A Simple Chaotic Circuit

1
A Simple Chaotic Circuit
Ken Kiers and Dory Schmidt Physics Department,
Taylor University, 236 West Reade Ave., Upland,
Indiana 46989 J.C. Sprott Department of Physics,
University of Wisconsin, 1150 University Ave.,
Madison, WI 53706
EQ. 1 Differential equation represented by the
circuit shown in Fig. 1.
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. 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
1?F 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?.
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
 
TABLE I Comparison of the experimental and
theoretical bifurcation points labeled in Fig. 5.
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. 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. 8 Experimental first- and second-return
maps for Rv 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.
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.
FIG. 4 Block diagram of the complete setup with
circuit connected to several power supplies and
the digital potentiometers sending output to a
computer.
Chaos 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.