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Structural changes in dynamical systems are often related to the appearance or disappearance of orbits connecting two stationary points (either heteroclinic or homoclinic). Homoclinic orbits typically arise in one-parameter problems when on a branch of periodic solutions the periods tend to infinity (e.g. Guckenheimer & Holmes, 1983). We develop a direct numerical method for the computation of connecting orbits and their associated parameter values. We employ a general phase condition and truncate the boundary-value problem to a finite interval by using on both ends the technique of asymptotic boundary conditions; see, for example, de Hoog & Weiss (1980), Lentini & Keller (1980). The approximation error caused by this truncation is shown to decay exponentially. Based on this analysis and additional numerical investigations we set up an adaptive strategy for choosing the truncation interval.