We show that electric motors and dynamos can be used to illustrate most elementary instabilities or bifurcations discussed in courses on nonlinear oscillators and dynamical systems. These examples are easier to understand and display a richer behavior than the ones commonly used from mechanics, electronics, hydrodynamics, lasers, chemical reactions, and population dynamics. In particular, an electric motor driven by a dynamo can display stationary, Hopf, and codimension-two bifurcations by tuning the driving speed of the dynamo and the electric current in the stator of the electric motor. When the dynamo is driven at constant torque instead of constant rotation rate, chaotic reversals of the generated current and of the angular rotation of the motor are observed. Simple deterministic models are presented which capture the observed dynamical regimes.

Topics
Electrical properties and parameters, Oscillators, Magnetic fields, Electric motors, Mathematical modeling, Dynamical systems, Phase space methods, Chaotic regime, Hydrodynamics, Population ecology
1.
G.
 
Iooss
 and
D. D.
 
Joseph
,
Elementary Stability and Bifurcation Theory
(
Springer
,
Berlin
,
1980
).
Google Scholar
Crossref
Search ADS
 
2.
V.
 
Arnold
,
Geometrical Methods in the Theory of Ordinary Differential Equations
(
Springer
,
Berlin
,
1982
).
Google Scholar
 
3.
J.
 
Guckenheimer
 and
P.
 
Holmes
,
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields
(
Springer
,
Berlin
,
1983
).
Google Scholar
Crossref
Search ADS
 
4.
P. G.
 
Drazin
,
Nonlinear Systems
(
Cambridge U.P.
,
Cambridge
,
1992
).
Google Scholar
Crossref
Search ADS
 
5.
A. B.
 
Pippard
,
Response and Stability
(
Cambridge U.P.
,
Cambridge
,
1985
).
Google Scholar
 
6.
M. I.
 
Rabinovich
 and
D. I.
 
Trubetskov
,
Oscillations and Waves in Linear and Nonlinear Systems
(
Kluwer Academic Press
,
Dordrecht
,
1989
).
Google Scholar
Crossref
Search ADS
 
7.
A. A.
 
Andronov
,
A. A.
 
Vitt
, and
S. E.
 
Khaikin
,
Theory of Oscillations
(
Pergamon
,
Oxford
,
1966
).
Google Scholar
 
8.
G.
 
Nicolis
 and
I.
 
Prigogine
,
Self-Organization in Nonequilibrium Systems
(
John Wiley & Sons
,
New York
,
1977
).
Google Scholar
 
9.
S.
 
Chandrasekhar
,
Hydrodynamic and Hydromagnetic Stability
(
Oxford U.P.
,
Oxford
,
1961
).
Google Scholar
 
10.
R. J.
 
Stephenson
, “
Experiment with a unipolar generator and motor
,”
Am. J. Phys.
 
5
,
108
–
110
(
1937
).
https://doi.org/10.1119/1.1991194
Google Scholar
Crossref
Search ADS
 
11.
G.
 
Baker
 and
G.
 
Bradley
, “
A small scale dc motor experiment for the elementary laboratory
,”
Am. J. Phys.
 
21
,
689
–
690
(
1953
).
https://doi.org/10.1119/1.1998713
Google Scholar
Crossref
Search ADS
 
12.
S.
 
Bednarek
, “
Unipolar motors and their application to the demonstration of magnetic field properties
,”
Am. J. Phys.
 
70
,
455
–
461
(
2002
).
https://doi.org/10.1119/1.1412648
Google Scholar
Crossref
Search ADS
 
13.
Y.
 
Kraftmakher
, “
Experiments with a dc motor
,”
Eur. J. Phys.
 
31
,
863
–
870
(
2010
).
https://doi.org/10.1088/0143-0807/31/4/016
Google Scholar
Crossref
Search ADS
 
14.
P.-H.
 
Ng
,
S.-L.
 
Wong
, and
S.-Y.
 
Mak
, “
Efficiency measurement using a motor-dynamo module
,”
Phys. Educ.
 
44
,
639
–
643
(
2010
).
https://doi.org/10.1088/0031-9120/44/6/012
Google Scholar
Crossref
Search ADS
 
15.
C. W.
 
Siemens
, “
On the conversion of dynamical into electrical force without the aid of permanent magnetism
,”
Proc. R. Soc. London
 
15
,
367
–
369
(
1866–1867
).
Google Scholar
 
16.
M. A.
 
Witz
, “
Recherches sur les inversions de polarité des série-dynamos
,”
J. Phys. Theor. Appl.
 
8
,
581
–
586
(
1889
).
https://doi.org/10.1051/jphystap:018890080058101
Google Scholar
Crossref
Search ADS
 
17.
A.
 
Hugues
,
Electric Motors and Drives. Fundamentals, Types and Applications
, 3rd ed. (
Newnes
,
Oxford
,
2006
).
Google Scholar
 
18.
T.
 
Rikitake
, “
Oscillations of a system of disk dynamos
,”
Proc. Cambridge Philos. Soc.
 
54
,
89
–
105
(
1958
).
https://doi.org/10.1017/S0305004100033223
Google Scholar
Crossref
Search ADS
 
19.
M.
 
Berhanu
 et al., “
Magnetic field reversals in an experimental turbulent dynamo
,”
Eur. Phys. Lett.
 
77
,
59001
–
1
(
2007
).
https://doi.org/10.1209/0295-5075/77/59001
Google Scholar
Crossref
Search ADS
 
20.
F.
 
Ravelet
 et al., “
Chaotic dynamos generated by a turbulent flow of liquid sodium
,”
Phys. Rev. Lett.
 
101
,
074502
–
1
(
2008
).
https://doi.org/10.1103/PhysRevLett.101.074502
Google Scholar
Crossref
Search ADS
PubMed
 
21.
F.
Pétrélis
 and
S.
Fauve
 , “
Chaotic dynamics of the magnetic field generated by dynamo action in a turbulent flow
,”
J. Phys. Condens. Matter
 
20
(
49
),
494203
–
1
(
2008
);
Crossref
Search ADS
Google Scholar
 
F.
 
Pétrélis
 et al., “
Simple mechanism for reversals of Earth’s magnetic field
,”
Phys. Rev. Lett.
 
102
(
14
),
144503
–
1
(
2009
).
Crossref
Search ADS
[PubMed]
Google Scholar
 
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2012
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