Skiba points and heteroclinic bifurcations, with applications to the shallow lake system

Abstract Techniques from dynamical systems, specifically from bifurcation theory, are used to investigate the occurrence of Skiba points in one-state, one-co-state control systems, for which the effect of the control has a definite direction. A Skiba point is an initial state for which two different optimal solutions of the control problem exist. It is found that the parameter region for which Skiba points occur is bounded by heteroclinic bifurcation manifolds. A local criterion is given that ensures the existence of Skiba points in systems with small discount rates. The analysis is applied to the shallow lake system investigated by Maler et al. (in: K.-G. Maler, C. Perrings (Eds.), The Economics of Non-Linear Dynamic Systems, Resilience Network, 2000, in preparation). For this system, it is shown that for any given parameter value, there is at most one Skiba point.