What is chaos?

An experiment-based introduction
into chaotic phenomena

Interactive material for secondary school students
to be used in assisted or individual learning

Questions are set in italics.

Warming-up

A. Mixing thick paints of different colors is an every-day life experience:

After a few movements of stirring, intriguing filamentary patterns evolve. Further similar examples can be seen when milk is mixed in coffee, or cocoa in cakes.

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X Cake

Can you determine from such photographs the initial pattern before stirring?

B. Knowing that this picture

is a long-exposure photograph taken of a motion, could you guess what moving object the green led is fixed to?

Remember, from the trajectories discussed in the standard physics curriculum you can always figure out what motions they belong to.

What you'll learn? The questions above might appear to be unconventional, and the answers you tried to give are perhaps unusual or thought-provoking. The reason: these questions touch the essence of chaos. If you are interested to become acquainted with this mysterious phenomenon, we invite you to an exciting journey to discover the 4 main features of Chaos. By the end of this excursion, organized about question B, you shall have a fairly complete first understanding of what modern science calls chaos. The last section of the material illustrates, by elaborating on the subject related to question A, that we often come across with chaotic phenomena in our every-day life, and this is just one of many possible examples.

Discovering Feature 1
(temporal irregularity)

The following video provides the answer to question B.

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We see the motion of a ruler, fixed to an axis moved by the the action of an electric wiper motor. In the language of physics: this is a pendulum driven by the periodically moving suspension point:

There is nothing strange in the motion of an ordinary pendulum: it swings periodically. (On the long run, it slowly stops due to air drag and friction.) Surprisingly, a slight change, just making the suspension point moving periodically (with a period of about 1.2 s), alters the character of the motion radically.

Can you say, having watched the video for some time, the direction the ruler will rotate next (clockwise or counter-clockwise)?

The motion is rather strange (and funny). It is long lasting without any tendency to come to a rest (due to the energy input coming from the motor which compensates on average the action of friction), but the rotation was seen to have stopped in both a vertical and a tilted position for a moment. After these unstable states rotation can reemerge in any direction. The overall irregular character is the reason why we call such motions chaotic. Note that this scientific term has a meaning different from the every-day use in which chaos typically means spatial disorder. For a distinction, the term deterministic chaos can be used when we speak of motions with irregular temporal properties.

Returning to the photograph in question B, here is a video of the same system taken in darkness, when only the green led at the end of the ruler is lit.

Nowadays easily accessible technologies make possible that besides visual observation quantitative data can also be gained from the motion. We can measure, e.g., the velocity of the ruler as a function of time over long periods of time as this video illustrates:

The finding reinforces our first impressions: the function goes up and down, changes sign quite often, but is very much different from a cos or sin function. It has no recurrent piece, is not periodic at all! Following it for some time, one cannot imagine how it will be continued in the next seconds. This is a basic feature of any chaotic motions. If you are interested in the raw data gained from the sensor, click here.

Based on these observation, try to give an answer to the question: what is chaos? First answer

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Chaos is a long lasting motion of relatively simple systems, which is

  • irregular in time, truly aperiodic, and is not the superposition of periodic components

A home-made water wheel containing 12 small plastic coffee cups on a tilted disk which is able to rotate is another surprising set-up Video. The driving force is here water flowing from a tap with constant speed. Although there is no time-dependence in the drive, again it is not possible to predict for how long the wheel is going to rotate in one or in the other direction. The pattern of the motion does not repeat itself, not even for long runs. This wheel was built by a secondary school student. Bigger devices can be seen in science centers.

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Chaotic water wheel in front of the Amsterdam Science Center

Elements of such motions can be observed in every-day life: Example1 Example2 Example3

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Check your knowledge acquired: Quiz 1

Discovering Feature 2
(unpredictability)

An additional characteristic feature of chaos shows up when comparing motions initiated with nearly identical conditions. In the first experiment this can be demonstrated by fixing two rulers to the same periodically moving axis, so that the rulers can move in two parallel planes without collisions. The rulers are practically identical (are made by the same producer).

Do you expect any essential deviation between the motion of the two rulers?

This is what you observe.

