If the title of this blog post seems dramatic, that’s probably because it is intended to mimic a newspaper headline, pun included (although the subject is not exactly breaking news). According to the Oxford dictionary the word chaos characterises a “state of complete disorder or confusion”, which is precisely the situation in which physicists and mathematicians found themselves towards the end of the 19th century and, to add insult to injury, re-experienced much later in the 1970s. What we now know as chaos theory in physics and mathematics was what emerged from this chaos.

First contact

Briefly, there was a grand idea — inherited from Descartes and Newton —  that if one  specifies the state of a system (in some technical sense), then both the future and the past of that system are knowable through a diligent application of the laws of mechanics. In short, knowing the present plus a bit of hard work guaranteed that the future could also be known: this was one of the promises of the post-Newtonian understanding of the scientific revolution. This promise got crushed in the late 19th century. When studying the mechanics of planetary motion, Henri Poincaré realised that when he simply added an extra planet to the two-body case solved by Newton more than 2 centuries prior (with the Sun and Earth for example), he could not find any explicit exact solution to it in general. He even ventured to say that the solution to this problem, if it exists at all, would require new mathematical tools that had yet to be invented (for those who can read French here is an excerpt of the article he published then). The bottom line is that the future of this simple system could not be known no matter how hard people seemed to try.

A new hope…

Fast forward 70 years and digital computers are the trendiest thing around (for good reason). If you cannot solve your super complicated equations by hand, you can programme a computer to carry out the calculations for you. To understand how computers can complete mathematical problems that are too hard for us to resolve we will use Figure 1, shown below.

Figure 1: A difficult task can be completed with at least two strategies: “all-at-once”, when possible, or “one-subtask-at-a-time”.

The idea is that when confronted with a difficult task one may think of two possible strategies to complete it: either consider the task as a single unique task to complete “all-at-once” — which can be very hard or even impossible to do — or “slice” the difficult task into smaller task-bites that are very easy to solve individually but can be very many and usually need to be completed in a particular sequence. For example, if you live on the second floor of a building and the task is to get to the ground floor, you may accomplish it “all-at-once” by jumping out of the window (don’t try this at home!!) or you might prefer to take the stairs one step at a time starting from the 2nd floor and finishing safely when reaching the ground floor (you see here by the way that the order with which you take the steps matters to complete the task).

It turns out that the kind of mathematical problems that physicists and mathematicians had to solve in mechanics and related disciplines all had this nice property that, if the slicing is fine enough, they could be split into a multitude (millions or billions) of much simpler tasks to be resolved one at a time in a specific order by a machine. “Unsolvable” mathematical problems, including Poincaré’s inextricable 3-body problem, were unsolvable no longer thanks to computers… or so it seemed.

Chaos strikes back

In the 1960s an applied mathematician called Edward Lorenz was developing a mathematical model of the weather with a minimal number of variables. What he observed for some parameter models (setting up the “climate” of the weather model) quite accidentally is summarised in Figure 2 below.

Figure 2 realised with the help of Wolfram’s Mathematica: each point on one of these 3D graphs represents a given weather condition (like strength of updrafts, temperature gradient or temperature difference between up and down drafts). The drawn curves depict the succession of different weather conditions (or weather trajectories) obtained from an initial one as time progresses for a given climate setting. The trajectories are coloured in such a way that the color red corresponds to the early times of the trajectory while the brown color corresponds to the latest times of the trajectory. The figure shows 4 different trajectories ran on the same computer, that simulated the same weather system, for the same amount of “real” time and for the same initial conditions (initial weather data).  The only difference between each of the trajectories shown is the number of decimal places used for the initial condition. A red spot identifies the final point (weather condition) of the trajectory for each case. 

In Figure 2, 4 trajectories of Lorentz’s system are represented. You can tell they are different because:

  • The red spot indicating the end of the trajectory is not located at the same place in each graph, roughly meaning that they predict different weather for the same geographic location at the same date,
  • If you look closely enough the shapes of these trajectories are somewhat different, meaning that prediction of the weather for most dates between the start date and end date actually differ and
  • There is a strong difference in colouring of the trajectories illustrating that, even when two trajectories contain the same weather condition (say blue sky with no wind), it is predicted to occur at different dates in the same geographic location. 

