Georgia Institute of Technology College of Sciences

This talk contains two parts. First I will discuss our work related to causal modeling in hybrid systems. The idea is to model jump conditions as caused by impulsive inputs. While this is well defined for linear systems, the notion of impulsive inputs poses problems in the nonlinear case. We demonstrate a viable approach based on nonstandard analysis. The second part deals with dynamical systems with delays. First I will show an application of the maximum principle to a delayed resource allocation problem in population dynamics solving a problem in the model of a bee colony cycle. Next I discuss some problems regarding causality in systems with varying delays. These problems relate to the well-posedness (existence and uniqueness) and causality of the mathematical models for physical phenomena, and illustrate why one should consider the physics first and then the mathematics. Finally, I consider the post Newtonian problem as a problem with state dependent delay. Einstein’s field equations relate space time geometry to matter and energy distribution. These tensorial equations are so unwieldy that solutions are only known in some very specific cases. A semi-relativistic approximation is desirable: One where space-time may still be considered as flat, but where Newton’s equations (where gravity acts instantaneously) are replaced by a post-Newtonian theory, involving propagation of gravity at the speed of light. As this retardation depends on the geometry of the point masses, a dynamical system with state dependent delay results, where delay and state are implicitly related. We investigate several problems with the Lagrange-Bürman inversion technique and perturbation expansions. Interesting phenomena (entrainment, dynamic friction, fission and orbital speeds) not explainable by the Newtonian theory emerge. Further details on aspects of impulsive systems and delay systems will be elaborated on by Nak-seung (Patrick) Hyun and Aftab Ahmed respectively.