The approach of representing the behaviour of a vibrating system as a “flow” of trajectories in the phase plane is the starting point for the mathematician’s approach to nonlinear systems which leads to “chaos theory”. Single-degree-of-freedom oscillators with smooth nonlinearity cannot exhibit chaotic behaviour in their free motion (although their forced motion can sometimes do so: see the Wikipedia page on the Duffing equation for some examples). The reason is that non-intersecting trajectories in the plane don’t have enough room to manoeuvre: they cannot make patterns that are sufficiently complicated to suggest the description “chaotic”. However, a system with more degrees of freedom can do so.
The idea of the phase plane can be generalised to a phase space, with more dimensions. We saw in the previous section that an oscillator with a single mass needs to be represented in two dimensions (i.e. in the plane) in order to give the crucial result that trajectories cannot cross. A vibrating system with two masses can be treated in a similar way, but the corresponding phase space needs four dimensions: unfortunately that makes it virtually impossible to visualise the complete phase portrait.
So we will cheat. I will illustrate chaotic behaviour with an example that has nothing to do with music or vibration, but one that has a three-dimensional phase space so that we can plot pictures and visualise at least part of what is going on. Three dimensions is sufficient for more complicated behaviour to be possible, compared to what can happen when confined to a plane. Trajectories can intertwine, or be tied in three-dimensional knots. In fact, that’s just the start of it: the intertwining can become almost unbelievably complicated.
The example we will look at relates to something called the Lorenz equations. Edward Lorenz was a meteorologist interested in the dynamics of the Earth’s atmosphere. Way back in the 1960s, he formulated a very simple mathematical model of atmospheric convection. Computers were in their infancy back then, but he conducted computational studies based on his model, aided by his colleague Ellen Fetter. What they discovered in those studies laid the foundations for modern chaos theory. The equations seem quite innocuous (you can see them in the next link), but for some values of the parameters appearing in the equations they predict behaviour that is complicated beyond what anyone was expecting at the time. Their results have spawned a whole academic industry: hundreds of research papers and at least one book have been written about the Lorenz system.
We can get an inkling of this complexity from a computed example. Figure 1 shows the first part of a particular trajectory. The starting point is the obvious “loose end”, low down in the plot. This trajectory curves around, then seems to fall into a plane, where it circulates round and round, spiralling slowly outwards. Nothing obviously “chaotic” here, surely?
Figure 2 shows the same trajectory, computed for twice as long. Now what we see is that after spiralling outwards for a while, it took off into space, did a loop somewhat similar to the initial one, then dropped back onto the plane, in a different place from before, and started spiralling out again.
Finally, Fig. 3 shows the result for the same trajectory when we follow it for 5 times as long as in Fig. 1. The pattern begins to look quite complicated. It involves repeated episodes of the kind of behaviour we have just seen: spiral out for a while, then leap off, do a loop or two in a different place, then drop back on to the “spiralling plane”, but in a different place every time. The pattern never repeats: this one of the hallmarks of chaotic response. Figure 4 lets you see this same trajectory in 3D, with a rotating animation. This reveals that what we have called a “plane” is not in fact quite flat. It also shows that the loops on the right-hand side of the plot also map out a different “curvy plane”.
A crucial characteristic of a chaotic system is “sensitive dependence on initial conditions”. We can illustrate this directly with our Lorenz example. Figure 5 shows two trajectories. The one in blue is the same as in Fig. 3. Its starting point had coordinates (1, 1, 1) in the (x,y,z) space of these plots. The trajectory plotted in red started from the point (1, 1.001, 1): so close to the initial value of the blue trajectory that they are indistinguishable at this plot scale. In the early part of the evolution, both trajectories remain close. Because the red one was plotted on top of the blue one, that means that you can hardly see that the blue one is there. However, it is obvious in the plot that you don’t have to wait all that long before the red and blue lines are in different places, and both clearly visible. The tiny difference in starting points has been amplified, so that after a while the two trajectories look quite different. This is “sensitive dependence” in action.
