A lot of the technical literature on nonlinear systems uses an approach that allows some geometrical insight into the behaviour, at least for systems with smooth nonlinearity. This approach is based on an idea called phase space. We will give an introduction in this section, based on the simplest possible example. Suppose we have a nonlinear oscillator with a single mass, or more generally a single degree of freedom. For any such system, if you specify the position and the velocity at the moment when you set things off, that is sufficient information to determine the subsequent free motion. (The technical reason is that the motion will obey some version of Newton’s law, and so the governing equation will be of second order, requiring two initial conditions to determine a unique solution.)
We can picture this in a two-dimensional plot, with position and velocity as the two axes, as sketched in Fig. 1. Our initial position and velocity, at time $t=0$ say, is represented by a dot in this plane. The subsequent motion of the system, for $t>0$, can then be plotted as a trajectory starting from that point. If we think of running time backwards, we could create the other half of the trajectory, coming from times $t<0$ and reaching our point at $t=0$. But we could have chosen a different position and velocity for our initial spot, and we could have done that anywhere in this plane. So there is a trajectory passing through every point, and there is only one. To put that in different words, the diagram must be filled up with a pattern of trajectories, and they cannot cross. This diagram is called the phase plane, or more generally phase space. The pattern of trajectories is the phase portrait of the particular system we choose to study. It captures in one picture every possible motion of the system.
A good example to illustrate the idea is the pendulum discussed in section 8.2.1 and shown again here in Fig. 2. Its angle of swing is described by an angle $\theta$ (Greek “theta”, remember). We will use this angle for the horizontal axis of our phase portrait, and its rate of change $d\theta / dt$ as the corresponding “velocity” on the vertical axis.
The phase portrait for the pendulum appears in Fig. 3. This plot doesn’t show arrows on the trajectories, but it is easy to work out where the arrows need to point. When we are above the middle of the plot, that means that the velocity $d\theta / dt$ is positive. In other words, the angle $\theta$ is increasing, so all the arrows would point towards the right. In the lower half, $d\theta / dt$ is negative so $\theta$ is decreasing and all the arrows would point to the left.
The values marked on the horizontal axis may be confusing if you aren’t used to thinking of angles in radians, but the code is very simple. One complete revolution of the pendulum takes $\theta$ through an angle of $2\pi$ radians. The value $\theta=0$ corresponds to the pendulum hanging vertically downwards. The values $-2\pi$, $2\pi$ and so on are simply further representations of the same position, after a whole number of rotations. The values $\theta = -3\pi,-\pi,\pi,3\pi$ and so on all indicate the vertical-upwards position.
Figure 3 shows trajectories of two different shapes. Along the middle of the plot there are closed-loop trajectories, whereas in the upper and lower portions there are undulating lines that cross the entire plot. These correspond to the two types of motion you should expect for a pendulum. The closed loops show oscillation, back and forth about the position at the bottom of the swing. The undulating lines in the upper part (with “arrows of time” pointing to the right, remember) show the pendulum continuously rotating, in the direction of increasing $\theta$ and so anticlockwise in the view of Fig. 2. The undulating lines in the lower half have $\theta$ decreasing, so they describe clockwise rotation. These three types of motion are summarised in Fig. 4.
The curves that separate motion of different types are obviously important. These curves are trajectories with a very special property, but to see what that is we should first back off a little and address a more general issue. There are a few points in Fig. 3 which are special, because they correspond to possible equilibrium positions of the pendulum. The pendulum has two equilibrium positions. The obvious one is at $\theta=0$, where the pendulum can hang vertically downwards without any motion. But there is another, where the pendulum sits vertically upwards. In principle the pendulum could balance in that position without moving, but in practice you don’t see it doing so because this equilibrium position is unstable: you only have to tweak the position away from the perfect vertical by the smallest amount, and it will topple and start to swing. The mathematics behind stability and instability of the pendulum is explained in the next link.
These equilibrium positions each appear multiple times in Fig. 3, because we have “unwrapped” the angle $\theta$ to allow for the possibility of rotating motion. The stable equilibrium corresponds to the point at the centre of each of the sets of closed loops: at $\theta=0,2\pi,4\pi$ and so on, on the mid-line of the plot where $d\theta/dt=0$ because the pendulum is stationary. The unstable equilibrium also occurs on that mid-line, at the positions $\theta=-\pi,\pi$ and so on. These are the only points in the plot where we seem to have trajectories that cross. But I said earlier that this couldn’t happen, so what is going on? We will see in a moment that they don’t in fact cross, but they are important for another reason.
The behaviour of the phase portrait in the vicinity of an equilibrium point can be analysed — at least for a system like this one where the nonlinearity is smooth. The details are explained in the next link, but the main result can be shown in pictures. For an undamped system like our pendulum, there are only two types of behaviour that can occur near an equilibrium point (also called a singular point by the mathematicians). The two corresponding phase portraits are shown, in a rather general form, in Fig. 5. On the left, we see what is called a centre: it features a set of closed loops, and describes oscillation around the equilibrium point. Behaviour roughly like this will always occur near a stable equilibrium point of any undamped system. On the right we see a saddle point, which is the generic behaviour near any unstable equilibrium point.
