the discontinuous vector field arises. Three sources of an impasse come from (1)
reduction of models with multiple time scales to the slow time scale, (2) mechanical
systems with impacts and (3) the design of controllers like thermostats or relays that
have switches. Each of these settings suggests a different resolution of the impasse,
and the dynamics observed in each case is also qualitatively different. Thus the
dynamics of a hybrid dynamical system depend upon the structure embodied in the
details of how the continuous and discrete time components of the system are modeled.
Theory that classifies the different possibilities and characterizes the dynamics of
systems that are generic in each context would be very useful in engineering complex
systems.
The interplay of theory, experiment and computation will continue to be important
for the study of emergent properties of complex systems. All are needed in the discovery
of unifying principles that explain how, where and why emergent properties arise.
Empirical data is the beginning and the end: we want to understand and engineer the
real world. Simulation is an important tool for detailed study of specific models. With
diverse models from the abstract to the highly detailed we can explore the origins and
characteristics of emergent properties. Still, many simulation models are sufficiently
complex that they are difficult to analyze, so we need theory to provide a guide that
helps us interpret and organize simulation results. Theory also directs our attention to
interesting phenomena that might otherwise be overlooked, often by highlighting the
structural similarities between different systems.
Research programs on complex systems should maintain a balance for the mutual
contributions of theory, experiment and computation. Support for the engineering and
operation of complex systems that we increasingly rely upon in our daily lives should
recognize the value of cross-cutting principles even in work focused upon a particular
system.
Sample Recommendations
Here we list recommendations on research and training activities in the mathematical
sciences that should be promoted in light of the above discussion. The
recommendations are directed at various groups:
Mathematics Institutes and Scientific Societies:
- To broaden the partnership of scientific societies involved in the current joint
effort to promote mathematical research toward sustainability. The engagement
of the Institute for Mathematical Statistics (IMS), the American Statistical
Association (ASA), the Society for Industrial and Applied Mathematics (SIAM)
and of societies such as INFORMS (Institute for Operations Research and the
Management Sciences) and IEEE (Institute of Electrical and Electronics
Engineers) with a broad international footprint is pivotal. There is also a clear
need to reach out and involve national mathematical societies all over the world. - To organize multi-year research and training programs where undergraduate and
graduate students as well as postdocs could return year after year. It is important
to make sure that these programs engage regulators, policy makers, international
institutions and industry representatives.