Notes on the videotape
Nonlinear Dynamics and Chaos:
Lab Demonstrations
by Steven H. Strogatz
For more information, please write to me at Kimball Hall, Theoretical and Applied Mechanics,
Cornell University, Ithaca, NY 14853, or send e-mail to strogatz@cornell.edu
Please copy the videotape and give it to anybody who might be interested. Also, please send the
tape back to me after you’ve copied it, so I can send it to other people.
These notes are intended to summarize the main points in the tape, to provide background where
necessary, and to underscore the dynamical concepts that are being demonstrated.
1. Chaotic Waterwheel
(with Howard Stone, Division of Applied Sciences, Harvard)
A tabletop waterwheel, designed and built by Prof. Willem Malkus (Math. Dept., MIT), is
used to demonstrate chaos in a mechanical analog of the Lorenz equations. The waterwheel’s
rotational damping rate can be adjusted by tightening or loosening a brake. When the brake is
not too tight, the wheel settles into a steady rotation. Either direction of rotation is possible.
When the brake was tightened in an attempt to make the wheel go chaotic, unfortunately it
was tightened too much. Instead of settling into the desired pattern of irregular reversals, the
wheel fell into a simple pendulum-like motion, turning once to the left, then back to the right,
and so on. After the brake was loosened slightly, the motion became chaotic. (And the crowd
went wild...)
The waterwheel is an exact mechanical analog of the Lorenz equations (see Exercise 9.1.3 in
Strogatz (1994)). The corresponding Lorenz parameters are b = 1, σ ∝ ν, and r ∝ ν-1
, where ν is
the rotational damping rate. So as the brake is tightened, the parameters are constrained to move
along a hyperbola rσ = C in the (r, σ) parameter space.
Reference: Section 9.1 of S.H. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley
(1994).
2. Double Pendulum
(with Howard Stone, Division of Applied Sciences, Harvard)
A double pendulum is used to illustrate some basic ideas about chaos in Hamiltonian
systems. The small-amplitude motion of the double pendulum consists of in-phase or antiphase
normal modes, or linear combinations of these two modes. The resulting motion is tame: either
periodic or quasiperiodic with two frequencies.
But large-amplitude motions can be spectacular. The motion is chaotic if the pendula are
released from rest, starting from a configuration where both pendula are horizontal but in
antiphase. In contrast, if the pendula are released from a horizontal in-phase configuration, the
motion is merely quasiperiodic. Both initial conditions have the same total energy, yet the
antiphase initial condition leads to much more complicated dynamics. This shows that the
complexity of the motion depends on more than just its energy. (This demonstration also
illustrates the idea of a “divided phase space” in a Hamiltonian system; chaotic solutions and
quasiperiodic solutions can co-exist on the same energy shell.)
To show that chaotic motions depend sensitively on initial conditions, two (nearly) identical
pendula were released from (nearly) the same antiphase initial condition. The pendula swung in
unison for a while (this is interesting to watch in slow motion, if you have that setting on your
VCR) but soon their motions became totally different.
Reference: T. Shinbrot, C. Grebogi, J. Wisdom, and J.A. Yorke, “Chaos in a double pendulum”,
American Journal of Physics 60, 491 (June 1992).
3. Airplane Wing Vibrations
(with John Dugundji, Dept. of Aeronautics and Astronautics, MIT)
These demonstrations of aeroelastic instabilities serve at least two purposes. They illustrate
some fundamental ideas in nonlinear dynamics, including stable and unstable fixed points, limit
cycles, subcritical and supercritical Hopf bifurcations, and coexistence of attractors. They also
connect nonlinear dynamics to real-world phenomena -- see the NASA footage at the end of this
segment of the video.
3.1 Single degree-of-freedom system
A small wind tunnel is used in the lab demonstrations. There is a variable-speed fan at the
left end of the wind tunnel. The fan is pointed outward, sucking air through the tunnel from right
to left. The experiments are performed on a wing that is pivoted about its mid-chord so that it
has only a torsional degree of freedom. The wing is restrained by a linear torsional spring (not
visible in the video) that can be biased to provide the wing with various initial angles of attack.
The leading edge of the wing is marked by a dot, and there are angle lines marked on the board
behind the wing to aid in quantifying the wing’s motion.
