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We've been talking about the Lyapunov exponent since
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the very first unit of this course. And you remember the
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idea. It's the exponent that parametrizes the exponential
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growth of the separation between two points on a
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chaotic attractor as time goes forward.
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Now, there are actually n Lyapunov exponents in
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an n-dimensional dynamical system. The convention
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is that we call the largest one in this ordering
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lambda 1, the second largest lambda 2, and so on
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down to lambda n. If the system is chaotic, at
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least one of the Lyapunov exponents is positive.
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If the system is dissipative, the sum of the Lyapunov
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exponents is negative. Lyapunov exponents are
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dynamical in variance. That is, you can take an
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attractor and deform it, and bend it, and twist it
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and as long as you don't change the topology
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of the attractor, the Lyapunov exponent will be
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preserved. And that's part of why all that state
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space reconstruction stuff is so important.
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Because it allows you to reconstruct the dynamics
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up to diffeomorphism from a scalar time series
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data set. And diffeomorphisms are those
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transformations that bend and twist without
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changing the topology. And that means that you can
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compute the Lyapunov exponent of the reconstructed
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dynamics and be fairly sure that if you did it right,
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that lambda is true of the underlying dynamics as well.
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By the way, it also makes sense to think about and
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compute Lyapunov exponents of systems that
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do not have attractors - non-dissipative systems.
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That's outside of the scope of this course, but
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you should be aware that it's not only systems
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that have attractors that have Lyapunov exponents.
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This segment is about how to compute those
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exponents. If you know the system equations -
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the differential equations - then you can compute
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Lyapunov exponents using something called the
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variational equations. If you're interested in that,
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take a look at the web page for the semester-long
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version of this
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course that I teach at the University of Colorado
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under the "Liz's Written Notes" section and you'll
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find some notes on how to do that. The usual
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situation, however, is not that you have the equations.
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That's extremely rare. Usually you have time series
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data measured from the system and you want to
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compute the largest Lyapunov exponent. The first step
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in the procedure is to perform a delay coordinate
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embedding of that data to reconstruct the full dynamics.
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Again, if you do that right, the results are
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guaranteed to be diffeomorphic to the true dynamics
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and the lambdas are the same. The second step
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in the procedure of calculating Lyapunov exponents
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from data, after the delay coordinate embedding,
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is to operationalize this picture. This is a picture I've
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drawn several times now. It's the notion
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of the apple and the tennis ball in the eddy.
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You drop two points in a chaotic attractor, you
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watch where both of them go, and you track the
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distance between them, and that distance grows as
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e to the lambda t. Now, this is a real challenge when
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you're working with data because the data are fixed.
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You don't get to drop the points at will or let them
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go as long as you want, but rather, you have to work
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with what you've got. If all I had, for example,
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was the video from the field trip in the first unit, ie
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I could not go down to the creek and drop in more
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apples, but I was forced to use only the information
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in that video, I could only track the dynamics of the eddy
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where it was sampled by the apple and by the tennis
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ball which I could track. How to get traction on that
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problem? There are tons of approaches in the
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nonlinear dynamics literature. Algorithms for
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taking a trajectory from a system - that is, a finite
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number of points from a system, maybe noisy -
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and from that data, estimating the largest positive
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Lyapunov exponent. The original one, which is
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called Wolff's algorithm, was a direct operationalization
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of this picture. It took a trajectory of a dynanmical
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system, chose a point on that trajectory, looked for
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that point's nearest neighbor, watched where both
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those points went, tracked the distance between them,
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and watched how that distance grew with time.
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If you go to the webpage for the semester-long
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version of this course that I teach at CU and scroll
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down to the "Liz's Written Notes" section, there's
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a set of written notes on that algorithm. Here's
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the picture from that set of written notes. As you
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can see from this schematic, Wolff's algorithm
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tracks the distance between the points, not indefinitely
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but only until that distance grows to a certain level
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and then it does something called a renormalization
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by looking for the nearest neighbor of the endpoint
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and then repeating the whole operation. Here's
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the algorithm from that and here's the formula
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for backing the Lyapunov exponent out of the ratio
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of those different lengths. Now, this picture gets
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back to an issue that I raised a while back -
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that business about how can you have exponential
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growth in a bounded object? Back then, I waived
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my hands about the answer. Now, you can actually
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see that answer. Lambda 1, the largest positive
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Lyapunov exponent, captures the average
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long-term stretching as you move along the attractor.
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That is - kind of the transverse stretchiness as you
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walk along that original trajectory. There's a
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complication here that arises from the fact that
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there are multiple dimensions and as many
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Lyapunov exponents as there are dimensions.
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Remember though, that if you have a bunch of
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exponentials, and you let t go to infinity,
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the largest positive one will dominate.
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And what I just said before was that lambda 1
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captures the long-term average transverse stretching.
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There's an underlying assumption here and in the
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other algorithms for calculating Lyapunov exponents
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from a time series that can be a little confusing.
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Those 2 black points are both points on the same
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trajectory. That is, this guy, is where the system is
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at some time, t, and this so-called nearest neighbor
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is where the system is at some earlier or later time.
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So this notion of following them both forward in
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time is a little bit weird, but it's completely okay
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if your system is autonomous. That is, if the
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direction that is dynamically downhill at a given
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point is always the same, regardless of when or
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how the point got there. That assumption underlies
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pretty much all of methods for calculating
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Lyapunov exponents as I said. If it doesn't hold,
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that is, if your system is non-autonomous such that
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trajectories can go in different directions from the
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same state space point at different times, the
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Lyapunov definitions and algorithms don't apply.