A Simple Chaotic CircuitA Simple Chaotic CircuitKen Kiers and Dory Schmidt

Physics Department, Taylor University, 236 West Reade Ave., Upland, Indiana 46989

J.C. SprottDepartment of Physics, University of Wisconsin, 1150 University Ave., Madison, WI 53706

FIG. 1: Circuit diagram of our simple chaotic circuit. The box labeled D(x) represents an arrangement of diodes, resistors, and an operational amplifier that provide the necessary nonlinearity. All unlabeled resistors and capacitors have a nominal value of R = 47k and C = 1F respectively. The values of the other components are approximately V0 = 0.250V and R0 = 157k with the nodes labeled V1 and V2 representing -x’ and x’’ respectively. Also, Rv represents a variable resistor composed of a fixed resistor in series with eight digital potentiometers providing a range for Rv from approximately 50k to 130k.

FIG. 2: Subcircuit, D(x), shown in Fig. 1. The voltage at VIN corresponds to x while the voltage at VOUT = D(VIN) =-(R2/R1)min(VIN,0). For this study R2 6R1.

FIG. 3: Plot of D(x) vs. x showing the relationship between the voltage on the left side of the “box” in Fig. 1 and the voltage on the right side. By using the slightly more complicated arrangement of electrical components, the results obtained agree very well with theory. In contrast, a “bare” diode does not yield such precise results.

FIG. 5: Bifurcation plots of both experimental and theoretical data based on the circuits shown in Figs. 1 and 2 as well as a superimposed view of the two. The minute differences between the superimposed plot and the experimental or theoretical plot demonstrates the excellent agreement.

TABLE I: Comparison of the experimental and theoretical bifurcation points labeled in Fig. 5.

FIG. 6: Power spectral density plots from experimental data with insets showing the corresponding time series data. For each value or Rv there is a dominant frequency at approximately 3Hz although as Rv is increased, this peak shifts slightly to the right. In the top-most plot a theoretical curve has been super-imposed on top of the experimental data and the peaks agree to less than one percent.

FIG. 7: Experimental phase portraits for several different values of Rv. The upper-left and lower-right plots correspond to chaotic attractors with the latter representing a two-banded attractor. The upper-right and lower-left plots show the data from a period six region and period ten region respectively. The period six plot has a theoretical curve super-imposed over the experimental data and because of the inability to distinguish between the two curves, it is apparent that the experimental data and theoretical expectations agree to a high degree of precision.

FIG. 8: Experimental first- and second-return maps forRv = 72.1k. In each case, the intersection of the diagonal line with the return map gives evidence for the existence of unstable period one and period two orbits. The time data series in Fig. 9 shows examples of these unstable orbits.

FIG. 9: Experimental waveforms for Rv = 72.1k. This Rv value corresponds to a chaotic region and yet within the chaos, there are unstable regions of periodicity. The top plot shows an unstable period one region at 0.41V while the bottom plot portrays an unstable period two region with maxima oscillating between 0.10V and 0.57V. Both of these plots have maxima that correlate with the expected values from the return maps in Fig. 8.

CIRCUIT

RESULTS

FIG. 4: Block diagram of the complete setup with “circuit” connected to several power supplies and the digital potentiometers sending output to a computer.

The study of chaos provides an ideal avenue for understanding nonlinear systems. In this study, the chaotic behavior is provided by a simple nonlinear electronic circuit. It contains several common electronic components including resistors, diodes and operational amplifiers. Along with its simple structure, several key benefits include its stability as well as the agreement between experiment and theory of less than one percent for bifurcation points and power spectra.

00

VR

RxDxx

R

Rx

v

EQ. 1: Differential equation represented by the circuit shown in Fig. 1.

CONCLUSIONSChaos is a fascinating area of research that is very suitable for students at the undergraduate level. It provides a wide array of ways to view a single data set including bifurcation plots, phase portraits, and power spectra. Other quantities can also be calculated such as the Lyapunov exponent or the Kaplan-Yorke dimension. While we use a very detailed A/D system, it is also possible to digitize the data using a digital oscilloscope. Furthermore, replacing the digital potentiometers with an analog potentiometer is a another possible simplification that can be made. Also, the operating frequency of the circuit can be adjusted to the audible range by scaling the resistors and capacitors therefore providing a useful demonstration of periodic and chaotic behavior. While these are minor changes that can be made on the specific circuit shown here, the nonlinearity, D(x), can also be replaced providing a whole new path for further study. Because of its stability, precision when comparing experiment to theory and wide variety of ways to study the data obtained, this circuit is very appropriate for the undergraduate research lab.

Exp. (k) Theory (k) Diff. (k) Diff. (%)

a 53.2 53.15 0.05 0.1

b 65.1 65.1 0.0 0.0

c 78.8 78.7 0.1 0.1

d 101.7 101.6 0.1 0.1

e 125.2 125.6 -0.4 -0.3