Stability Analysis of Equilibrium Patterns in a Transportation

Stability Analysis of Equilibrium Patterns in a Transportation


G E Cantarella, University of Salerno; P Velonà, Mediterranean University of Reggio Calabria, IT



Assignment models allow to simulate demand-supply interaction, that is how flows, resulting from user behaviour, are affected by transportation costs, expressing the provided level of service, and vice versa. They are the basic tool to analyse and design transportation networks. Most existing models, and those commonly adopted in practical applications, follow an equilibrium approach, where it is assumed that mutually consistent flows and costs well describe the state of the system relevant for analysis and design (a recent review in Cantarella and Cascetta, 2001).

A more general approach is based on dynamic process models, which allow to explicitly simulate the evolution over time of the system, and the convergence to different types of attractors (a general framework in Cantarella, Cascetta, 1995). Several authors (main references are Smith, 1984; Horowitz, 1984; Cantarella, 1993; Cantarella, Velonà, 2000, 2001 e 2002; Cascetta, Coppola, Adamo, 2000) have contributed to this increasingly interesting field through applying and analysing deterministic process models derived from Non-Linear Dynamic Systems Theory (see among many others Glendinning P., 1999). Some authors have also proposed stochastic process models (Cascetta, 1989; Davis, Nihan, 1993; Hazelton, 2002; Watling 2001, 2002). Dynamic process models include an explicit simulation of user cost and choice updating processes underlying system evolution over time.

In this paper, a general deterministic process models is described, such that the equilibrium pattern is a fixed-point state. A simple but effective approach, based on exponential smoothing, is followed to model both user cost and choice updating processes. Then, conditions assuring the stability of a fixed-point state, which means that it is an attractor, are analysed with respect to main parameters, such demand, behaviour dispersion, derivatives of link cost functions, etc. as well as habit and yesterday experience weight. As reported in previous papers, results obtained from numerical simulations pictorially confirm theoretical expectations.

In this paper hence a formal stability analysis has been carried out, first concerning:
* attractor definition, when different from fixed-points, through Poincarè characteristic multipliers;
* a-periodic attractor identification, through fractality measures.

Then a bifurcation analysis has been carried out for fixed-point attractors, to investigate equilibrium pattern stability. Obtained results, confirming numerical results, show that when a fixed-point attractors loses its stability several types of attractors may occur, such as k-periodic, quasi-periodic, a-periodic, depending on conditions on dynamic process parameters. The effect on the length of transients before convergence to an attractor as well as the role of the starting state are also commented. In addition, the case of multiple fixed-point states, some of them stable other unstable, may occurs and it has been investigated through results from catastrophe theory (Thom, 1974). Relationship with equilibrium existence and uniqueness conditions has also been discussed.

The results in this paper indicate that the equilibrium approach, which does not allow for an explicit stability analysis, may fail to effectively described the state of the system relevant to analysis and design.


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