Attractors in continuous-time switching networks

We consider a system of equations with discontinuous right hand side, which arise as models of gene and neural networks. We study attractors in R-4 which lie in a set of orthants in the form of figure eight. We find that if the attractor is symmetric with respect to these two loops, then the only possible attractor is a periodic orbit which traverses both loops once. We show that without the symmetry the set of admissible attractors include periodic orbits which follow one loop k times and other loop once, for any k. However, we also show that no trajectory in an attractor can traverse both loops more then once in a row.