# Stability of underdetermined dynamical systems

## Complex Systems Fund project:

One of the three projects that has received a grant from the 2020 Complex Systems Fund focuses at the role of networks in the dynamics of random dynamical systems. The research proposal was written by Dr. Ivan Kryven, Dr. Mara Baudena and Dr. Anna von der Heydt, who give an introduction to the aims and ambitions of their project.

We aim to construct a general framework for studying a network’s role in the dynamics of random dynamical systems. This question will be addressed on the level of mathematical foundations, as well as by studying several real-world examples where networks are anticipated to play a crucial role in systems dynamics. These include Earth system models and ecological systems.

## Imaginary dice

For this research, we need to be able to construct networks with random structure. This is not as straightforward as it seems. Scientist of different background are very keen on working with random numbers, for example by writing numbers on faces of an imaginary dice. However, this becomes much more complex if we try to construct random networks with an imaginary dice having different networks on its faces. We would end up with a dice of an immense complexity in its structure, having large number of faces without any natural order. What is even more exciting, is that the behaviour of such objects turns out to feature extreme sensitivity to their description. We call this unexpected behaviour phase transitions.

## Taming complex behaviour

Another important part of the study is the link to dynamical systems. Some random dynamical systems have a clear network structure, as can often be seen in ecology, for example. Could the phase transitions in random networks explain qualitative change in systems’ dynamics, such as bifurcation points and their proximity? If so, we may obtain a very general tool to tame complex behaviour and cast predictions about dynamical systems that are uncertain or random in their parts or entirety. Finally, we need to understand the complex mechanisms that may influence the description of random networks. In its essence, this question transcends the knowledge about the system itself, and requires unpacking the context encapsulating the system, that is its environment.

## Ecology, Earth systems and Math

Several disciplines come together in this research proposal. Both ecology and Earth systems science span the space of open problems that motivate this research. Networks have long since been central to ecology, while Earth systems models do not conceal an obvious network. However, they are affected by uncertainty and feature critical transitions. There is also a flavour of statistical physics, which has developed methods for reasoning about large combinatorial spaces, just as our immensely complex dice with networks on its faces. These topics are all sewn together with a silver lining of applied mathematics -- the core discipline in this project.

## Project start-up and vacancy

We estimate that the project can begin in late 2020, and the vacancy for a PhD position is expected to be opened soon. Mathematical maturity and motivation are the two most important features of the PhD candidate we are looking for. The candidate should be willing to tackle core mathematical problems in the area of network research, whilst also taking initiative to communicate the importance of the new findings to fellow researches in applied disciplines. The project gives the PhD candidate the rare opportunity to both create new mathematics and watch as it changes the world.