Generalized Differential Calculus and Hybrid Dynamical Systems
Howard Blair, David Jakel and Angel Rivera, Electrical Engineering and Computer Science, Syracuse University
Understanding the evolving state of a dynamical system is currently a piecemeal affair that depends on the nature of major components of a system: e.g. discrete vs. continuous components of state, discrete vs. continuous time, local vs. distributed clocks, etc., or else represents differing types by encodings and translations on an ad hoc basis.
Differential calculus on convergence spaces - among which are all directed graphs, all topological spaces, and ramified hybrids of these spaces - will be presented. The uniform approach to differentials serves to unify continuous dynamical systems with discrete state transition systems, and the calculi that are obtained generalize the differential calculus of classical analysis on real and complex vector spaces and function spaces.