Advances in the theory and applications of projected dynamical systems and double-layered dynamics

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This proposal is meant to advance both the theory and the applications of the infinite-dimensional projected dynamical systems (PDS), as well as to advance the theory and applications of multiple time-scale dynamics. PDS were introduced in the mathematical literature in the 70's in the form of differential inclusions (in the context of convex variational analysis and its economic applications) and then evolved to their present form via their treatment as a class of discontinuous differential equations on Hilbert spaces. In general, a PDS describes the time evolution of certain constraint problems from disequilibrium to equilibrium. PDS share an intimate relation to variational inequality problems and constitute a mathematical theory of interdisciplinary interest. They have been used in areas such as engineering, transportation science, operations research, economics, finance and game theory. We propose to advance the theory of PDS by continuing to study the question of existence and stability of periodic cycles and the implications of the new results for applied problems. The first results in this direction have been obtained (and published) by the author of this proposal. The current study in this direction also involves a graduate student. We propose to extend the present theory of PDS, as intervenes in that of double layered dynamics-DLD (by extending PDS to Lp-spaces), and to extend the theory and areas of DLD (see current publications) to networks interaction and internet traffic problems. We started using, for the first time, PDS theory in epidemiology, by modelling vaccination strategies games for groups of populations with distinct infection/ vaccination risk assessments. We plan to further this investigation towards a DLD model for such games.     Last but not least, given the CFI High Performance Computing Facility grant the author of this proposal received (as PI), we propose the development of numerical simulations for the applications of PDS/DLD.