Nonlinear two-point boundary value problems (BVPs) may have none or more than one solution. For the singularly perturbed two-point BVP εu″ + 2u′ + f(u) = 0, 0 < x < 1, u(0) = 0, u(1) = 0, a condition is given to have one and only one solution; also cases of more solutions have been analyzed. After attention to the form and validity of the corresponding asymptotic expansions, partially based on slow manifold theory, we reconsider the BVP within the framework of small and large values of the parameter. In the case of a special nonlinearity, numerical bifurcation patterns are studied that improve our understanding of the multivaluedness of the solutions.

Road networks are becoming more vulnerable to incidents as a result of the increasing level of congestion. An innovative empirical method is proposed and applied to analyze the delays caused by about 490,000 incidents that occurred in the Netherlands in the period 2007 to 2009. The method was applied to the motorway network of the Netherlands for which detailed loop detector data were available and for which incidents were registered. The method contributes to the existing literature by explaining how delays that are caused by incidents can be computed by selecting a reference day and by tracing the effects of incidents. The method considers upstream delays on the road where the incident occurred, spillback effects to other roads, and rubbernecking effects. The method provides insight for the locations that are most vulnerable to incidents and a basis for taking robustness measures and understanding the potential benefits of these measures.

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