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On the stability of a boundedly rational day-to-day dynamic

In: Networks and Spatial Economics, 15(3), 537-557; DOI: 10.1007/s11067-014-9233-y

Authors: Xuan Di, Henry X. Liu, Xuegang Jeff X. Ban, Jeong Whon Yu

Although the boundedly rational (BR) day-to-day dynamic proposed in Guo and Liu (Transp Res B 45(10):1606–1618, 2011) managed to model drivers’ transient behaviour under disequilibrium in response to a network change, its stability property remains unanswered. To better understand the boundedly rational (BR) dynamic, this paper initiates the stability analysis of the BR dynamic. As we will show, the BR dynamic is a piecewise affine linear system consisting of multiple subsystems. The conventional Lyapunov theorem commonly used in the literature cannot be applied and thus a multiple Lyapunov method is adopted. The multiple Lyapunov method requires that the Lyapunov values decrease when trajectories evolve as time elapses and the decreasing rate is bounded above. We can show that within each subsystem, the Lyapunov function decreases at an exponential rate. Meanwhile, when trajectories reach boundaries between subsystems, they can either slide or switch and the Lyapunov value also changes across subsystems at a negative rate. Therefore, the BR dynamic is stable. A small network example is given to illustrate this method.