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We study deterministic continuous-time lossy dynamical flow networks withconstant exogenous demands, fixed routing, and finite flow and buffercapacities. In the considered model, when the total net flow in a cell---consisting of the difference between the total flow directed towards itminus the outflow from it--- exceeds a certain capacity constraint, then theexceeding part of it leaks out of the system. The ensuing network flow dynamicsis a linear saturated system with compact state space that we analyze usingtools from monotone systems and contraction theory. Specifically, we prove thatthere exists a set of equilibria that is globally asymptotically stable. Suchequilibrium set reduces to a single globally asymptotically stable equilibriumfor generic exogenous demand vectors. Moreover, we show that the criticalexogenous demand vectors giving rise to non-unique equilibria correspond tophase transitions in the asymptotic behavior of the dynamical flow network.

Stability and phase transitions of dynamical flow networks with finite capacities