Order in chaos: mathematician looks for patterns in unpredictable systems
mathematics
What do chemical reactions, epidemics and vegetation have in common? More than you might think, says mathematician Mark van den Bosch (Leiden University). In his PhD research, he shows that even in systems that behave in chaotic and erratic ways, a surprising amount of order and structure can be found.
The world is full of patterns and structures. Some are easy to see, such as how vegetation spreads across an area. Others are more hidden, like fluctuations in the number of white blood cells in patients with leukaemia. ‘Patterns and structures are really two sides of the same coin,’ Van den Bosch explains. ‘The world is full of order, but sometimes you have to look carefully to recognise it.’
In his work, he does exactly that: trying to understand why patterns form, when they persist, and how sensitive they are to disturbances. Real-world systems are never perfect. They are constantly influenced by small, unpredictable fluctuations — what mathematicians call ‘noise’.
How simple rules create complex forms
A key part of his research focuses on so-called reaction–diffusion systems. This may sound complicated, but the idea is quite intuitive: substances react with each other while also spreading out in space. In mathematical terms, this can be summarised as:
what happens over time = reaction + diffusion
‘That combination is enough to generate patterns,’ says Van den Bosch. This applies not only in chemical experiments, but also to processes such as the spread of diseases or signals in the nervous system.
In the lab, these patterns can actually be seen. In a dish of chemicals, moving structures can appear on their own, such as waves and spirals. After studying these patterns mathematically for five years, Van den Bosch also carried out experiments himself and recreated them in the lab.
Small disturbances, large effects
Once you add noise, the picture changes. And noise is always present. ‘We live in a world full of noise. Think of small fluctuations in light, temperature or other influences that you cannot fully control.’ Models can never capture everything, which is why scientists include noise as a way to represent all the factors that cannot be measured or predicted precisely.
If these disturbances are added to the equation, it becomes:
what happens over time = reaction + diffusion + noise
This extra term turns out to be far from negligible. ‘These small disturbances can sometimes have a big impact,’ says Van den Bosch. They can affect the speed of a moving pattern, slow it down or even destroy it altogether. ‘The stronger the noise, the more fragile the pattern. That may sound obvious, but we have also proven it mathematically.’
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Alongside noise — which is always present — chaos also plays a role, but it is something quite different. While noise comes from outside the system, chaos is built into the system itself. It means that small differences at the start can grow into large differences later on, making the system extremely sensitive. ‘The weather is a good example. You can never predict it exactly, no matter how good your model is.’
Order in a chaotic system
Perhaps the most important conclusion of the research is that even in chaos, there is order. Van den Bosch, for example, studied populations of blowflies. At first sight, their numbers seem to fluctuate randomly, but a closer look reveals that the population always stays within a certain structure. ‘Statistically, that underlying distribution does not change. We call this invariant behaviour.’
At first sight, their numbers seem to fluctuate randomly, but a closer look reveals that the population always stays within a certain structure
Understanding complex systems
This insight is not just interesting in theory. It also helps us better understand complex systems, from heart rhythm disorders to climate models.
‘In systems that are highly sensitive, you cannot ignore noise. Small disturbances can have major consequences.’ By understanding how these disturbances work, models can be improved and predictions made more reliable.
Knowing where to look
What fascinates Van den Bosch most is that nature is not simply random. ‘You might think some systems are truly unpredictable, but if you look closely, there is always some structure behind them.’