3.1.1 Angle of attack = 0°.
In the absence of any wind, if the wing is disturbed slightly, its free vibrations are
underdamped. (The eigenvalues are complex and in the left half plane.) When the wind is
turned on, at around 10-15 ft/sec the wing is still stable at the horizontal position (0° angle of
attack), but now its free vibrations take much longer to decay. The wind has caused the
eigenvalues to move closer to the imaginary axis. The system is becoming less stable.
At a wind speed V ≈18 ft/sec, the wing still has a stable angle of attack, but now it has rotated
to around 5°. A sufficiently large disturbance sends the wing into a violent oscillation,
corresponding to a large-amplitude limit cycle. However, small disturbances die out. This
shows that the system has two co-existing attractors: a large-amplitude limit cycle, and a static
equilibrium at around 5°. This coexistence is one of the hallmarks of an impending subcritical
Hopf bifurcation (the mechanism by which the static state will go unstable at a slightly higher
wind speed.)
For a wind speed a little above 18 ft/sec, the static state loses stability and the wing
spontaneously goes into a limit cycle with fluctuations of ± 90° (keep your eye on the white dot
to see this). At even higher wind speeds, the amplitude is about ±120°. As the wind speed is
brought back down, the amplitude of the limit cycle appears to decrease continuously, at least for
a while, and then the wing jumps back to static equilibrium.
3.1.2 Angle of attack = 5°
By adjusting the torsional spring, the initial angle of attack is now biased to 5°. When the
wind is turned on, the wing immediately rotates to a new static position (unlike the experiments
above, where the wing stayed near its initial 0° over a range of wind speeds.)
At V ≈17 ft/sec, we find a number of remarkable phenomena. First of all, the wing starts
jiggling slightly. This low-amplitude limit cycle is completely different from the violent
oscillations seen earlier -- it is born in a supercritical Hopf bifurcation. Furthermore, it’s a selfexcited
vibration; it doesn’t need a kick to get it started.
At the same wind speed, a large kick sends the wing into a violent oscillation of ± 70°.
Therefore, at V ≈17 ft/sec the system has two co-existing stable limit cycles. (You should think
about what the phase portrait would look like.)
Finally, when the large-amplitude vibration is manually suppressed, the system
spontaneously reverts to the small-amplitude cycle.
3.2 Two degree-of-freedom system
This system has both bending and torsional degrees of freedom; it can move up and down, as
well as rotate. It is therefore more like a section of an airplane wing, whereas the single degreeof-freedom
airfoil used earlier is more like a turbine blade.
Without any wind, the free vibrations are underdamped for both torsional and bending
motions. At a wind speed of 20 ft/sec, the vibrations take much longer to die down, but the wing
is still stable.
At V = 25 ft/sec, the static equilibrium is still stable to very small disturbances, but a
sufficient kick brings the system into a limit cycle that combines torsional and bending
oscillations (so again we have two co-existing attractors). Next a somewhat smaller kick is used,
one that is just barely large enough to trigger the oscillations. Note how the oscillations build up
before saturating at their final amplitude. When a tiny kick is used, nothing much happens; the
wing goes back to equilibrium. Finally, at high wind speed, the wing is easily kicked into a
violent oscillation.
3.3 Real airplanes
This marvelous and unintentionally campy footage shows some scenes from a NASA film
about aeroelastic phenomena (NASA 1973). One scene shows a small airplane undergoing tail
flutter, and another shows a sailplane exhibiting, in the words of the announcer, “...classical
aileron flutter. The aileron motion couples with wing bending.”
In a part of the film that we edited out (but shouldn’t have), the announcer emphasizes that
commercial airplanes are carefully designed and tested to ensure that these dangerous vibrations
could never occur under normal conditions, so you shouldn’t worry.
References:
K. Aravarnudan and J. Dugundji, “Stall flutter and nonlinear divergence of a two-dimensional
flat plate”, Aeroelastic and Structures Research Laboratory, MIT, Cambridge, Massachusetts,
ASRL-TR-l59-6, Air Force Office of Scientific Research, AFOSR-TR-74-1734, July 1974.
E.H. Dowell and M. Ilgamova, Studies in Nonlinear Aeroelasticity, Springer-Verlag (1988).
NASA Technical Film Serial L-1 128, “Aeroelastic phenomena and related research”, NASA
Langley Research Center, Hampton, VA 23365, June 1973.