Even if initially similar, the movement of the two rulers become, after a short time, completely different!

The result is the same when initiating more than two motions under similar conditions. This can best be followed by means of a computer simulation.

If you are interested in how motions can be followed by means of a computer in an easy way, click to this interactive elementary introduction here.

The velocity of the endpoints of 11 rulers started in the same position with slightly different initial speeds looks like this

After about five cycles of the electric motor, all rulers behave completely differently. This is not like with ordinary motions. We can thus conclude: in chaos tiny differences in the initial state lead to dramatic deviations in the final outcomes. In other words, such motions are unpredictable since the final state cannot be predicted due to the small uncertaintes in the preparation of the initial state, which are unavoidable both in experiments and numerical simulations. A widely spread popular expression for this property is 'butterfly effect'. It reminds us in an allegoric way on how small initial uncertaintes leading to dramatic consequences can be (like e.g. the flutter of a butterfly). More on 'butterfly effect' here.

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The American meteorologist Edward Lorenz gave a lecture in 1972 with the title: 'Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?'. The term 'butterfly effect' went into general use due to J. Gleick's popular book entitled 'Chaos: Making a New Science', implicitly suggesting that the answer to the question is affirmative. Outside scientific literature this is often interpreted as if modern sciences would claim that everything is related to everything, and we could not therefore be sure of anything. In contrast to this, the analysis of chaotic systems shows that unpredictability is limited, it only holds on the chaotic attractor. As far as motion before reaching the attractor is concerned, we know for sure that it converges to a very small (but extended) set of zero volume: the attractor. Nearby orbits do not deviate before reaching the attractor. It is worth noting that Lorenz referred in his talk to the difficulties of weather forecasting, but did finally not give an answer to the particular question raised in the title.

In the language of science we speak of a sensitive dependence on initial conditions.

Based on both features discussed, try again to answer to the question: what is chaos? Refined answer

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Chaos is a long lasting motion of relatively simple systems, which is

  • irregular in time, truly aperiodic, and is not the superposition of periodic components

and

  • sensitive to initial conditions: small initial differences are strongly magnified, and are, therefore, unpredicable

It is interesting to see that modern meteorological forecasts are also based on ’plume diagrams’ summarizing the results of an ensemble of 50 different simulations starting from nearly identical states of the atmosphere. Predicted surface temperature for Budapest within a given time interval can be seen on the chart below

During the first two days, all curves run practically together, but then they start to deviate: a clear demonstration of the 'butterfly effect'. The forecast can thus be considered to be reliable only in the first 3-4 days. For any longer period, weather is unpredictable.

Interested in finding the current ensemble forcast for your city? Visit the public German page Wetterzentrale. First, klick on the 3rd, green line in the middle: Europe (Stadte), next on the line below: Europe (Karte). A map of Europe appears and after klicking on the desired geographical location, the ensemble graphs for the temperature and precipitation of the chosen location becomes visible for the next two weeks.

The fact that the 'plume diagrams' are very similar to what we had in our physics experiment, reinforces the view that the motion of a pendulum with a moving suspension point is unpredictable (although the system is much simpler than the atmosphere)!

The length of time interval during which a bunch of initially nearby trajectories stay together is the prediction time $t_p$. For the pendulum experiment $t_p$ is about 6 s, while for the atmosphere $t_p$ is 3-4 days. If you want the learn more about the meaning of the prediction time and its typical order of magnitude, consider the problem:

Imagine a chaotic system in which the initial uncertainty, or relative error, $\Delta r_0$ doubles every time unit. After $n$ time units it is $\Delta r(n)=\Delta r_0 2^n$, the growth is exponential. Compare this with a regular system in which the uncertainty grows only linearly, say as $\Delta r(n)=\Delta r_0 (1+2 n)$. Determine the prediction time $t_p$ as the time at which the error is unity: $\Delta r(t_p )=1$. Evaluate $t_p$ in both cases for $\Delta r_0=10^{-6}$ first, and for a much smaller initial uncertainty: $\Delta r_0=10^{-9}$ next. Solution

