Okay, here's something that's certainly occured to many of you. ODE solvers make errors at every step. That is, they introduce small changes in the system's state at every step, and chaotic systems are sensitively dependent on small changes in state. Remember the wood chip in the eddy or the tennis ball? So how in the world can we say that the trajectories that we get from our ODE solvers are actually trajectories of a chaotic system? And do we ever really see a chaotic attractor? There's a very important result about this. It's called the shadowing lemma and this is what it says. If you add noise to a trajectory on a chaotic attractor, you'll land on something called a shadow trajectory. In other words, noise bounces you onto an attractor thread that you would have gotten to eventually anyway in forward time or in backward time. But there is an important caveat here. Unless the noise is big enough to bounce you out of the basin of attraction of the attractor. If that's true, this doesn't hold. And the shadowing lemma does not hold if there's noise in the parameters. Like the r in the lorenz system. It only holds if there is noise in the state variables. So the answer to my earlier rhetorical questions is that ODE solvers produce points on the attractor, but maybe not in the right order, because at every step, the trajectory gets bounced around by all those little errors we've been talking about. The shadowing lemma saves us by saying that those small changes don't bounce you off the attractor, although if they're a little bigger, then there may be some transient to die out and if they're really big, they can bounce you out of the basin of attraction. But even if they're small enough not to bump you off the attractor, they do mess up the time base. If you're using a reasonable solver - step size, etc - these effects happen on tiny scales. So the flow of the dynamics in the pictures that I've been showing you are correct at the scale of our vision. But if you zoomed in, you'd see something like this. Here, the blue things are attractor threads. Of course, you wouldn't see those if you were doing this. I've just drawn them so that you can see kind of what the schematic is. The red points are the output of the ODE solver. It starts at point 1 and you would think it would go to the next point along the trajectory - maybe there - but instead, it goes over here because of numerical error. Again, the shadowing lemma says that that blue curve is a piece of the same attractor. At the next time step, again, you would expect the trajectory to move right along the attractor, but the noise bumps it over to that red point labeled 3. The shadowing lemma only applies technically to chaotic attractors. But some of the ideas in today's discussion apply to other attractors. If you're at a fixed-point attractor, for instance, a small perturbation will shrink. That's the definition. If you're on a limit cycle, you'll see something like this so that this state might get bounced off the limit cycle because of the numerical error, but the difference will shrink because of the contraction of state space of the transverse stable manifold that slopes downhill towards that limit cycle. A lemma, by the way, is something that you prove as a step along the way towards proving something else. Something that's "bigger" or more important. No one remembers what the shadowing lemma was a step on the way to, but the lemma itself ended up having much more impact. Now, a note to those of you who want to dig further into this, which requires a fair bit of math background: the shadowing lemma requires hyperbolicity which requires an intersection of a stable and unstable manifold. Chaotic attractors, by definition, have at least one of each as we'll talk about in the next unit of this course. That unit is our last one about flows, and there we will hit some of the fun capstone examples and ideas like Lyapunov exponents, unstable periodic orbits, and fractals and fractal dimension.