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

本研究旨在促进无限维投影动力系统的理论与应用，并促进多时间尺度动力学的理论与应用。PDS在70年代的数学文献中以微分包含的形式被引入（在凸变分分析及其经济应用的背景下），然后通过将其作为希尔伯特空间上的一类不连续微分方程的处理而发展到现在的形式。一般来说，PDS描述了约束问题从非均衡到均衡的时间演化过程。PDS与变分不等式问题有着密切的关系，构成了一个跨学科的数学理论。它们已被应用于工程、交通科学、运筹学、经济学、金融学和博弈论等领域。我们建议通过继续研究周期循环的存在性和稳定性问题以及新结果对应用问题的意义来推进PDS理论。这个建议的作者已经获得（并发表）了这个方向的第一个结果。目前这个方向的研究也涉及到一名研究生。我们建议扩展当前的PDS理论，作为对双层动态DLD的干预（通过将PDS扩展到lp空间），并将DLD的理论和领域（参见当前出版物）扩展到网络交互和互联网流量问题。我们首次在流行病学中使用PDS理论，通过为具有不同感染/接种风险评估的人群群体建模接种策略游戏。我们计划进一步研究此类游戏的DLD模型。最后但并非最不重要的是，鉴于本提案作者收到的CFI高性能计算设施拨款（作为PI），我们建议为PDS/DLD的应用开发数值模拟。

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.