Why studying random dynamical systems matters

Goodwin Gibbins

Goodwin Gibbins

I'm doing my MRes and PhD on the mathematics of climate change at the moment, trying to step back and look at the system as a whole. Current main areas of interest are paleoclimate, thermodynamics of the earth system in general, entropy in particular and what scientists can do to help society best with this global warming problem.
Goodwin Gibbins

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If I said I was studying multistability in random dynamical systems driven by Brownian motion, why should you care?

Martingales, filtrations, ergodicity, synchronisation, Monte Carlo methods: mathematics is full of words which invoke colorful ephemeral imagery for heavy, abstract concepts. Even if we explained the concepts and brought them to life, it wouldn’t be obvious how they were related to your world. Mathematics is powerful exactly because it exists a few steps away from reality, in a place where anything which obeys its structured rules is possible. But this means that to answer the question of why we study what we study (apart from that it’s beautiful and fun) we have to step out of mathematics and back firmly into the physical.

When we think about the Planet Earth mathematically we often think about understanding and capturing each of its parts, studying technical topics like turbulence, boundary currents, atmospheric waves, sub-gridscale processes or atmospheric chemistry. These are the building blocks of the Earth and investigating them shows us patterns and connections which let us project their behaviour forward a few days into the future by running them artificially faster inside our super-computers, predicting the weather or the dissipation of volcanic ash.

There is another powerful way to think about our Earth, though. For the past decade or so, biologists, for example, have stepped away from the reductionist  approach of looking at the components of their system (protein and DNA) to start looking at the high-level systems behavior. The pieces slot together intricately into such a complex system that some of its behavior (consciousness, for an extreme example, or an ant colony) is nearly impossible to discern by looking at the parts. They find that sometimes it’s better to drop the details and start at the top and work down, characterising as much of the emergent behaviour as possible since it’s part of understanding the animal, even if we can’t quite see where it comes from.

In the study of Planet Earth we’re lucky that we’ve been so driven early on by prediction even while we’ve been studying the pieces. We’ve always had an eye on the whole system, trying to simulate it in weather models since predictability is equivalent to safety and success as we try to flourish as a species. These models do an amazing job of mirroring the massive weather beast, especially as we tweak them year on year, checking them against reality and improving them. But what we don’t often do is explore theoretically the range of behaviour the complicated highly non-linear and multi-dimensional system we live within is capable of. We don’t often look at it from the top without the baggage of the details of the pieces. This is the truly “systems” approach.

Perhaps the simplest example of what you might see if you took a systems perspective is multistability: the idea that the Earth might have different states (or rhythms) it could comfortably fall into which would self-perpetuate themselves, even if in the same conditions another state would have been possible. The mathematical example is always a ball rolling down into a valley with another valley next door; the ball could stop at the bottom of either valley, both are stable states, but it can’t get from one to another easily. Ecosystems exhibit multi-stability – the same patch of land could be a birch forest or a grazed field and stay that way indefinitely. Even bodies exhibit multistability if you look at them from the right perspective: something alive generally keeps itself alive, but dead also stays dead.

A schematic view of bistability, in which the “ball” can be in either “valley” happily, but finds it difficult to transition between the two. Source: Wikimedia Commons.

Whether or not our climate has multiple stable states is vitally important because if it does and we start to push it, we could very possibly push it out of one and into another and it might be very hard to return. How would we know if it did? One way (the reductionist way) might be to carefully build a replica of it, to the very best of our ability, then start pushing it around and see if it switches. Not a bad idea, but with obvious drawbacks too: our model might not be quite perfect and it takes a long time to run these pushing experiments on a computer, so it’d be hard to draw solid conclusions. Another way is to think theoretically about what is required of a system to be a jumping system then study whether the Earth satisfies the conditions, looking back at the record we have about our real Earth system to see if we can see signatures of jumping. To do that, however, we need to know what jumping looks like.

That’s why we’re interested in a class of mathematical constructs called random dynamical systems. They’re an abstract construct and could be interpreted into the real world in lots of different ways so, like most everything in mathematics, they’re probably useful in a lot of fields. What they can do for us, though, is to represent the Earth like a point moving between states according to some update rule, complicated or simple, which summarises the evolution of our climate in time. What makes them even more complicated than the dynamical systems people like Ed Lorenz have been studying for decades is that we now allow the update rule to be a little bit random thanks to a Brownian motion term: to send the Earth up the valley when it should have gone down, occasionally. Suddenly, a whole new class of behaviour is exposed: with bistabilitity in the update scheme and additional noise comes stochastic resonance, the seemingly regular jumping of our proverbial ball between valleys depending on which way the wind (which wouldn’t usually be enough to move a ball between valleys!) blows. They also might simplify dynamical systems, with noise-induced synchronisation meaning that even if we don’t know where exactly the point started, it’ll end up in a similar place if it undergoes the same noise.

It’s not that the Earth is a random dynamical system (it’s a planet, not an abstract mathematical concept), it’s that we hope that by trying to represent it as one, we’ll learn more about it. Climate certainly has some random elements: weather provides unpredictable and sometimes large change in local climate, not to mention volcanoes. We also have stochastic resonance-looking jump-like events recorded in the history: the Glacial-Interglacials and maybe Dansgaard-Oeschger events in the past million years. If we find that to some extent our reality fits well into this concept, we will be able to draw conclusions from the concept back out to the real world. It probably won’t be as revealing as complex wave functions were for quantum mechanics, but having a new way to account for the world around us is almost surely a good thing.

When it comes to complex systems that we count on for our survival, no one approach can be relied on. We don’t have a reason to believe that there’s a single bit of mathematics that nature obeys on this scale, but there will be patterns and properties at all scales which fit into mathematical frameworks. Random dynamical systems let us look at the pattern of stability and large changes in the whole system and is, as both a mathematical field and a climate perspective, relatively new and so very exciting.

One thought on “Why studying random dynamical systems matters”

  1. I enjoy what you guys are usually up too. Such clever work
    and reporting! Keep up the very good works guys I’ve included you guys to
    my personal blogroll.

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