what is really happening in nature.
It’s such computer models that are at the heart of trying to answer the question Poincaré first tackled when he discovered chaos: will there even be a stable Earth orbiting the Sun for evolution to continue playing its game of dice? How safe is our planet from the vagaries of chaos? Is our solar system stable and periodic, or do I have to worry about a grasshopper disrupting our orbit around the Sun?
A BUTTERFLY CALLED MERCURY
Poincaré wasn’t able to answer the King of Sweden’s question about the solar system: whether it would remain in a stable equilibrium or might fly apart in a catastrophic exhibition of chaotic motion. His discovery that some dynamical systems can be sensitive to small changes in data opened up the possibility that we may never know the precise fate of the solar system much in advance of any potentially devastating scenario unfolding.
It is possible that, like population dynamics with a low reproductive rate, the solar system is in a safe predictable region of activity. Unfortunately, the evidence suggests that we can’t console ourselves with this comforting mathematical hope. Recent computer modelling has provided new insights which reveal that the solar system is indeed within a region dominated by the mathematics of chaos.
I can measure how big an effect a small change will have on the outcome using something called the Lyapunov exponent. For example, in the case of billiards played on differently shaped tables, I can give a measure of how catastrophic a small change will be on the evolution of a ball’s trajectory. If the Lyapunov exponent of a system is positive, it means that if I make a small change in the initial conditions then the distance between the paths diverges exponentially. This can be used as a definition of chaos.
With this measure several groups of scientists have confirmed that our solar system is indeed chaotic. They have calculated that the distance between two initially close orbital solutions increases by a factor of ten every 10 million years. This is certainly on a different timescale to our inability to predict the weather. Nevertheless, it means that I can have no definite knowledge of what will happen to the solar system over the next 5 billion years.
If you’re wondering in despair whether we can know anything about the future, then take heart in the fact that mathematics isn’t completely hopeless at making predictions. There is an event that the equations guarantee will occur if we make it to 5 billion years from now, but it’s not good news: the mathematics implies that at this point the Sun will run out of fuel and evolve into a red giant engulfing planet Earth and the other planets in our solar system in the process. But until this solar blowout engulfs the solar system, I am faced with trying to solve chaotic equations if I want to know which planets will still be around to see that red giant.
This means that, like the predictions of the weather, if I want to know what is going to happen, I am reduced to running simulations in which I vary the precise locations and speeds of the planets. The forecast is in some cases rather frightening. In 2009 French astronomers Jacques Laskar and Mickael Gastineau ran several thousand models of the future evolution of our solar system. And their experiments have identified a potential butterfly: Mercury.
The simulations start by feeding in the records we have of the positions and velocities of the planets to date. But it is difficult to know these with 100% accuracy. So each time they run the simulation they make small changes to the data. Because of the effects of chaos theory, just a small change could result in a large deviation in the outcomes.
For example, astronomers know the dimensions of the ellipse of Mercury’s orbit to an accuracy of several metres. Laskar and Gastineau ran 2501 simulations where they varied these dimensions over a range of less than a centimetre. Even this small perturbation resulted in startlingly different outcomes for our solar system.
You might expect that if the solar system was going to be ripped apart it would have to be one of the big planets like Jupiter or Saturn that would be the culprit. But the orbits of the gas giants are extremely stable. It’s the rocky terrestrial planets that are the troublemakers. In 1% of simulations that they ran, it was tiny Mercury that posed the biggest risk. The models show that Mercury’s orbit could start to extend due to a certain resonance with Jupiter, with the possibility that Mercury could collide with its closest neighbour, Venus. In one simulation, a close miss was enough to throw Venus out of kilter, with the result that Venus collides with Earth. Even close encounters with the other planets would be enough to cause such tidal disruption that the effect would be disastrous for life on our planet.
This isn’t simply a case of abstract mathematical speculation. Evidence of such collisions has been observed in the planets orbiting the binary star Upsilon Andromedae. Their current strange orbits can be explained only by the ejection of an unlucky planet sometime in the star’s past. But before we head for the hills, the simulations reveal that it will take several billion years before Mercury might start to misbehave.
INFINITE COMPLEXITY
What of my chances to predict the throw of the dice that sits next to me? Laplace would have said that, provided I can know the dimensions of the dice, the distribution of the atoms, the speed at which it is launched, its relationship to its surrounding environment, theoretically the calculation of its resting point is possible.
The discoveries of Poincaré and those who followed have revealed that just a few decimal places could be the difference between the dice landing on a 6 or a 2. The dice is designed to have only six different outcomes, yet the input data ranges over a potentially continuous spectrum of values. So there are clearly going to be points where a very small change will flip the dice from landing on a 6 to a 2. But what is the nature of those transitions?
Computer models can produce very good visual representations that give me a handle on the sensitivity of various systems to the starting conditions. Next to my Vegas dice I’ve got a classic desktop toy that I can play with for hours. It consists of a metal pendulum that is attracted to three magnets, coloured white, black and grey. Analysis of the dynamics of this toy has led to a picture that captures the ultimate outcome of the pendulum as it starts over each point in the square base of the toy. Colour a point white if starting the pendulum at this point results in it ending at the white magnet. Similarly, colour the point grey or black if the ultimate destination is grey or black. This is the picture you get:
As in the case of population dynamics, there are regions which are entirely predictable. Start close to a magnet and the pendulum will just be attracted to that magnet. But towards the edges of the picture I find myself in far less predictable terrain. Indeed, the picture is now an example of a fractal.
There are regions where there isn’t a simple transition from black to white. If I keep zooming in, the picture never becomes just a region filled with one colour. There is complexity at all scales.
A one-dimensional example of such a picture can be cooked up as follows. Draw a line of unit length and begin by colouring one half black and the other white. Then take half the line from the point 0.25 to 0.75 and flip it over. Now take the half in between that and flip it over again. If we keep doing this to infinity then the predicted behaviour around the point at 0.5 is extremely sensitive to small changes. There is no region containing the point 0.5 which has a single colour.
There is a more elaborate version of this picture. Start again with a line of unit length. Now rub out the middle third of the line. You are left with two black lines with a white space in between. Now rub out the middle third of each of the two black lines. Now we have a black line of length 1⁄9,
