Abstract
We are interested in time evolution of systems that switch their modes of operation at discrete moments of time. The intervals between switching may, in general, vary. The number of modes may be finite or infinite. The mathematical setting for such systems is variable time step dynamics with choice. We have used this setting previously to study the long term behavior of such systems. In this paper, we define and study the continuous time dynamics whose trajectories are limits of trajectories of discrete systems as time step goes to zero. The limit dynamics is multivalued. In the special case of a switched system, when the dynamics is generated by switching between solutions of a finite number of systems of ODEs, we show that our continuous limit solution set coincides with the solution set of the relaxed differential inclusion.