Lyapunov Time

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In their recent episode of the VALUE: After Hours Podcast, Taylor, Brewster, and Carlisle discuss Lyapunov Time. Here’s an excerpt from the episode:

Jake: Yeah, sure. [laughs] All right, so, this topic is called the Lyapunov time. That’s spelled L-Y-A-P-U-N-O-V. That was a Russian mathematician. This has to do with complexity theory and chaos. Basically, the amount of time that we can successfully predict the state of a chaotic system depends upon three things. Number one, it depends upon how much error we’re willing to tolerate in the forecast. Number two, how precisely we can measure the initial conditions of the system. And then number three, a timescale beyond our control is called the Lyapunov time and it limits the predictability of a system.

Roughly speaking, we can only predict up to that Lyapunov time and after that, the errors start to snowball to where it’s basically exceeded any allowable tolerance that we have. So, you have then only two options. You can lower your standards of how much you’re willing to tolerate in the err of the forecast or you can improve your initial measurements to extend that Lyapunov kind of horizon and allow you to then predict longer as to what a chaotic system will do. But what’s difficult about this is that it’s nonlinear. So, let’s say that you wanted to be able to look out twice as long with the same accuracy of a chaotic system, it’s not going to take 2x the effort of getting better initial measurements, it’s 10x. If you want 3x further out, it’s not 3x or 6x, it’s 100x. If you want 4x, it’s 1,000x that you need in initial measurement improvements.

It’s this nonlinear, basically, like a factor of 10. It grows exponentially. You can apply this then to different domains. So, the chaotic electric circuit has the predict like, it’s Lyapunov window, measurement is like 1/1000th of a second is about how far out you can predict before the chaos, the errors accumulate in a way where you can no longer predict what the system is going to look like based on the initial starting conditions.

Bill: How long?

Jake: 1/1000th of a second.

Bill: Wow. That’s not long.

Tobias: What’s a chaotic electrical circuit?

Bill: Thank you, Toby.

Jake: Yeah. You can get them in a lot of different ways but if you put in certain elements that will create a randomness of syncing between things, let’s say, it turns on or off depending on the other conditions of what the other elements in it are doing and it can cascade out and you’re not exactly sure. It’s that same like grain of sand analogy where you’re never sure exactly when it’s going to run away from–

So, now, when it comes to weather, we’re roughly probably about a few days for that Lyapunov window, like, where we can see the initial starting conditions. We know the cloud cover, we know high and low pressure, and that gives us about a couple days’ worth of accurate predictions at the most.

Tobias: We can predict weather for a couple of days? That’s pretty good.

Jake: I think most of the time. I think they’re reasonably accurate at this point, don’t you? Now, when it comes to the solar system, it’s about 5 million years. So, what that means is that, during our lifetimes, the motion of the planets appears to be perfectly measurable and perfectly predictable. However, in actually over the entire known astronomy, it has been true that they have been predictable. But over more than 5 million years, there’re enough errors that accumulate where we actually can’t make very good predictions as to where a specific planet might be 5 million years from now.

Now, it gives this illusion of predictability though because it feels like, Jupiter is where we think it’s going to be during all of our lifetimes but it’s not actually true. Then, Pluto’s orbit for instance, for whatever reason, we have about a 20-million-year Lyapunov window for that. For some reason, we know where Pluto is going to be with a relative accuracy 20 years from now. So, let’s bring this back to maybe something that we can all use. It’s interesting as a framework but I’m trying to imagine what are the Lyapunov windows for macro predictions or even like this AMC or whatever that something is if people are really hot on and interested in.

We can work at getting better initial known elements of the chaotic system. But within however amount of time like that, you lose that understanding and the chaos unfolds to the point where the errors accumulate to where you just don’t even know what the answer is going to be. You don’t know how the system is going to change. So, my hypothesis or my hunch is that, the Lyapunov window is much shorter than anyone imagines for a lot of this like macro stuff. We’d like to think, we could look out a year. Even if you knew a lot of the initial conditions and we’ve probably been getting better at that over time, like, data collection, I would imagine.

But remember, it’s exponential, the amount of data collection that you need the effort required to extend that window out. So, if you need 100x data collection and accuracy to get doubling the window, I think that we’re probably a little bit like, our intuition is off on that. We think if we measure everything in the whole goddamn economy that we can then make these accurate predictions as to what it needs putting on my Federal Reserve hat right now. But really that window is probably much shorter than– [crosstalk]

Tobias: It means the lower rates.

Jake: Oh, there we go.

Tobias and Jake: Problem solved.

Jake: Anyway, I don’t know. Is anything popped to mind on that of like, what applications might be from thinking through how chaotic systems devolve away from what we could ever predict?

Tobias: It’s interesting. Yeah. I wonder is it impacted further by a system like the stock market or a system like the economy where people are reacting based on the signals that they’re getting too. So, a prediction in one direction will make you behave in one way.

Jake: Right. There’s a Schrödinger’s cat kind of situation there where you don’t know the feedback loops and you don’t know–

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