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Answer: In the chaotic case $\Delta r(t_p)=\Delta r_0 2^{t_p}=1$, from which $t_p=\log_2(1/\Delta r_0)$. In the nonchaotic case $t_p=(1/\Delta r_0 -1)/2$. With initial uncertainty $\Delta r_0=10^{-6}$, one finds the prediction time to be $t_p$ approx 20 time units in the chaotic case (since 210 is approximately 1000), while in the regular case $t_p$ approx 5 105. An overwhelming difference! The latter indicates that regular motions can be predicted for any time occuring in practice. The situation becomes more shocking with $\Delta r_0=10^{-9}$. The prediction time grows up to only $t_p=30$ time units for chaos, but becomes 1000 time larger, $t_p=5\ 10^8$ for usual motions. We thus have to conclude: the prediction time of chaotic systems is impossible to increase in practice!

Check your knowledge acquired: Quiz 2

Discovering Feature 3
(regular pattern in a specific view)

This feature can be demonstared in precise experiments or numerical simulations only, run for a long time. It is worth the trouble to use a view of the motion which plots the velocity in a given moment as a function of the position. As time passes, the point representing these two data on a plane moves on, and a curve is traced out.

This curve is nothing but plotting the velocity as a function of the position (angle) at any instant in which our sensor records a value (with a time difference of 0.1 s). A complicated coil is drawn up, illustrating the irregular nature of chaos. Red dots indicate the instants when a full period of oscillation is over (when the suspension point is in its rightmost position). These points appear to show up randomly,

If, however, this sampling is applied for a long time, and only the red points are plotted, an intricate pattern evolves.

The initially seemingly random appearance of points starts, after some time, accumulate in certain regions, and avoiding certain others. It is remarkable to see that a kind of order appears in this view of chaos: a huge number of points are allowed to occur in this plane during the chaotic motion (but many others - the points of the white regions - are forbidden).

More details of the mentioned order in chaos is unfolded when following the motion of our ruler by means of a computer simulation (where the lack of sensor-related restrictions allow us to take the time difference much smaller than 0.1 s). The point representing the velocity-position pairs on a plane moves on in the simulation as shown here

With this better resolution, in the snapshots taken in the rightmost position of the suspension point one also discovers that the points trace out complicatedly interwoven filaments with empty regions between them, down to fine scales:

If you are interested in seeing this picture again, obtained in a longer simulation with more points, artistically colored click here.

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The pattern provides an example of what is called a chaotic attractor. The object is called an attractor since all motions are attracted to it. Due to friction, the system forgets its initial state, the same set of points emerges in this sampling procedure after some time, irrespective of where the motion was intitated. There is an 'infinite' number of permitted states at the sampled instances, extending over a large range. This is in harmony with the fact that the chaotic motion is aperiodic. It is surprising, however, that the plethora of the permitted states is ordered in some sense. Such complicated patterns are examples of fractals, an unusual class of geometric objects.

If we blow up a square from the pattern of the chaotic attractor and repeat this blowing-up once again

the latter ones appear to be nearly identical. We say, the geometry is self-similar. This self-similarity is the definitive property of fractal objects. This picture illustrates that the chaotic attractor is a bundle of approximately parallel line on smaller and smaller scales. Chaos is thus related to filamentary fractals.

The fractal pattern identified also indicates that chaos is not as random as noise. This is so since a motion is always a consequence of some laws, such as the Newtonian equations of motion in physics.

Based on the three features discussed, try to give an anser as detailed as possible to the question: what is chaos? Complete answer

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Chaos is a long lasting motion of relatively simple systems, which is

  • irregular in time, truly aperiodic, and is not the superposition of periodic components

and

  • sensitive to initial conditions: small initial differences are strongly magnified, and is unpredicable

and

  • in an appropriate view not fully irregular, appears to be structured but complex, exhibits a fractal pattern.

As the conjunctive and indicates, the listed characteristics are present simultaneously: when a simple system is aperiodic over a long time, its evolution must be unpredictable, and can be represented by a fractal structure in suitable coordinates.

Note that none of the features explored here holds true for motions learnt in your physics curriculum. We also saw that a slight modification of a familiar case (e.g. letting the suspension point to move) converts the motion into chaos. If you wish to have an overview of all possible motions of simple systems, you cannot avoid becoming acquainted with the unusual features of chaos.

