  Complex environments, processes and systems may exhibit several distinct
modes of physical behaviour or operation. Thus, for example, in their design, a
set of mathematical models may be needed, each model having its own domain of
application and representing a particular mode of behaviour or operation of
physical reality. The models may be of disparate kinds { discrete or continuous
in data, time and space. Furthermore, some physical modes may not have a
reliable model. Physical measurements determine modes of operation. We explore
the question: What is a mode of behaviour? How do we specify algorithms and
software that monitor or govern a complex physical situation with many modes?
How do we specify a portfolio of modes, and the computational problem of
transitioning from using one mode to another mode as physical modes change? We
propose a general definition of an analogue-digital system with modes. We show
how any diverse set of modes { with or without models { can be bound together,
and how the transitions between modes can be determined, by constructing a
topological data type based upon a simplicial complex. We illustrate the ideas
of physical modes and our theory by reflecting on simple examples, including
driverless racing cars.
