Having seen how unpredictable the motion of the pendulum is, it might seem not too far fetched to abandon all hope and let the case rest. This, however, is a bit too pessimistic. It is possible to make some fairly accurate predictions concerning the motion, represented through the deflection angle θ even in the chaotic regime! They are, however, not too concerned with predicting the full pattern and time evolution, but are a bit more qualitative. To see some first glimpse of it consider the phase space plots below. There we show the trajectories of the pendulums with two different driving forces in a θ-ω plane.
The pattern is clear: With a small driving force (left panel), the trajectory in phase space is easy to understand. At the beginning there's some transient time, in which the eigen-frequency decays, but after that the pendulum quickly sets into a regular orbit corresponding to the oscillatory motion in both θ and ω. It can be shown that this is independent of the initial conditions, after a few cycles the pendulum will be there - in agreement with what the negative Lyapunov exponent for small driving force suggests.
In the chaotic regime (right panel), the behaviour is certainly more surprising: We note the existence of orbits that seem to be nearly closed and are thus populated time and again, at least nearly. Of course, however, after a few cycles such orbits are left, only to be eventually revisited at a later point. While this pattern is definitely not simple, it seems also not to be completely random. This is a common finding for many systems of that kind: They often exhibit trajectories in phase space with significant structure.
Let us now examine these trajectories in a slightly different manner, by plotting ω vs. θ only at times that are in phase with the driving force. In other words, we only display points where ΩDt = 2nπ, where n is an integer. This type of presentation is known as Poincare section, and quite a useful tool. Pictorially speaking, it works similarly to a stroboscope. The key point in such representations is that we analyse/watch the system at a rate that matches the problem. For a driven pendulum, this obviously is the frequency of the driving force. The result for the chaotic regime is displayed below:
It turns out that the result is very different for the chaotic and the non-chaotic regime. While the former forms a proper surface in phase space, called the strange attractor, the latter is more or less a single point, which is smeared out by the transient regime at the beginning only. The "fuzziness" of the strange attractor in the chaotic regime is not due to numerical uncertainties or plotting errors - it is a property of the system. The strange attractor itself in fact is largely independent of the initial condition. It is worth noting that it has a fractal structure, something we will encounter in Lecture 7.