and constructing the actual circuit to replicate numericalresults with experimental data. The physical circuit demonstrates that one can create chaoticbehavior in a relatively simple system.Leon Chua invented the circuit in 1983 to demonstrate chaos in an actual physical modeland to prove that the Lorenz double-scroll attractor is chaotic.1 The electronic circuit suits thestudy of chaos well because one can precisely control its parameters and can readily observe theresults on an oscilloscope. The circuit became popular to study because it is easy to construct,and many people have built the circuit using ordinary electronic components. In fact, one canmodel the circuit using only resistors, capacitors, inductors, op-amps, and diodes. In 1986,Chua, Komuro, and Matsumoto proved rigorously that the Chua attractor is indeed chaotic.2CROSS’S VERSION OF THE CIRCUIT AND ITS EQUATIONSAs mentioned above, people have created versions of the circuit that do not rely on anyspecially produced parts (the original included a laboratory-made “Chua diode”); rather, one canbuild versions of the circuit using off-the-shelf components. Michael Cross proposed the versionof the Chua circuit used in this project and shown in Figure 1.3 The right-hand side of this circuit(all the components to the right of C1 in the diagram) simulates the Chua diode, providingnonlinear negative resistance. The operational amplifier and its resistors have a negative1 http://www.scholarpedia.org/article/Chua_Circuit.2 Ibid.3 Cross. http://www.cmp.caltech.edu/~mcc/chaos_new/Chua.html.3resistance of size –R1, and the diodes provide nonlinearity.4 The left side of the circuit acts as anRLC circuit, which would simply produce damped oscillations without the right-hand side.Figure 1: Cross’s Version of Chua’s Circuit5To derive the three differential equations for the system, we choose three variables thatchange over time: V1, the voltage across capacitor C1; V2, the voltage across capacitor C2; andI, the current through the inductor. We apply Kirchhoff’s first law—which states that the currententering a node equals the current leaving a node—to the nodes above C1 and C2 in the diagram.Next we apply Kirchhoff’s second law—which states that the sum of the voltage around a loopequals zero—to the loop containing the inductor and C2. Thus, Kirchhoff’s laws give thefollowing equations for the circuit:C1(dV1/dt) = (V2-V1)/R – g(V1)C2(dV2/dt) = -(V2-V1)/R + IL(dI/dt) = -rI – V24 Cross. http://www.cmp.caltech.edu/~mcc/chaos_new/Chua_docs/works.html.5 This diagram comes from Cross’s website. http://www.cmp.caltech.edu/~mcc/chaos_new/Chua.html.4Note that g(V1) is the current through the nonlinear part of the circuit. Michael Cross transformsthe above equations into the following nondimensional system of equations:6dX/dt = a(Y-X) – G(X)dY/dt = s[-a(Y-X) + Z]dZ/dt = -c(Y + pZ)wherea = R1/R b = 1-R1/R2 c = (C1*R1^2)/L s = C1/C2 p = r/R1and G(X) is the normalized current-voltage characteristic for the right part of the circuit:7G(X) =-X, |X|≤1-[1+b(|X|-1)]*sign(X), 1<|X|≤10[10(|X|-10)-(9b+1)]*sign(X), |X|>10The current-voltage characteristic for the negative nonlinear resistance in normalized terms is:To find the equilibrium points, we set the three differential equations equal to zero and letG(X)=+(1-b)-bX to derive (0,0,0) and:X = + (1-b)/[a/(1+pa) – b] Y = [ap/(1+ap)]*X Z = -Y/pThese fixed points will be used to plot the bifurcation diagram and a 2D attractor.6 Cross. http://www.cmp.caltech.edu/~mcc/chaos_new/Chua_docs/chua_eq.pdf.7 Cloyd 9. Cloyd noticed sign errors in the G(X) function published by Cross. G(X) here includes Cloyd’s corrections.5NUMERICAL ANALYSES IN MATLABMatlab’s ode45 can numerically model the three normalized differential equations toproduce attractors for the circuit. For the numerical analyses, we will use the values for R, R1,R2, C2, L, and r that are listed in Figure 1. We will vary C1 to change the behavior of thesystem. The default initial condition will be (0.1, 0.15, 0.05). Figure 2 shows a periodic orbit atC1=4.28e-9 and a chaotic orbit at C1=4.32e-9.8 In order to accurately model the attractors, I hadto adjust the ode45 settings of relative and absolute tolerance to 1e-6 and 1e-9, respectively.Figure 2: Orbits when C1=4.28e-9 and C1=4.32e-9To better understand the behavior of the circuit as C1 varies, we can examine abifurcation diagram. To create the bifurcation diagram, I considered a projection of the orbitonto the X-Y plane (which corresponds to the normalized V1-V2 plane) and where the orbits8 Cloyd identified these values for C1 as nonchaotic and chaotic; a bifurcation diagram also justifies these values.6intersect the projection onto the X-Y plane of the line which passes through the equilibriumpoints.9 Figure 3 shows a visualization of these intersection points. To find the points in Matlab,I used a flag variable that triggered if a given Y-value of the orbit was below the line betweenequilibrium points and the next Y-value was above it, or vice versa. The program then calculatedthe intersection of two lines projected onto the X-Y plane: the line between the equilibriumpoints and the line between the two consecutive points in the orbit that crossed the equilibrium-point line. Figure 3: Procedure to Create Bifurcation DiagramThe bifurcation diagram used the X (normalized V1) values of the intersections andplotted them along the vertical axis. The varying values of C1 were plotted along the horizontalaxis. To examine a wide range of behavior—including a period doubling route to chaos—I letC1 vary between 4.2e-9 and 4.5e-9. In order to balance resolution with computing time, I used1000 intervals. Figure 4 represents the end-result of an overnight computation: a clearbifurcation diagram for the values of C1. The diagram illustrates the periodic orbit at C1=4.28e-9 The idea to use this method came from Cloyd’s paper; the Matlab code was my own.79