Singular dynamical systems and Picard solutions of PDE

The research proposal is framed in the theory of Colombeau generalized smooth functions and its applications. In the first part of the present project (work package 1), we propose to start developing the theory of singular dynamical systems, i.e. systems described by equations involving both the state of a system and its derivatives, and where we have to study abrupt changes of this state, e.g. like in car crashing or ruptures and damages of several systems. In work package 2, we plan to apply this theory of singular dynamical systems to idealized models of car crash modeling. This type of examples will also be studied as complex systems in the theory of interaction spaces, a universal mathematical theory of complex systems developed by the principal investigator of the project. In work package 3, we propose to extend some recent results about the Picard-Lindelöf theorem for partial differential equations, i.e. where the involved derivatives concern one or more variables. In work package 2, we first want to develop a general theory of hierarchy of complex systems using category-theory functors preserving cause-effect relations in two interaction spaces. Some examples of hierarchical complex systems are: the brain, urban systems, the immune system, organisms in biology, social systems, etc. In this way, we would have a further far-reaching generalization of dynamical systems theory outlined through this link with complex systems modeling and relevant examples. We also plan to consider a hierarchical description of car crash modeling, by using this general theory by first representing a single car using a model for the nonlinear behavior of car crash mechanics. Then, using freely available software code, we finally plan to couple the previous description of a vehicle with a traffic flow model (higher hierarchical description as complex systems). In work package 3, we plan to develop a general notion of Picard iteration, with several applications to linear and nonlinear singular partial differential equations. The project will deliver: a general theory of dynamical systems including the classical one but also singular differential equations. A model of car accidents in traffic flow as hierarchic complex systems, and apply to this system the general theory of work package 1. Primary researchers involved are Priv.-Doz. Dr P. Giordano and Prof. M. Kunzinger as senior researchers working on all the work package of this project. The employment of a post-doc researcher is planned to help in developing work package 1 and work package 2.