A self-oscillator generates and maintains a periodic motion at the expense of an energy source with no corresponding periodicity. Small perturbations about equilibrium are negatively damped, leading to instability of the linear equations of motion. Non-linearity accounts for the steady-state oscillation and for the ability of coupled self-oscillators to exhibit spontaneous synchronization ("entrainment") and chaos. The theory of self-oscillators has achieved its greatest sophistication in control theory and in the theory of ordinary differential equations. I shall explain why an understanding better suited to the needs of engineers and physicists is based on considerations of energy, efficiency, and irreversibility. After reviewing how forced and parametric resonances are described by energy flow into a driven oscillator, I shall explain how a simple example of self-oscillation (the swaying of the London Millennium Footbridge in 2000) results from positive feedback between the oscillation and an external energy source (the motion of pedestrians on the bridge). I shall explain how the nonlinearity of self-oscillators connects to entrainment and to the thermodynamic irreversibility of motors, before commenting on how this approach throws new light on concrete problems, like the maintenance of the Chandler wobble in geophysics. This presentation will be based on the review article A. Jenkins, Phys. Rep. 525, 167 (2013), and a book currently in preparation.

1. The energetics of
self-oscillators
Alejandro Jenkins
U. de Costa Rica &
Academia Nacional de Ciencias
International Conference on Advances in Vibrations
U. do Porto, Portugal
30 March 2015

2. References
• Self-oscillators maintain regular,
periodic motion, at expense of
power source with no
corresponding periodicity
• Positive feedback between
oscillation and power modulation
• AJ, “Self-Oscillation”, Phys. Rep.
525, 167 (2013)
• AJ, The Physical Theory of Self-
Oscillators, (in preparation)
2
This article appeared in a journal published by Elsevier. The attached
copy is furnished to the author for internal non-commercial research
and education use, including for instruction at the authors institution
and sharing with colleagues.

3. Control theory
“A distinguishing feature of
this new science is the total
absence of considerations
of energy, heat, and
efficiency, which are so
important in other natural
sciences.”
- Qian Xuesen, Engineering
Cybernetics (1954)
3
Qian Xuesen (1911 – 2009)

4. Rayleigh
4
¨q ↵ ˙q + ˙q3
/3 + !2
q = 0
Combines linear anti-damping α
2nd ed. of Theory of Sound (1894-6)
models “maintained oscillations”,
including wind musical instruments,
by
Ploss = ˙qFdamp = m ˙q4
/3
Pgain = ˙qFanti damp = ↵m ˙q2
with non-linear damping β

5. Van der Pol
5
¨V ↵ V 2 ˙V + !2
V = 0
Van der Pol (1920) uses eq. equivalent to Rayleigh’s:
V0 = 2
r
↵
Steady amplitude:
Directly implementable as electric circuit:
Vin Vout
R
L C
I0
V0
Idiode
V
I
I0
V0
Vin = g · Vout

6. Limit cycles
¨x ↵ 1 x2
˙x + x = 0
⇢
˙x = ↵ y + x x3
/3
˙y = x/↵
Liénard transformation:
-2 -1 1 2
x
-2
-1
1
2
y
↵ = 0.2
10 20 30 40 50 60
t
-2
-1
1
2
Vx
6

7. -2 -1 1 2
x
-1.0
-0.5
0.5
1.0
y
5 10 15 20 25 30
t
-2
-1
1
2
Vx
↵ = 5
- 3 -2 -1 1 2 3 4
x
- 3
-2
-1
1
2
y
↵ = 0.2 10 20 30 40 50 60
t
- 3
-2
-1
1
2
3
4
Vx
7

8. Positive feedback
• Vout amplified & fed
back to Vin
• Resistance effectively
negative
• Exponential growth
limited by amplifier’s
saturation (nonlinearity)
• All clocks work on this
principle
Vin = g · Vout
¨Vout +
1 g
RC
˙Vout +
1
LC
Vout = 0
8
Vin Vout
R
L C

9. Millennium Footbridge
(London)
http://www.youtube.com/watch?v=eAXVa__XWZ8
9

10. Kelvin-Helmholtz
⇠ ⇠ exp [ik(x vt)] v = !/k
Galileo:
@⇠
@t
!
@⇠
@t
+ V
@⇠
@x
= i!⇠ + V · ik⇠
= i!⇠
✓
1
V
v
◆
See: Zel’dovich, JETP Lett. 14, 180 (1971)
10
Air
Water v= wave velocity, seen by the water
V = water velocity, seen by the air
2 /k

11. Relaxation oscillation
• Pearson-Anson flasher
(1922)
• Period not associated
to resonance
• Non-linear switching at
thresholds
• Theory by Van der Pol
& Friedländer (1926)
• e.g., heart & neurons
Vout
R
CV0
neon
lamp
11
Vout
Von
t
Voff

12. Synchronization
• Huygens (1665) noticed that two adjacent pendulum
clocks, mounted on wooden partition, ended up moving
in anti-phase
• Locking of frequency, mode, or phase (synchronization)
possible because non-linear oscillator’s frequency
depends on its amplitude
• Amplitude may vary until phase relative to forcing
motion makes power input match dissipation
• See discussion of “Duffing problem” in Sargent, Scully
& Lamb, Laser Physics (1974), sec. 3-2
12

13. Forcing
• Relaxation oscillators
are particularly easy
to entrain
• Also show:
A. demultiplication
B. quasi-periodicity
C. chaos
¨x 3 1 x2
˙x + x = 5 cos(1.788t)
10 20 30 40 50 60
t
-2
-1
1
2
x
x(0) = 0.1 , ˙x(0) = 0
x(0) = 0.1 , ˙x(0) = 0.01
13

14. Chandler wobble
• Extinctions, followed by phase
jumps, in 1850s, 1920s &
2000s
• Not associated with obvious
geophysical events
• Work in progress: wobble as
self-oscillation, powered by
fluid circulations
• Turned on & off by stochastic
perturbations (Hopf bifurcation)
14
Malkin & Miller, Earth Planets Space 62, 943 (2010)
Singular spectrum analysis (SAS) filtered:

15. Classical engine
• Two heat baths, working
fluid, piston
• Piston modulates power
from working fluid, via fly-
wheel & valves
• Positive feedback
between piston & valve
action allows net work
extraction
15
Andronov, Vitt & Khaikin, Theory of Oscillators
(Dover, 1987 [1966]), ch. VIII, sec. 10

16. ħΩ
Quantum engine
• Recent work by Alicki et al. on
quantum engines
• Solar cell: baths at phonon
(room) temperature + incident
photons at high effective
temperature (~1000 K)
• Plasma oscillation at p-n
junction may act as piston
• Ω/2π ~ 1 THz
• Maintains cyclic DC current
16
Alicki, Gelbwaser-Klimovsky & Szczygielski,
arXiv:1501.00701 [cond-mat.stat-mech]

17. Summary
• Self-oscillation usually studied within mathematical
theory of non-linear dynamical systems
• Energetics dispels needless obscurities
• Intermittent self-oscillation (Hopf bifurcation) may
account for some heavy-tailed distributions &
other complex phenomena
• Picture of motors as self-oscillators useful to
thermodynamics
17