We've been talking about the Lyapunov exponent since

the very first unit of this course. And you remember the

idea. It's the exponent that parametrizes the exponential

growth of the separation between two points on a

chaotic attractor as time goes forward.

Now, there are actually n Lyapunov exponents in

an n-dimensional dynamical system. The convention

is that we call the largest one in this ordering

lambda 1, the second largest lambda 2, and so on

down to lambda n. If the system is chaotic, at

least one of the Lyapunov exponents is positive.

If the system is dissipative, the sum of the Lyapunov

exponents is negative. Lyapunov exponents are

dynamical in variance. That is, you can take an

attractor and deform it, and bend it, and twist it

and as long as you don't change the topology

of the attractor, the Lyapunov exponent will be

preserved. And that's part of why all that state

space reconstruction stuff is so important.

Because it allows you to reconstruct the dynamics

up to diffeomorphism from a scalar time series

data set. And diffeomorphisms are those

transformations that bend and twist without

changing the topology. And that means that you can

compute the Lyapunov exponent of the reconstructed

dynamics and be fairly sure that if you did it right,

that lambda is true of the underlying dynamics as well.

By the way, it also makes sense to think about and

compute Lyapunov exponents of systems that

do not have attractors - non-dissipative systems.

That's outside of the scope of this course, but

you should be aware that it's not only systems

that have attractors that have Lyapunov exponents.

This segment is about how to compute those

exponents. If you know the system equations -

the differential equations - then you can compute

Lyapunov exponents using something called the

variational equations. If you're interested in that,

take a look at the web page for the semester-long

version of this

course that I teach at the University of Colorado

under the "Liz's Written Notes" section and you'll

find some notes on how to do that. The usual

situation, however, is not that you have the equations.

That's extremely rare. Usually you have time series

data measured from the system and you want to

compute the largest Lyapunov exponent. The first step

in the procedure is to perform a delay coordinate

embedding of that data to reconstruct the full dynamics.

Again, if you do that right, the results are

guaranteed to be diffeomorphic to the true dynamics

and the lambdas are the same. The second step

in the procedure of calculating Lyapunov exponents

from data, after the delay coordinate embedding,

is to operationalize this picture. This is a picture I've

drawn several times now. It's the notion

of the apple and the tennis ball in the eddy.

You drop two points in a chaotic attractor, you

watch where both of them go, and you track the

distance between them, and that distance grows as

e to the lambda t. Now, this is a real challenge when

you're working with data because the data are fixed.

You don't get to drop the points at will or let them

go as long as you want, but rather, you have to work

with what you've got. If all I had, for example,

was the video from the field trip in the first unit, ie

I could not go down to the creek and drop in more

apples, but I was forced to use only the information

in that video, I could only track the dynamics of the eddy

where it was sampled by the apple and by the tennis

ball which I could track. How to get traction on that

problem? There are tons of approaches in the

nonlinear dynamics literature. Algorithms for

taking a trajectory from a system - that is, a finite

number of points from a system, maybe noisy -

and from that data, estimating the largest positive

Lyapunov exponent. The original one, which is

called Wolff's algorithm, was a direct operationalization

of this picture. It took a trajectory of a dynanmical

system, chose a point on that trajectory, looked for

that point's nearest neighbor, watched where both

those points went, tracked the distance between them,

and watched how that distance grew with time.

If you go to the webpage for the semester-long

version of this course that I teach at CU and scroll

down to the "Liz's Written Notes" section, there's

a set of written notes on that algorithm. Here's

the picture from that set of written notes. As you

can see from this schematic, Wolff's algorithm

tracks the distance between the points, not indefinitely

but only until that distance grows to a certain level

and then it does something called a renormalization

by looking for the nearest neighbor of the endpoint

and then repeating the whole operation. Here's

the algorithm from that and here's the formula

for backing the Lyapunov exponent out of the ratio

of those different lengths. Now, this picture gets

back to an issue that I raised a while back -

that business about how can you have exponential

growth in a bounded object? Back then, I waived

my hands about the answer. Now, you can actually

see that answer. Lambda 1, the largest positive

Lyapunov exponent, captures the average

long-term stretching as you move along the attractor.

That is - kind of the transverse stretchiness as you

walk along that original trajectory. There's a

complication here that arises from the fact that

there are multiple dimensions and as many

Lyapunov exponents as there are dimensions.

Remember though, that if you have a bunch of

exponentials, and you let t go to infinity,

the largest positive one will dominate.

And what I just said before was that lambda 1

captures the long-term average transverse stretching.

There's an underlying assumption here and in the

other algorithms for calculating Lyapunov exponents

from a time series that can be a little confusing.

Those 2 black points are both points on the same

trajectory. That is, this guy, is where the system is

at some time, t, and this so-called nearest neighbor

is where the system is at some earlier or later time.

So this notion of following them both forward in

time is a little bit weird, but it's completely okay

if your system is autonomous. That is, if the

direction that is dynamically downhill at a given

point is always the same, regardless of when or

how the point got there. That assumption underlies

pretty much all of methods for calculating

Lyapunov exponents as I said. If it doesn't hold,

that is, if your system is non-autonomous such that

trajectories can go in different directions from the

same state space point at different times, the

Lyapunov definitions and algorithms don't apply.