Section 8.2 of S.H. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley (1994).
4. Chemical Oscillators
(with Irving Epstein, Chemistry Dept., Brandeis University)
The Briggs-Rauscher reaction is an oscillating chemical reaction that cycles between blue
(oxidized) and yellow (reduced) states. Its recipe is given below, courtesy of Irving
Epstein:
Three solutions are required:
A. 0.14M K1O3
B. 3.2M H2O2 (start from 30% solution)
0.17M HC1O4
C. 0.l5M malonic acid (CH2(COOH)2)
0.024M MnSO4
l0g/l soluble starch
Mix equal volumes of A, B, C and stir.
The system is closed and so it eventually goes to thermodynamic equilibrium, but en route it
oscillates between blue and yellow states. As the demonstration proceeds, the reaction spends
more and more time in the blue state, and so the oscillation period gradually lengthens. Finally,
when the magnetic stirrer is turned off, the system spends a distressingly long time in the blue
state before finally changing color again. During the transition, notice the spatial structure in the
system, made possible by the lack of stirring.
References:
T.S. Briggs and W.C. Rauscher, Journal of Chemical Education 50, 496 (1973).
I.R. Epstein, K. Kustin, P. De Kepper, and M. Orban, “Oscillating chemical reactions”, Scientific
American 248 (3), 112 (1983).
Section 8.3 of S.H. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley (1994).
5. Synchronized Chaos and Private Communications
(with Kevin Cuomo, MIT Lincoln Laboratory)
Using an analog circuit implementation of the Lorenz equations, this demonstration shows
that two chaotic systems can become synchronized. The demonstration also indicates that chaos
might have the potential to be useful in certain applications to private communications (although
it is still too early to tell just how useful it might be; the work described here is only a few years
old, and there are many competing methods that are well-established and tested).
5.1 Lorenz circuit
First a single Lorenz circuit is studied. The plot on the oscilloscope screen shows x(t) vs.
z(t), and the trace resembles the familiar strange attractor for the Lorenz system. The Lorenz
parameters r, σ, and b can be varied independently by adjusting the corresponding variable
resistor in the circuit. As one of these resistors is varied, the circuit is seen to exhibit a sequence
of different attractors: first a fixed point, then a small-amplitude limit cycle that grows
continuously out of the fixed point (indicating that a supercritical Hopf bifurcation has occurred),
then a period-doubled limit cycle, then a strange attractor of Rössler type, and finally a Lorenz
attractor.
In addition to being seen, the attractors and bifurcations can also be heard -- when the signal
x(t) is played through a loudspeaker, the limit cycle sounds like a pure tone. When perioddoubling
occurs, another tone becomes audible, about an octave higher. The Rössler and Lorenz
attractors sound noisy, as expected for chaotic signals.
If you find it hard to believe this bifurcation sequence, see footnote 5.4 below. But for most
readers, it is probably better not to be distracted by footnote 5.4 at this stage.
5.2 Synchronized chaotic circuits
Now two nearly identical Lorenz circuits are studied; one acts as the transmitter and the other
as the receiver. The transmitter parameters are such that the circuit is in the Lorenz chaotic
regime. The transmitter signal x(t) is fed into the receiver into certain way, with the result that
the receiver quickly synchronizes to the transmitter, starting from any initial conditions (see
below for mathematical details). This synchronization is demonstrated experimentally by
plotting x(t) vs. the corresponding receiver variable xr(t). The oscilloscope trace shows a blurry
45° diagonal line, indicating that x(t) ≈ xr(t). The blur is visibly reduced when the transmitter’s
parameters are tuned to match the receiver’s. You can also hear the improvement -- when the
synchronization error ex(t) = x(t) - xr(t) is played through the loudspeaker, the hiss gets quieter as
the tuning improves. Similar results would be obtained for the other state variables y(t) and z(t).
In more mathematical detail, the governing equations for the transmitter are
( )
bzxyz
xzyrxy
xyx
−=
−−=
−=
⋅
⋅
⋅
σ
while the receiver’s state (xr, yr, zr) evolves according to
( )
( ) ( )
( ) rrr
rrr
rrr
bzytxz
ztxytrxy
xyx
−=
−−=
−=
⋅
⋅
⋅
σ
Notice that the transmitter signal x(t) is fed into the second and third equations of the receiver.
