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.