Yet the expectation of Lorenz was that these 4 trajectories should have been at least approximately identical (especially the top left and bottom right) because all 4 were meant to complete the exact same task but simply with various degrees of accuracy by changing the precision on the initial condition (initial weather data). For example, imagine that one number of this initial condition is Pi. Depending on the required precision in your calculation you could try 3.14159. This seems accurate enough, no? The problem is that there are infinitely many numbers that can be constructed with the same starting 5 decimals; and only one of them is Pi. For example 3.14159123456 or 3.1415995141 are not Pi. “Fair enough” you may say, you just need to include all the digits of Pi to be sure. The problem is that this is not possible, since Pi has infinitely many digits and no periodicity: it is said to be an irrational number (it is also a transcendental number) in the mathematics jargon. As an example see the figure below, where Pi has been estimated up to its first 4000 digits.

Screenshot 2019-05-23 at 06.07.46
Figure 3 relapsed with Wolfram’s Mathematica. Screen print of Pi up to its first 4000 digits.

That numbers can have infinitely many digits is usually not a problem since a vast majority of mathematical systems would give very similar outcomes when started from very similar places. But Lorenz had simply found a system where this was not always so. Nowadays we say that such systems displays sensitivity to initial conditions. This means that no matter how small the imprecision in your initial condition may be it will inevitably result in a totally different outcome after some characteristic time called the Lyapounov time. Indeed Figure 4 below illustrates that some agreement between the trajectories is possible since they all share the same red branch at the beginning of the trajectory.

Figure 4. Same as Figure 2 but the Lorenz model is ran for a much shorter time. We see that the end point (red spot) appears at roughly the same position for all four runs. They therefore agree on their weather predictions at least up to that time or date. Depending on the climate parameter model and initial weather data, this time window of agreement can range from 1 hour to a few days in real time language. 

The butterfly effect

Lorenz’s discovery (do we discover things in mathematics by the way?) inspired people to find many new chaotic systems or identify previously known systems as being chaotic in the sense described above, one of them being Poincaré’s 3-body nightmare. There is now a plethora of such systems that have been proved to be chaotic.

In popular culture chaos theory has been summarised by the famous catchphrase “The Butterfly Effect”. This phrase possibly originated more than half a century prior to the 2004 film of the same name, when Ray Bradbury imagined that going back in time and stepping on a single butterfly could have such far-reaching implications as changing who was elected president, in his 1952 short story ‘A Sound of Thunder’. Whether or not Edward Lorenz was aware of this work when he gave a talk entitled ‘Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?‘ in 1972 is not clear (in his papers he refers to sea gulls rather than butterflies). Regardless, if you ask a physicist (or mathematician) where the phrase comes from, they are likely to credit Lorenz with the ideas. In fact, the actual words ‘the butterfly effect’ were first used by James Gleick in his book ‘Chaos – making a new science‘. Looking at Figure 2 it is also interesting to note that the trajectories drawn by Lorenz’s weather model do resemble the wings of a butterfly, which probably helped to burn the butterfly effect into our subconscious minds.

To finish we will illustrate chaos one more time with a pedagogy of physics work about the chaotic double pendulum created by a student from the School of Mathematics and Physics at the University of Lincoln.

Minority report

So, can we know the future? Well, not quite. We can go on for quite a long time here but if knowledge is to be considered as “Justified True Belief”, then what chaos theory shows is that there are systems for which the outcomes of calculations cannot give any consistent answer (leaving us with no justification to hold a particular belief about the weather in the case of Lorenz’s model for example) and therefore the only thing that we can know about the future of such systems is that there is a time “window of prophecy” beyond which it is precisely unknowable. Chaos is predictably unpredictable!

This post was written jointly with Matthew Booth, PhD from the School of Mathematics and Physics at the University of Lincoln.


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