Also, the receiver has the same parameter settings r, σ, b as the transmitter. One can prove that
the receiver synchronizes with the transmitter, in the sense that xr (t) → x(t) as t → ∞, where x =
(x, y, z) and xr = (xr ,yr, zr). The synchronization is exponentially fast for all initial conditions.
The proof uses a Liapunov function for the difference dynamics -- see Cuomo and Oppenheim
(1993) and Strogatz (1994).
Some people are surprised to learn that chaotic systems can synchronize, despite their
sensitive dependence on initial conditions. This oft-mentioned “paradox” is based on a
misunderstanding -- sensitive dependence holds only for the uncoupled systems. When driven
by the transmitter, the receiver is actually completely insensitive to its initial conditions, in the
sense that xr(t) always converges to x(t), no matter what the initial conditions are.
5.3 Application to private communications
Suppose you want to send a private message over the airwaves to a friend. For instance,
Princess Diana might have preferred more privacy when she whispered sweet nothings to her
boyfriend James Gilbey over a cellular phone, only to find those conversations later reported in
the media as the embarrassing “Squidgy” tapes. One approach would be to mask that message
with much louder static, so that an eavesdropper would only hear the static, and would have a
hard time detecting the underlying message. Of course, for this “signal masking” approach to be
useful, James needs to have a way of subtracting off the static covering Diana’s message.
That’s where synchronized chaos comes in. Diana has the transmitter and James has the
receiver. They agree to use the same parameter values (which could be changed from day to
day), but he doesn’t know what message she’s going to send. Suppose m(t) is her message signal
and the chaotic transmitter signal x(t) is the much louder mask. The hybrid signal s(t) = m(t) +
x(t) is sent over the airwaves and used to drive James’s receiver, i.e., s(t) replaces x(t) in the
receiver dynamics above. The hope is that the variable xr(t) regenerated at the receiver will be
close enough to x(t) that it can be used in place of the unknown original mask. Then subtracting
xr(t) from s(t) gives an estimate of the message: ( )tm
)
= s(t) - xr(t).
The demonstration tests this approach in real circuits, although wires are used in place of the
airwaves. The song “Emotions” by Mariah Carey is used as the message signal m(t). After
hearing the original, we next hear the hybrid signal s(t), which sounds like a meaningless hiss
because the music is buried under the chaotic mask. On the oscilloscope trace, we see x(t) vs.
xr(t). The diagonal line indicates that the receiver and transmitter circuits are maintaining
synchronization. Departures from the diagonal occur whenever the music is relatively loud --
notice that the beat of the music is visible in the fluctuations. Finally, when the reconstructed
message ( )tm
)
is played, we hear the song again, although it’s a little fuzzy because xr(t) is not a
perfect reproduction of x(t). (In fact, the synchronization error is comparable to the message
itself, which leads one to wonder why the signal recovery works at all. See Cuomo, Oppenheim,
and Strogatz (1993) for an analysis of this issue.)
5.4 A footnote for experts
As soon as we finished taping this demonstration, we realized we had seen some unexpected
phenomena in Demonstration 5.1. For instance, the nontrivial fixed points in the Lorenz system
are normally supposed to lose stability in a subcritical, not a supercritical, Hopf bifurcation.
Hence it was a surprise to see a small stable limit cycle bifurcating from the fixed point. And the
period-doubling route to chaos is not the standard route in the Lorenz system.
Here is our best guess at the explanation. We know that the parameter values used were not
the traditional values σ = 10 and b = 8/3 (where r is adjustable). In the original experiments
reported in Cuomo and Oppenheim (1993), the parameters were r = 45.6, σ = 16, and b = 4, but
those experiments took place many months before this video was made, and some of the circuit
components may have drifted. Furthermore, it appears now that Cuomo may have been varying
b, not r as claimed in the tape. Finally, the circuit equations may need to include an imperfection
term ε ≈ 0.01, caused by a small but measurable voltage offset at the output of the two analog
multipliers in the circuit. With this effect included, and with the equations scaled as in Cuomo
and Oppenheirn (1993), the circuit equations become
( )
( )
( ) bzxy5z
xy20yrxy
xyx
−+=
+−−=
−=
⋅
⋅
⋅
ε
ε
σ
This ε term breaks the symmetry (x,y) → (-x,-y) of the Lorenz equations, but it does not affect
the synchronization performance of the circuits.
