A nonlinear electronic circuit comprising of three nodes with a feedback loop is analyzed. The system has two stable states, a uniform state and a sinusoidal oscillating state, and it transitions from one to another by means of a Hopf bifurcation. The stability of this system is analyzed with nonlinear equations derived from a repressilator-like transistor circuit. The apparatus is simple and inexpensive, and the experiment demonstrates aspects of nonlinear dynamical systems in an advanced undergraduate laboratory setting.

1.
S. H.
Strogatz
,
Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering
, 2nd ed. (
Westview Press
,
Boulder, CO
,
2015
), pp.
251
256
.
2.
M.
Heinrich
,
T.
Dahms
,
V.
Flunkert
,
S. W.
Teitsworth
, and
E.
Schöll
, “
Symmetry-breaking transitions in networks of nonlinear circuit elements
,”
New J. Phys.
12
,
113030
(
2010
).
3.
J. C.
Sprott
, “
Simple chaotic systems and circuits
,”
Am. J. Phys.
68
,
758
763
(
2000
).
4.
T.
Mishina
,
T.
Kohmoto
, and
T.
Hashi
, “
Simple electronic circuit for the demonstration of chaotic phenomena
,”
Am. J. Phys.
53
,
332
334
(
1985
).
5.
E. H.
Hellen
, “
Real-time finite difference bifurcation diagrams from analog electronic circuits
,”
Am. J. Phys.
72
,
499
502
(
2004
).
6.
D.
Goswami
and
S.
Ray
, “
Feature-rich bifurcations in a simple electronic circuit
,” e-print arXiv:1705.07101v1.
7.
A.
Sack
,
J. G.
Freire
,
E.
Lindberg
,
T.
Pöschel
, and
J. A. C.
Gallas
, “
Discontinuous spirals of stable periodic oscillations
,”
Sci. Rep.
3
,
3350
(
2013
).
8.
C.
Cabeza
,
C. A.
Briozzo
,
R.
Garcia
,
J. G.
Freire
, and
J. A. C.
Gallas
, “
Periodicity hubs and wide spirals in a two-component autonomous electronic circuit
,”
Chaos, Solitons Fractals
52
,
59
65
(
2013
).
9.
A. S.
Sedra
and
K. C.
Smith
,
Microelectronic Circuits
, 5th ed. (
Oxford U. P
.,
New York
,
2004
)
10.
A.
Verdugo
, “
Hopf bifurcation analysis of the repressilator model
,”
Am. J. Comput. Math.
8
,
137
152
(
2018
).
11.
James A.
Blackburn
,
H. J. T.
Smith
, and
N.
Grønbech-Jensen
, “
Stability and Hopf bifurcations in an inverted pendulum
,”
Am. J. Phys.
60
,
903
908
(
1992
).
12.
J. P.
Sharpe
and
N.
Sungar
, “
Supercritical bifurcation in a simple mechanical system: An undergraduate experiment
,”
Am. J. Phys
78
(
5
),
520
523
(
2010
).
13.
P. G.
Drazin
and
W. H.
Reid
,
Hydrodynamic Stability
, 2nd ed. (
Cambridge U. P
.,
England
,
2004
), p.
405
.
14.
C. R.
Wallis
and
S. W.
Teitsworth
, “
Hopf bifurcations and hysteresis in resonant tunneling diode circuits
,”
J. Appl. Phys.
76
,
4443
4445
(
1994
).
15.
D. N.
Rim
,
P.
Cremades
, and
P.
Kaluza
, “
A simple electronic device to experiment with the Hopf bifurcation
,”
Rev. Mex. Fís. E
65
,
58
63
(
2019
).
16.
J.
Guckenheimer
and
P.
Holmes
,
Nonlinear Oscillations Dynamical Systems, and Bifurcations of Vector Fields
, 1st ed. (
Springer-Verlag
,
New York
,
1983
), pp.
151
152
.
17.
S. M.
Stewart
, “
A new elementary function for our curricula?
,”
Aust. Senior Math. J.
19
(
2
),
8
26 (2005)
.
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