Dynamical Systems Conversation
09 September 2022
On 22 September 2022, PRiSM doctoral researcher Simon Knighton will resume his collaboration with Professor Lasse Rempe (Professor of Pure Mathematics, University of Liverpool) with a concert of music inspired by science, maths and the natural world.
The concert will take place at 7:30pm, St John’s Church Waterloo, London. It is curated by Knighton as part of his Associate Composer project at Nonclassical.
Drawing connections between everyday life and the phenomena of Dynamical Systems, it will feature music by Martin Suckling, Salvatore Sciarrino, Rolf Wallin, Emily Howard, Kaija Saariaho, Maja Solveig Kjelstrup Ratkle, an installation by RNCM student composers Sam Longbottom and Tanguy Pocquet, alongside world premieres of two new works by Knighton himself and sound artist Chihiro Ono.
Ahead of this exciting occasion, Knighton caught up with Professor Rempe and they had a Q&A session on everything Dynamical Systems. This PRiSM blog is therefore an edited transcription of their conversation, which took place on 22 March 2022 at the PRiSM Lab, Royal Northern College of Music.
Dynamical Systems Conversation
By Simon Knighton and Lasse Rempe
SK: What is a dynamical system?
LR: A dynamical system is a system that changes and evolves over time, according to rules that remain fixed. Think, for example, of planetary bodies, whose movement is determined by the laws of gravity.
Such a system may act in a very simple manner, such as a pendulum, but more often one encounters much more complicated, unpredictable and “chaotic” behaviour. The latter occurs even for seemingly simple examples like a double pendulum (two pendula attached to each other), as well as for much more complicated systems such as the weather.
Dynamical systems have two core elements:
- The set of possible states of the system at any given point in time. Take a pendulum: its state is described by its angular displacement together with the velocity at which the pendulum is moving. So, you can ask ‘what is the current state of pendulum?’ And the answer is given by two numbers: the pendulum’s current angle and its velocity. Another example could be the motion of the planets in our solar system: the state of this system is described by the positions of the bodies in relation to the sun, and their velocities.
- The “evolution rule” of a dynamical system describes how the system changes over time; this change should depend only on the current state of the system, and in particular not depend on time! For example, the movement of a pendulum is determined entirely by the laws of gravity.
SK: What is a regular system? And what is a chaotic system?
LR: Regular systems show stable behaviour. This means that the long-term behaviour does not change drastically when we change our starting value a little (at least for most starting values.) Such a system may, for example, head towards a stable equilibrium over time. An unstable system shows the opposite behaviour, which means that it is essentially unpredictable: A small change can lead to dramatic consequences over time: A famous saying of Edward Lorenz is that ‘the flap of a butterfly’s wings in Brazil may cause a tornado in Brazil months later’; this is often described as “chaotic” behaviour. A very interesting fact, and the subject of some of my research, is that in certain circumstances such chaotic behaviour is not preserved under small perturbations of the system itself: No matter how chaotic such a system may be, there are always also some regular systems nearby. Emily Howard’s piece ‘Orbits’ artistically works with this idea on a musical level.
In the examples of dynamical systems that I described so far, time was a “continuous” variable, which is natural from the point of view of real-world applications. However, the systems that appear in my research are instead modelled in discrete time.
SK: Can you elaborate more on what is meant by a discrete dynamical system?
LR: In a continuous dynamical system, we consider time to take continuous values; that is, one can ask what the state of the system is in three days, in 42 minutes, in 1257 milli-seconds etc. To model such a system, one needs to use the language of calculus.
In contrast, in a discrete dynamical system, we restrict to looking at how the system changes in discrete time steps, e.g. from one second to the next, or from one day to the next, etc. This means that we can describe the evolution rule by what is called a function: ‘here is our state today and this is how we work out what the next state will be’. In other words, there is a formula that allows us to calculate the next state given the current one.
In actual systems, this function might be very difficult to compute. However, it turns out that it is possible to observe chaotic behaviour even when it is given by a very simple formula.
SK: But if you had some huge computer with all the data in the world could you predict the weather?
LR: No. It’s not an issue of how good our models are and how much information can we compute – it’s the system itself that produces chaos and unpredictable behaviour – the flap of a butterfly’s wings will eventually produce chaos. Even if you had perfect equations and infinite computing abilities with perfect accuracy (which you don’t have – calculations always have errors and here is no such thing as a perfect measurement), you could not predict this beyond a relatively short time frame, because any measurement of the current of state of the system can only be approximate.
SK: So, is a dynamical system a purely mathematical thing? Or is it a physical thing? Or both?
LR: Well, a dynamical system is a mathematical object, which may or may not be introduced with the aim of modelling a physical process. What I tend to do as a pure/theoretical mathematician is look for simple cases where I can really understand what’s happening in the hope that this will inform our understanding also in more generally applicable circumstances. I’m not necessarily interested in directly representing anything that happens in the natural/physical world, I’m more interested in having a system that encapsulates some of the key things that we see in natural systems, but with extraneous and non-relevant variables stripped away that would otherwise make the problem too difficult to solve. One might say that the goal is to create mathematical “laboratory conditions” to have a chance of really understanding what is going on.
SK: How does this contrast with the approach that an applied mathematician might take?
LR: An applied mathematician might say, ‘we think this theory is likely to be true, so what are the useful consequences of this? For example, they might be really interested in some swimming organism, or some cancer cell behaviour, and they don’t necessarily need to prove a theorem which says exactly what is going on with absolute certainty – which would be nice, but might not be realistic! Instead, getting enough of an idea of what is going on may be enough to start making predictions about the type of behaviour that is going to occur. This is the same principle as in physics and in chemistry. In pure maths you have a higher standard of rigour to establish results that applied mathematicians can then use and rely on. But necessarily this means that progress can be slower, and there are many problems that pure mathematicians are still trying to solve.
So, there can be very different approaches to dynamical systems theory depending on whether you apply a more theoretical or a more applied viewpoint. However, one will encounter the same underlying themes, such as stability and instability; chaotic behaviour; different forms of “bifurcations” (changes in the system under changes of parameters) and so on.
SK: How do the concepts around unpredictable system behaviour relate to determinism?
LR: The dynamical systems I study tend to be deterministic – in the sense that if you know the state of a system exactly, then you know what the next state is going to be. In practice you can never know a state exactly, and therefore a system may look random despite the fact that it is fully deterministic! So we see that it is essentially impossible to tell, by observing a system, whether it is deterministic or not.
SK: This might be a random tangent, but does this say anything about human free will and self-determinism?
LR: This is not my area, so I hesitate to speculate. I am of course aware that it is an important topic in philosophy. As I said above, I think the theory of dynamical systems can tell us that there may be any observable difference between a system that is deterministic and one that is not. So we might ask – if a world where free will exists would look the same as one where it does not, does the question matter or even make sense? As a mathematician, I would say one first needs a clear definition of what “free will” means in order to make the question meaningful. Of course, quantum mechanics means that we do not view the universe as being deterministic, in any case!
Lasse Rempe biography
Lasse Rempe is a mathematician whose work concerns the dynamics of functions of one complex variable. He received his doctoral degree from the University of Kiel, Germany, in 2003, and currently holds the endowed Chair of Pure Mathematics (created in 1882) at the University of Liverpool, where he has worked since 2006. As well as being a mathematician, Lasse is an amateur violinist with the Liverpool Mozart Orchestra.