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Dynamic Stabilization and Control of Material Flows in Traffic and Manufacturing Networks by Means of Phase Synchronization

Reik Donner, Johannes Höfener, Stefan Lämmer, Dirk Helbing
We study self-organization and optimization of material flows on complex networks. Two rather typical examples from the field of (socio-)economics are considered: vehicular traffic in an urban area and the production and supply of goods between cooperating manufacturers.

In the case of vehicular traffic networks, traffic lights, roundabouts or other regulation facilities control the flows at the intersections (nodes) and lead to an oscillatory switching between ’’on’’ and ’’off’’ states of the throughput in a particular direction. Recently, a model for self-organized control of such intersections has been suggested in terms of a demand-dependent switching of the permeability of the different directions [1]. Here, we use a generalization of this model which explicitly allows for turning and study its performance on networks of intersecting roads.

Similar to traffic networks, production rates and inventory levels in complex manufacturing systems show generic instabilities which lead to immediate oscillations [2]. Although these oscillations do not necessarily lead to intermittent breakdowns of the material flow like in the case of vehicular traffic, they have a severe influence on the network performance and may significantly decrease the productivity of the overall system.

The emerging of synchronization is an intensively studied phenomenon which is known to stabilize the dynamics of ensembles of mutually interacting self-sustained oscillators. In the case of material flows on networks, phase synchronization of the respective flows at the different intersections (nodes) may crucially improve the overall system performance. Recently, it has been demonstrated that this approach may be used for the development of a decentralized control with potential applications to vehicular traffic networks [3].

For the quantification of phase synchronicity in a network, we apply synchronization cluster analysis as well as a recently developed measure for collective behaviour in multivariate time series, the LVD dimension density [4,5], which is in this case based on the eigenvalues of the matrix of pairwise phase synchronization indices. It is shown that a combined analysis with both approaches is superior to the consideration of only one concept in the case of spatially disordered systems like general complex networks.

For both types of systems considered in this study, the mutual interrelationships between the optimization of the network performance in terms of minimal waiting times and low inventory levels, respectively, and the presence of (phase) synchronization in the system are systematically analyzed.

[1] D. Helbing, J. Siegmeier, S. Lämmer: Netw. Het. Med., in press.
[2] D. Helbing, S. Lämmer, U. Witt, T. Brenner: Phys. Rev. E 70, 056118 (2004).
[3] S. Lämmer, H. Kori, K. Peters, D. Helbing: Physica A 363, 39-47 (2006).
[4] R. Donner, A. Witt: Int. J. Bif. Chaos, in press.
[5] R. Donner, A. Witt: Nonlin. Proc. Geophys. 13, 485-497 (2006).
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