Check your knowledge acquired: Quiz 3

Exploring Feature 4 (probabilitic view)
or How to cope with unpredictable systems?

It is certainly an unusual feature that the state of chaotic systems cannot be predicted for a time interval longer than the prediction time. What is then a proper long term scientific description of such systems? The answer follows from the 3 basic features discussed above. The existence of the chaotic attractor implies that only certain states, the fractal set of points of the attractor (to which all motions converge to), are permitted. We do not know which of the permitted states will belong to a given motion, but we can ask about the probability with respect to which a permitted state (e.g. the state with the fatest possible rotation of the ruler) occurs among all possible motions.

The analysis of chaotic systems teaches us that from a statistical point of view the motion developing on the attractor can be described with full accuracy. Even if individual motions are unpredictable, the statistical properties of all possible motions can be predicted! The somewhat pessimistic view of interpreting the 'butterfly effect' as not being able to be sure of anything, should be counterbalanced by the potentially perfect ability for a statistical characterization of chaos. (We are then not surprised by seeing weather [or most recently even climatic!] forecasts moving in the direction of providing likelyhoods rather then precise predicted data - this is the proper treatment of such problems.)

If interested in the probability with which different regions on the attractor of the chaotic pendulum is visited, click here.

For a more artistic view, the angle is not restricted here to -180, +180 degress, it grows continuously. The distributon is rather uneven: the probability of a given state might be completely different from that of the neighboring pixel. This fine dependence on the states is an additional property of chaos. Note that such distributions are the consequences of the underlying equations. In our pendulum example from mechanics, the existence of the probability distribution follows from the Newtonian equation of motion!

Check your knowledge acquired: Quiz 4

Relaxation: An application: mixing of dyes,
spreading of pollutants

Looking back to point A and the photographs seen there, you might realize that stirring and mixing are chaotic processes. The fact that the initial state cannot be traced back from long time observations is a sign of unpredictability, and the filamentary patterns exhibit fractal features, indeed.

Further examples with fringerprints of stirring processes are ceramics1, ceramics2 and even abstract artistic paintings. You can read more on aesthetic aspects in chaos teaching here.

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Folk-art bowl from Korond, Transylvania (Romania), about 2008

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Double Gourd Tiffany Vase from 1900

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S. Hartung: ’Mesélj még’ ('Tell some mor' detail), oil, 1989

Surprisingly, similar patterns can also be seen on satellite images on continental scales (here drift ice along the coast of Kamchatka)

Clearly, both the dye patterns and the forms in which material in our environment is advected are related to chaotic processes. A particularly worrisome case is the spreading of pollutants.

When looking at a simulation like this, the vortices can be not only stirrers in a house-hold mixer, but also oceanic or atmospheric eddies. The initially small black loop can thus represent for example both a region of high alcoholic content and a patch of concentrated pollution. The fact that this small loop becomes stretched and folded very rapidly and extends, in a filamentary, fractal-like pattern, farther than the distance of the stirrers, is a clear demonstration of the sensitive dependence on initial conditions (butterfly effect). From the point of view of our house-hold applications, chaos is an advantageous property, ensuring proper mixing, a property utilized in dough kneading. The same mechanism, however, is disadvantageous from the point of view of environmental protection since pollution on large scales spreads very rapidly, along complicatedly folding filaments which evolve in an unpredictable way.

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This second vortex-based simulation shows that the advected material can cover long distances while being stretched and folded, illustrating that the pollution might travel in a chaotic way huge (perhaps geographic) distances.

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If you would like to learn how ash spreads after a volcanic eruption in the real, measured winds of the atmosphere,

click here to download a chaos-related 'computer game'.

Literature: T. Tél, M. Gruiz, Chaotic Dynamics, An introduction based on classical mechanics, Cambridge University Press, Cambridge, 2006

This material is dedicated to the memory of Márton Gruiz

Copyright: T. Tél, 2019

This work was funded by the Content Pedagogy Research Program of the Hungarian Academy of Sciences. The technical assistance of A. Tél in setting up the pendulum experiment and carrying out the sensoric measurements is acknowledged.