Numerical simulations with r = 45.6, σ = 16, ε = 0.01 show all the phenomena seen in the
video. There is a supercritical Hopf bifurcation at b = 0.4028, a stable limit cycle for b = 0.5, a
period-doubled limit cycle for b = 0.65, and a Rössler attractor for b = 0.68, all as observed on
the oscilloscope.
Despite any remaining uncertainties about the equations governing the circuit on the day that
the video was made, it is clear that the system still worked! The chaotic circuits successfully
synchronized, and the Mariah Carey song was successfully masked, transmitted, and recovered.
References:
K.M. Cuomo and A.V. Oppenheim, “Circuit implementation of synchronized chaos with
applications to communications”, Physical Review Letters 71, 65 (1993).
L.M. Pecora and T.L. Carroll, “Synchronization in chaotic systems”, Physical Review Letters 64,
821 (1990).
K.M. Cuomo, A.V. Oppenheim, and S.H. Strogatz, “Robustness and signal recovery in a
synchronized chaotic system”, International Journal of Bifurcation and Chaos 3, 1629
(1993).
Section 9.6 of S.H. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley (1994).
6. Musical Variations from a Chaotic Mapping
(with Diana Dabby, Department of Electrical Engineering, MIT)
Diana Dabby has developed a chaotic mapping for generating musical variations of an
original work (Dabby 1995). In this demonstration, the method is applied first to Bach’s Prelude
in C Major, from the Well-Tempered Clavier, Book I, and then to Gershwin’s Prelude 1.
Since Diana explains her artistic motivation very clearly on the videotape, and the variations
speak for themselves, the only remaining thing to do here is to explain how the method works.
The Lorenz equations are integrated numerically, starting from some reference initial condition.
Let the sequence {xi} denote the x-values obtained after each time step. Each xi is then paired
with a pitch pi from the pitch sequence {pi} of the notes or chords heard in the original piece
(Figure 1). The first pitch p1 is assigned to x1, and p2 is assigned to x2, and so on, continuing
until each pitch has been assigned a value of x.
Figure 1 shows a small subset of the x-axis configured in this way, with the corresponding
pitches plotted along it like fenceposts. For clarity, only five points are shown. In practice, the
method might involve hundreds of points along the x-axis.
i610543
xi10.786.826.403.672.11
piG3E3E4C4G3
Figure 1.
To generate a variation, a trajectory is started from a different initial condition. Let jx′ denote
its x-components after successive time steps in the numerical integration. For each , assign it
the pitch p
jx′
g(j), where g(j) denotes the index i of the smallest xi such that . To express this
rule in more visual terms, imagine that the original pitches p
ji xx ′≥
i define a musical landscape along
the x-axis, as in Figure 1 above. The rule says that each pi controls the interval of the x-axis
between itself and the pitch to its immediate left. Then when a new trajectory comes along, its
components fall into these various intervals, thereby triggering the pitches at the righthand
endpoints and generating a variation. As a hypothetical example, if
jx′
5x′ = 6.53, then the new
pitch would be E3, whereas if = 6.38, then the new pitch5p′ 5x′ 5p′ would be E4, the same as p5
in the original piece.
It may be helpful to think of the method this way: the mapping takes the original music and
maps it onto the back of the strange attractor. The temporal sequence of pitches is linked to the
temporal sequence of points along a certain Lorenz trajectory. When a new trajectory is
launched, it travels through this musical landscape in a different way, thereby generating a
variation. But because the landscape incorporates features of the original musical piece, the
resulting variations tend to evoke the source.
The method tempers the variations in several ways. For instance, only pitch is varied, and no
variation can include a pitch not present in the original. By extending the mapping to the y and z
axes, variations could be generated that differ in rhythm and loudness as well as pitch. Other
factors affecting the extent and nature of variation are the choice of initial condition, step length,
length of integration, and the amount of truncation and round-off applied to the trajectories.
References:
D.S. Dabby, “Musical variations from a chaotic mapping”, Chaos 6, 95 (1996).
I. Peterson, “Bach to chaos: Chaotic variations on a classical theme”, Science News 146, 428
(December 24 & 31, 1994).