Universal equation for explosive phenomena

Climate change, a pandemic or the coordinated activity of neurons in the brain: in many physical models, explosive phenomena appear at a critical tipping point in the process. This means, that at a certain point, the initial structure changes into a new state.

Christian Kühn, professor of multiscale and stochastic dynamics at TUM, and Dr. Christian Bick, of the Vrije Universiteit Amsterdam and Hans Fischer Fellow at the TUM Institute for Advanced Study (IAS), have discovered a universal mathematical structure behind such so-called tipping points. This structure forms the basis for a better understanding of the behavior of interconnected systems.

Their results are described in the paper "A universal route to explosive phenomena" in Science Advances.

Tipping points: when do things get critical?

This is an essential question for scientists of all fields: how can changes in networked systems be predicted and influenced? "In biology, one example is the modelling of coordinated neuron activity", explains Professor Christian Kühn. But such models are also used in other disciplines, such as to better understand the spread of disease or the effects of climate change.

All critical changes in networked systems have one thing in common: a tipping point where the system makes a transition from a base state to a new state. This may be a smooth shift, where the system can easily return to the base state. Or it can be a sharp, difficult-to-reverse transition where the state of the system can change abruptly or "explosively." In this manner, a virus can, depending on its infectiousness, either die out naturally or dramatically spread and remain circulating amongst the population over generations. In the field of climate change, transitions of this kind also occur, for example in connection with the melting of the polar ice caps.

In many cases, the transitions result from the variation of a single parameter. In the case of climate change, one such parameter is the increase in concentrations of greenhouse gases in the atmosphere. Nonlinear mathematics, a central discipline of applied mathematics, deals with such structural transitions, so-called bifurcations. The scientists research how changing parameters influence the qualitative structures of nonlinear systems and when they trigger a tipping point.

Similar structures in many models

In some cases – such as climate change – a sharp tipping point would have extremely negative effects, while in others it would be desirable. Consequently, researchers have used mathematical models to investigate how the type of transition is influenced by the introduction of new parameters or conditions. "For example, you could vary another parameter, perhaps related to how people change their behavior in a pandemic. Or you might adjust an input in a neural system," says Kühn. "In these examples and many other cases, we have seen that we can go from a continuous to a discontinuous transition or vice versa."

Kühn and Bick studied existing models from various disciplines that were created to understand certain systems. “We found it remarkable that so many mathematical structures related to the tipping point looked very similar in those models,” says Bick. "By reducing the problem to the most basic possible equation, we were able to identify a universal mechanism that decides on the type of tipping point and is valid for the greatest possible number of models."

The scientists have thus described a new core mechanism that makes it possible to calculate whether a networked system will have a continuous or discontinuous transition. "We provide a mathematical tool that can be applied universally – in other words, in theoretical physics, the climate sciences and in neurobiology and other disciplines – and works independently of the specific case at hand," says Kühn.

Bifurcation theory of complex systems

However, mathematical tools for tipping points or bifurcations are in many respects still in the early stages of a long development. Particularly in complex systems, many mathematical questions are still completely open because here, not only classical network structures play a part, but also influences from stochastic, interconnections with higher dimensional geometric structures or non-local connections in time and space. In this respect, fields in which the theories can be applied to distinct problems, such as in neurobiology, climate studies or epidemiology give an increased motivation and offer ideal testing ground.

The mathematics group "Multiscale and Stochastic Dynamics" is currently active in such a field and is, for example, developing new bifurcation techniques within the large interdisciplinary European project "Tipping Points in the Earth System (TiPES)" in cooperation with partners from the fields of physics and climate studies.