Symmetries in timed continuous Petri nets

The dynamics of timed continuous Petri nets under infinite server semantics can be expressed in terms of a piecewise linear system with polyhedral regions. In this article, Petri nets with symmetries are considered where symmetry is understood as a permutation symmetry of the nodes. We establish connections between the qualitative dynamical behavior of the continuous marking and the symmetries. In particular, it is shown that such a symmetry leads to a permutation of the regions and to equivariant dynamics. This allows us to identify special flow-invariant sets which can be used for reductions to systems of smaller dimension. For general piecewise linear systems with polyhedral regions, it is shown that equivariant dynamics always implies a permutation of the regions.