Download PARTICLE COARSENING IN - University of Wisconsin–Madison

A Simple Chaotic Circuit
Ken Kiers and Dory Schmidt
Physics Department, Taylor University, 236 West Reade Ave., Upland, Indiana 46989
J.C. Sprott
CIRCUIT
R
R
x    x  x  Dx    V0
 Rv 
 R0 
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.
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.
RESULTS
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.
a
Exp. (k) Theory (k) Diff. (k) Diff. (%)
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. 4: Block diagram of the complete setup with “circuit” connected to several
power supplies and the digital potentiometers sending output to a computer.
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. 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 topmost plot a theoretical curve has been super-imposed on top of the
experimental data and the peaks agree to less than one percent.
CONCLUSIONS
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.
FIG. 8: Experimental firstand 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.