The dynamics of a ring network with switched connections

In this work we analyse transitions between symmetric and asymmetric regimes in a ring network with periodically forced connections. In particular, the network consists of a ring where the connections are periodically switched (ON/OFF) with a circular law. We consider, as an example, a sequence of n reactors where the feed position is periodically shifted according to a permutation law. We analyse the symmetry-breaking phenomena which are consequence of interaction between the natural and external forcing action. As the main parameters are varied due to the presence of Neimark-Saicker bifurcations, the system exhibits periodic regimes where the periods are exact multiples of the period of the forcing or quasi-periodic regimes. In addition to the standard phenomenon of frequency locking, we observe symmetry breaking transitions. While in a symmetric regime all the reactors in the network have the same time history, symmetry breaking is always coupled to a situation in which one or more reactors of the ring exhibit a greater temperature than the others. We found that symmetry is broken when the rotational number of the limit cycle, which arises from the Neimark-Saicker bifurcation, is an specific ratio. Finally, symmetry locking and resonance regions are computed through the bifurcational analysis to detect the critical parameters which mark the symmetry-breaking transitions.