The paper considers the dynamical system which arises when a responsive control system is utilised in an urban road network. In this case current route-flows change current green-times (according to some responsive control policy) and current green-times change current delays and hence current flows (according to drivers route-choices).

The paper considers (under natural assumptions) the stability of such a dynamical system. Initially a simple symmetrical network with two routes and a traffic signal which follows the equisaturation policy for any given flows is studied. The symmetrical equilibrium, with equal flows on both routes, is unstable. The paper shows that, within the simple network considered, bottlenecks may be added which makes the symmetrical equilibrium stable for certain loads. Nonetheless the analysis shows that, even with the "stabilising" bottlenecks, standard responsive control policies may introduce the rather complicated behaviour familiar in complex systems: some equilibria will be stable and some will be unstable; in the neighbourhood of critical points small changes in network load may change the attractive equilibrium dramatically; and certain trajectories of network load give rise to hysteresis. Finally the paper considers stability of a similar network when the special P0 policy (see Smith (1980)) is utilised. Under natural conditions the simple network considered becomes stable under this policy, and then there is no need to seek to improve that stability by adding bottlenecks which restrict network capacity. 

Finally the paper shows in outline that under natural conditions the P0 policy maximises network capacity for an arbitrary network.