Bearing in mind that Lorenz was a meteorologist, this kind of behaviour points towards the famous “butterfly effect”. The traditional version is that a butterfly flaps its wings somewhere in the Amazon basin, and a few days later the weather in Europe is quite different from what it would have been without the butterfly. It is now fully accepted that the world’s weather does indeed show this kind of sensitive dependence. This has led to a different strategy for computer-based weather forecasting: you must accept that no single run of a computer model, however sophisticated, will correctly capture the actual progress of the weather, because the observations on which the models are based have finite accuracy so that sensitive dependence is bound to make the predictions wrong in detail. Instead, they run the model many times with slightly different starting points, then combine the results using statistical ideas. That is why a modern weather forecast gives you a probability of rain at a particular time tomorrow, rather than a definite yes/no prediction.
So can we get at least an idea of where this sensitive dependence comes from? We can see the beginnings of such sensitivity in the systems we looked at in section 8.3 based on the 2D phase plane, even though they are not chaotic. One key ingredient is the behaviour near a saddle point, which in mechanical terms describes an unstable equilibrium. Figure 6 reminds you of what happens near a saddle point. This plot illustrates the fact that two trajectories lying close together but on opposite sides of the separatrix heading into a saddle point will, after they have passed close to the saddle point, be widely separated into different regions of the phase plane.
To see the relevance to our example of the Lorenz equations, Fig. 7 shows a different view of the trajectory from Fig. 3. This time, we are looking at it from vertically above, down the z-axis. If you look in the centre of this plot, you can see the characteristic pattern of trajectories near a saddle point. An outward-spiralling trajectory will sooner or later come close to this saddle point, and that may result in it being diverted towards the second “curvy plane”. The details of that diversion will be sensitive to exactly how close it comes to the saddle point. Now, this is a gross over-simplification of the full complexity of this system’s phase portrait, but it gives an inkling. In a phase space with at least three dimensions, the separatrix itself can make its way back to the vicinity of the saddle point. In fact, it can do so repeatedly, and be distorted into very convoluted shapes: that’s exactly what happens with the Lorenz system. Following this train of ideas is one route into the mathematics of chaos, but we won’t try to go any deeper for the purposes of this introduction.
Systems capable of showing chaotic behaviour often share a feature in their phase space behaviour. You can think of the trajectories in phase space as if they were streamlines of a “phase fluid”. If you start with a concentrated “blob” of this fluid and then follow it forwards in time, you will often seen that the blob gets repeatedly stretched and folded. Well, repeated stretching and folding is a very good way to create intricate fine structure. A homely example is the way you make puff pastry. You start with a block of dough, and you spread butter on the top. You then fold it over, then roll it out thinner again with your rolling pin. You do that several times over. The result, most obvious once you cook the pastry, is the familiar texture of thin, crumbly layers. What has happened is that the original single layer of butter has been turned into a very large number of thin layers, separated by thin layers of pastry dough. The more times the stretching and folding is repeated, the finer the structure becomes. A cook will only do a small number of folding operations, but the mathematical system of equations does it infinitely many times.
A more technological example of a similar process comes from traditional sword-making. The modern version of “Damascus steel” is made by bonding together sheets of two different types of steel, and then repeatedly heating it, folding it over and hammering out thinner. The result is a fine layered structure of harder and softer forms of steel: the hard layers will hold a very fine edge, while the softer layers make the sword blade less brittle. In recent times, Japanese traditional sword-makers have turned to making woodworking tools, so we have a tenuous musical instrument link: many violin makers will have Japanese tools in their armoury. Figure 8 shows a Japanese knife blade: the surface shows a characteristic wood-grain or watered-silk pattern. If you look carefully at the bevelled edge in the close-up, you can see layers within the thickness of the metal.
