We describe a simple nonlinear electrical circuit that can be used to study chaotic phenomena. The circuit employs simple electronic elements such as diodes, resistors, and operational amplifiers, and is easy to construct. A novel feature of the circuit is its use of an almost ideal nonlinear element, which is straightforward to model theoretically and leads to excellent agreement between experiment and theory. For example, comparisons of bifurcation points and power spectra give agreement to within 1%. The circuit yields a broad range of behavior and is well suited for qualitative demonstrations and as a serious research tool.
REFERENCES
1.
T.
Mishina
, T.
Kohmoto
, and T.
Hashi
, “Simple electronic circuit for the demonstration of chaotic phenomena
,” Am. J. Phys.
53
, 332
–334
(1985
).2.
K.
Briggs
, “Simple experiments in chaotic dynamics
,” Am. J. Phys.
55
, 1083
–1089
(1987
).3.
R.
Zimmerman
, S.
Celaschi
, and L. G.
Neto
, “The electronic bouncing ball
,” Am. J. Phys.
60
, 370
–375
(1992
).4.
T.
Mitchell
and P. B.
Siegel
, “A simple setup to observe attractors in phase space
,” Am. J. Phys.
61
, 855
–856
(1993
).5.
M. T.
Levinsen
, “The chaotic oscilloscope
,” Am. J. Phys.
61
, 155
–165
(1993
).6.
B. K.
Clark
, R. F.
Martin
, Jr., R. J.
Moore
, and K. E.
Jesse
, “Fractal dimension of the strange attractor of the bouncing ball circuit
,” Am. J. Phys.
63
, 157
–163
(1995
).7.
B. K.
Jones
and G.
Trefan
, “The Duffing oscillator: A precise electronic analog chaos demonstrator for the undergraduate laboratory
,” Am. J. Phys.
69
, 464
–469
(2001
).8.
P. K.
Roy
and A.
Basuray
, “A high frequency chaotic signal generator: A demonstration experiment
,” Am. J. Phys.
71
, 34
–37
(2003
).9.
T.
Matsumoto
, L. O.
Chua
, and M.
Komuro
, “The double scroll
,” IEEE Trans. Circuits Syst.
32
, 797
–818
(1985
).10.
T.
Matsumoto
, L. O.
Chua
, and M.
Komuro
, “Birth and death of the double scroll
,” Physica D
24
, 97
–124
(1987
).11.
G. Chen and T. Ueta, editors, Chaos in Circuits and Systems (World Scientific, London, 2002).
12.
T. P.
Weldon
, “An inductorless double scroll chaotic circuit
,” Am. J. Phys.
58
, 936
–941
(1990
).13.
A. S.
Elwakil
and M. P.
Kennedy
, “Generic RC realizations of Chua’s circuit
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
10
, 1981
–1985
(2000
).14.
A. S.
Elwakil
and M. P.
Kennedy
, “Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices
,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl.
48
, 289
–307
(2001
).15.
J. C.
Sprott
, “Simple chaotic systems and circuits
,” Am. J. Phys.
68
, 758
–763
(2000
).16.
J. C.
Sprott
, “A new class of chaotic circuit
,” Phys. Lett. A
266
, 19
–23
(2000
).17.
K. Kiers, T. Klein, J. Kolb, S. Price, and J. C. Sprott, “Chaos in a nonlinear analog computer,” Int. J Bifurcations Chaos (to be published).
18.
H. P. W.
Gottlieb
, “Question #38. What is the simplest jerk function that gives chaos?
,” Am. J. Phys.
64
, 525
(1996
).19.
S. J.
Linz
, “Nonlinear dynamical models and jerky motion
,” Am. J. Phys.
65
, 523
–526
(1997
).20.
J. C.
Sprott
, “Some simple chaotic jerk functions
,” Am. J. Phys.
65
, 537
–543
(1997
).21.
S. J.
Linz
and J. C.
Sprott
, “Elementary chaotic flow
,” Phys. Lett. A
259
, 240
–245
(1999
).22.
J. C.
Sprott
and S. J.
Linz
, “Algebraically simple chaotic flows
,” Int. J. Chaos Theory Appl.
5
, 3
–22
(2000
).23.
E.-W.
Bai
, K. E.
Lonngren
, and J. C.
Sprott
, “On the synchronization of a class of electronic circuits that exhibit chaos
,” Chaos, Solitons Fractals
13
, 1515
–1521
(2002
).24.
K. M.
Cuomo
and A. V.
Oppenheim
, “Circuit implementation of synchronized chaos with applications to communications
,” Phys. Rev. Lett.
71
, 65
–68
(1993
).25.
The circuit also differs from that in Ref. 15 in a few other respects. For example, the present circuit does not use a passive integrator.
26.
For the particular circuit that we study only the combination is relevant, and it only affects the amplitude of the signal (as long as the op amps do not saturate). Thus cannot be used as a control parameter for this circuit. Such is not necessarily the case if different nonlinearities are employed.
27.
P. Horowitz and W. Hill, The Art of Electronics, 2nd ed. (Cambridge University Press, New York, 1989).
28.
Note that because the output of an integrator is the integral of its input, its input is also the derivative of its output.
29.
Note that this expression assumes that zero current leaves the output, a situation that holds more rigorously if the output is followed by a voltage follower amplifier. We do not use a voltage follower in the experiment. As a result, Eq. (8) is slightly altered once the subcircuit is inserted into the circuit itself, with the factor on the left-hand side replaced by The approximation quoted in Eq. (9) is not affected by this change.
30.
In a detailed study of the simpler single-diode circuit of Ref. 15, we employed a piecewise fit of the diode’s curve (using several values of α and . Even with the careful fit to the curve, the agreement between theoretical and experimental bifurcation points was not as good as in the present case, and the theoretical and experimental bifurcation diagrams contained some noticeable qualitative differences.
31.
http://www.css.tayloru.edu/∼knkiers
32.
Note that the data set is AC-coupled before the FFT to eliminate a strong peak near zero Hz.
33.
One might expect the theoretical curve to contain a series of delta functions. The finite sample size (8192 points) leads to the observed nonzero widths of the peaks in this curve.
34.
D. P.
Lathrop
and E. J.
Kostelich
, “Characterization of an experimental strange attractor by periodic orbits
,” Phys. Rev. A
40
, 4028
–4031
(1989
).35.
C.
Flynn
and N.
Wilson
, “A simple method for controlling chaos
,” Am. J. Phys.
66
, 730
–735
(1998
).36.
E. Ott, Chaos in Dynamical Systems (Cambridge University Press, New York, 1993).
37.
A.
Wolf
, J. B.
Swift
, H. L.
Swinney
, and J. A.
Vastano
, “Determining Lyapunov exponents from a time series
,” Physica D
16
, 285
–317
(1985
).38.
J. C. Sprott and G. Rowlands, Chaos Data Analyzer: The Professional Version (Physics Academic Software, Raleigh, NC, 1995).
39.
J. C. Sprott, Chaos and Time-Series Analysis (Oxford University Press, Oxford, 2003).
40.
J. Kaplan and J. Yorke, in Lecture Notes in Mathematics, edited by H.-O. Peitgen and H.-O. Walther (Springer, Berlin, 1979), Vol. 370, pp. 228–237.
This content is only available via PDF.
© 2004 American Association of Physics Teachers.
2004
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
