The Steady and Oscillating States of a Proportioning Network
Armen H. Zemanian · SIAM Journal on Applied Mathematics · 1978
A steady state for a proportioning network is a dynamic state in which neither the node values nor the branch values vary with time. However, it is not required that the flows in opposite directions in a given branch be the same, as is the case for a balanced state. Every proportioning network has at least one steady state. Those branches that have nonzero flows under a given steady but not balanced state induce a bipartite subnetwork. A necessary condition for a proportioning network to have an infinity of steady states for every choice of its (positive) node values is that it contain a complete bipartite subnetwork M with at least two nodes on each side of M. A sufficient condition is that it contain such an M in such a fashion that every node not in M that is adjacent to a node of M is also adjacent to another node not in M. On the other hand, in order for a proportioning network to have no more than a finite number of steady states for each choice of its node values, it is necessary and sufficient that each of its components have no even loops and at most one odd loop. An oscillating state is a dynamic state in which every branch flow keeps reversing direction but remains fixed in magnitude as time progresses. A tree cannot have an oscillating state. Necessary and sufficient conditions on the initial branch values are established for the existence of an oscillating state. Also, to every unbalanced oscillating state there corresponds one or more balanced states, and those balanced states are asymptotically unstable.