Much of the world around us is characterized by intervals of steady change, interrupted by abrupt upheavals or discontinuities. These may come in the form of parameter switching, like the jump in value of the earth's surface reflectivity between frozen and ice-free zones. They may come in the form of sharp events, like mechanical impacts, electronic switches, or economic crashes. The way we model these changes, and the effects they have on the systems they occupy, are the subject of nonsmooth dynamics. Once considered crude, nonsmooth models are now frequently encountered in applied mathematics publications and conferences, and increasingly, the question of how to model discontinuous jumps realistically is being asked.
Discontinuities are easily overlooked as being fleeting events with no internal structure. They are inconvenient for differential equations, at best meaning that standard methods must be patched together ad hoc across a discontinuity, at worst leading to ambiguity or non-uniqueness of solutions. We are now beginning to understand that there is much more to discontinuities than this. As a system passes through a discontinuity it can encounter novel attractors, or can briefly gain access to multiple possible onward trajectories, and most importantly, rigorous ways to describe such behaviour are at last being found.
This provides new ways to analyze abrupt changes in diverse applications, from contact mechanics in automotive engineering, to flows across discontinuous parameter boundaries in climate models. We shall address a particular problem, frequently asked but unanswerable until recently, of whether the typical methods used to simulate discontinuities are adequate, and whether they can be improved. Using recently described "hidden" dynamics of switching, models will be improved by considering how best to reproduce observed data, and where possible, how best to fit underlying physical laws.
This is a substantial opportunity that will require wide collaborations. The last decade has seen many old ideas of nonsmooth dynamics overturned, and a wealth of new theory and methods taking their place. This comes at the same time that greater emphasis on mathematical modeling, in industry as well as in academia, is bringing plentiful nonsmooth models to the fore. By bringing theory and application together in a large strategic effort, we stand to learn enormously about the role of discontinuities in real systems.
We will spearhead this effort with a project doing precisely that, bringing the latest theoretical breakthroughs together with well studied but complex engineering devices, starting with power transmission devices, and looking forward to more exotic applications. The first of these is crowd following behaviour, a current big topic in network science, where the role of how we model switching (for example switching allegiance) is largely overlooked. Looking even further ahead, we will engage with earth scientists who require the new theory to understand the role of discontinuities in climate dynamics.
Our initial focus will be on forming more precise models of discontinuities caused by contact between hard components, such as powertrain shafts rattling or slipping against the metal rings that support them. Using new theoretical ideas we will investigate how much of the irregular behaviour of these devices is due to environmental noise, and how much is due to the underlying complexity of the contact itself. This will help us in forming improved models of discontinuities in wider applications, in social and climatological models in particular, where the underlying physics is either more complex or less understood. The project will take advantage of certain networking activities in 2016, which will be the largest to happen in the nonsmooth dynamics community, to make preliminary steps engaging in a range of applications where discontinuities play a role.
|