We can immediately recognise these two patterns in Fig. 3. Now look closely at the right-hand plot of Fig. 5, for the saddle point, and pay close attention to the pattern of arrows. The plot does indeed seem to show a pair of trajectories crossing in the middle, but the arrows tell a different story. One of these lines has inward-pointing arrows either side of the “crossing”, while the other has outward-pointing arrows. In the context of our pendulum, we can describe what is happening. The two inward-point arrows indicate the trajectories in which you propel the pendulum upwards with just enough energy that it approaches the unstable equilibrium, but never quite gets there. You could do that either from the left-hand side or from the right-hand side: these are the two inward-pointing segments. The outward-pointing pair show the converse behaviour: the pendulum toppling either to the right or to the left, after an infinitesimally small nudge away from the unstable equilibrium. So what we have is four trajectories, not two, and there is no crossing. The equilibrium point at the centre is a trajectory all by itself.
Now look back at Fig. 4: the curves dividing the different types of behaviour are precisely the trajectories we have just been talking about, coming into and out of the saddle points marking the unstable equilibrium of the pendulum. Such a trajectory is known in the jargon of the subject as a separatrix, precisely because they always have this property of separating behaviour with different qualitative descriptions. We begin to see the power of the phase portrait: it gives a way to encapsulate everything the system is capable of doing, and it has a built-in method for sorting out regions with interestingly different behaviour.
We can take our pendulum example a little further. What happens if we introduce some damping to our pendulum, so that the motion will eventually die away? The phase portrait must change to reflect this fact. Nothing much happens near the saddle points: they are still unstable, and the qualitative behaviour stays the same. But the centre that previously described the behaviour near the stable equilibrium turns into a stable spiral. The corresponding phase portrait is illustrated in general form in Fig. 6: it should be pretty much what you were expecting, showing trajectories spiralling in towards the equilibrium point.
A portion of the phase portrait of the damped pendulum is shown in Fig. 7. Only one “cell” of the repeating pattern is shown, because you need to be able to see the pattern clearly enough to follow some of the trajectories. It remains true that all arrows in the upper half point to the right, and all arrows in the lower half point to the left. Now look carefully to see what has happened to the separatrix coming out of the saddle point in the middle of the left-hand side of the plot. Instead of heading towards the next saddle point, as it did in Fig. 3, it spirals inwards. That is exactly what we expect on physical grounds. Without damping, if you nudge the pendulum very gently away from the unstable equilibrium, it has just enough energy to do one complete revolution and come (more or less) back to the vertical position. But with damping, it loses energy as it moves so that it doesn’t have enough to get back to the vertical. It swings back and forth with gradually decreasing amplitude.
So what about the claim that the separatrices (that’s the plural of “separatrix”) divide regions with different qualitative behaviour? Well, you have to think rather carefully about what Fig. 7 would look like if we extended it to include more “cells” of the repeating pattern in the phase plane. There is a string of saddle points, and each one issues a separatrix heading upwards and rightwards in the top half of the plot. But these no longer look as if they join up as they did in Fig. 4: instead, they nest inside each other. In the region of the phase plane between two successive separatrices of this type, you might describe the motion as “doing two complete anticlockwise rotations before spiralling in”. The next region above this would do three complete rotations, then four, and so on. There is a matching set of nesting separatrices in the lower half of the phase plane, and of course those mark out regions with two, or three, or four clockwise rotations before spiralling in.
Finally, it is interesting to return to the Duffing equation, discussed in section 8.2, and look at its phase portrait. The mathematical details are given in the next link, but the key thing to remember is that this equation describes an oscillator with a spring that has a cubic term added to the linear term describing a linear Hooke’s law spring. For the case with a hardening spring, the phase portrait is rather boring: the only equilibrium point is at the origin in the phase plane, and the whole plane is filled with closed loops characteristic of a centre. Figure 8 shows a plot.
But if we allow the linear component of the spring to have negative stiffness, something dramatic happens. There are now three equilibrium points: a saddle point and two centres. The resulting phase portrait is illustrated in Fig. 9. The two centres represent stable equilibrium, but the saddle point is an unstable equilibrium. Now, you may think this is a silly example because negative springs are non-physical. You would be right, if you are thinking purely about mechanical systems. But if you allow the system to include magnets, you can easily make something that behaves like this. Imagine a pendulum with a magnet at the bottom, swinging over a table which has a second, repelling, magnet fixed to it, right underneath the pivot of the pendulum. The magnetic repulsion acts like a negative spring, and if it is strong enough the pendulum can’t hang vertically downwards. But it can sit either to the left or to the right, and it could oscillate about either of those positions: just what Fig. 9 suggests.
We can learn something interesting about the behaviour in Figs. 8 and 9 if we plot a graph of the position of the equilibrium points against the stiffness of the linear component of the spring. The result is plotted in Fig. 10. When the spring stiffness is positive, there is only one solution as we saw in Fig. 8. When it is negative there are three, as we saw in Fig. 9. The outer pair are stable, shown in solid lines. The middle one, shown as a dashed line, is the unstable equilibrium associated with the saddle point. At the centre of the plot we see the transition between the two. This pattern is called, reasonably enough, a pitchfork bifurcation. Plots like Fig. 10 are called bifurcation diagrams, and they are often useful to summarise the different regimes that a nonlinear vibrating system can show.