We can use the Lorenz equations to illustrate one more qualitative phenomenon. In Fig. 5 we emphasised the fact that a small change in initial position leads to a drastic divergence of trajectories. But now look at Fig. 9. This shows three trajectories (in different colours), starting from three very different positions. What we see here is that although the three trajectories remain different in detail, all three of them look remarkably similar in terms of a qualitative description. After a rather short initial phase, all three of them seem to settle in to the two “curvy planes” which we saw in the earlier examples.
To understand this (at least in a very vague way), we need to introduce a new concept. To start, let’s go back to the simple linear oscillator, with damping. For that system, the entire phase plane is filled with the pattern of a stable spiral: Fig. 10 reminds you of the kind of pattern that involves. Every trajectory, whatever its starting point, spirals in towards the centre of the plot. This is simply the graphical view of something that is physically obvious: no matter how you start the oscillator off, if you wait long enough the initial energy will all be dissipated by the damping, and it will settle back towards it equilibrium position. The position (0,0) in the phase plane is called an attractor, for obvious reasons.
There are other possible types of attractor. Some nonlinear systems show self-excited oscillation (we’ll be looking at that in the next section). A steady periodic oscillation makes a closed loop in phase space, usually called a limit cycle in this context. Such a limit cycle can also be an attractor : for at least some range of initial conditions, the trajectories may all move towards it, so that the system settles down into the periodic vibration pattern. We will see some examples in Fig. 1 of section 8.5.
Before the work of Lorenz and Fetter, and the other pioneers of chaos theory, equilibrium points and limit cycles were the only kinds of attractors people were aware of. But Fig. 9 suggests that the Lorenz equations, with the particular parameter values used here, have some kind of attractor, which is neither an equilibrium point nor a limit cycle. You can sense something of the puzzlement of the pioneers when I tell you that the name they gave this kind of thing is a “strange attractor”. That piece of jargon has stuck. It is indeed the case that all (or, strictly, almost all) trajectories of the Lorenz equations with these particular parameter values tend towards a single strange attractor. This attractor is a very complicated beast: it has what the mathematicians call a fractal structure: the more you zoom in on it, the more fine details are revealed, with no limit. If you remember seeing graphics of the “Mandelbrot set” that would give you the right kind of impression.
After that digression to the Lorenz equations, we can return to vibration and look at a simple system that exhibits chaotic behaviour. Back in section 8.3 we looked at the behaviour of a pendulum: there was no trace of chaotic behaviour there. But we only have to extend the idea of a pendulum in a simple way to find a chaotic system. Figure 11 shows a double pendulum: two identical rods, hinged together and hung from a pivot at the top.
The equations governing the motion of a double pendulum like this are derived in the next link. Armed with those equations, it is easy to solve them in the computer to illustrate how the system behaves: Fig. 12 shows an example. The pendulum has been launched from a particular position and allowed to move freely. It is easy to believe, surely, that this motion might be chaotic.
Sometimes one or other of the two component pendulums goes “over the top” to perform a rotation. In doing so, the system passes near to a saddle point describing the unstable equilibrium when one or other (or both together) balances in the vertical-upwards position. These saddle points are embedded in a four-dimensional phase space, so we can’t easily visualise them, but the generic behaviour is similar to what we saw earlier with the Lorenz system. Provided the motion has enough initial energy that at least one of the pendulums can occasionally do a loop, nearly every motion trajectory exhibits chaotic behaviour.
You might be thinking “But this is just a computer simulation: does a real double pendulum behave like that?” Figure 13 shows an example based on measured motion of a double pendulum: this one is used in a laboratory experiment for second-year Engineering undergraduates at Cambridge. Unlike the simulated version in Fig. 12, the real pendulum has some energy dissipation so that if you leave it for long enough it will eventually end up hanging vertically downwards. But you can see in the figure that before that happens, the motion looks similarly chaotic.
Figure 14 shows a direct demonstration of “sensitive dependence” using this laboratory pendulum system. The operator has launched the system three times, from approximately the same initial position. All three subsequent motions are combined in this plot. You can see that for a while the three cases move in a rather similar way, but they diverge progressively, and by the end the three cases are moving in quite different ways.