DOI: 10.1007/s00332-003-0565-x J. Nonlinear Sci. (2004): pp. 469–503 © 2005 Springer Science+Business Media, Inc. Bifurcations to Periodic Solutions in a Production/Inventory Model A. Steindl 1 and G. Feichtinger 2 1 Institute for Mechanics and Mechatronics, Vienna University of Technology, Wiedner Haupt- str. 8-10, A-1040, Vienna e-mail: Alois.Steindl@tuwien.ac.at 2 Institute for Econometrics, Operations Research and Systems Theory, Vienna University of Technology, Argentinierstr. 8, A-1040, Vienna ex-mail: Gustav.Feichtinger@tuwien.ac.at Received January 30, 2003; accepted October 7, 2004 Online publication January 7, 2005 Communicaed by C. H. Hommes Summary. Total production costs sometimes show an S-shaped form. There are several ways in which a plant with given capacity can be adapted to a specific demand rate, one of them being adaptation of intensity per work hour. In this paper we present an appli- cation of the Hamilton-Hopf bifurcation to an inventory/production intensity splitting model with a nonconvex cost function. Our analysis provides a new proof that persis- tent oscillations may be optimal for arbitrary small discount rates. For zero discounting a “Hamilton Hopf bifurcation” occurs, leading to a family of periodic solutions bifurcating from a steady state. If the discount rate becomes positive, almost all periodic solutions vanish; only a unique branch of periodic solutions is obtained. AMS Subject Classification. 34C23, 34C20, 34C25, 37G10, 37C80, 49J15 Key words. Hamilton Hopf bifurcation, optimal control, perturbation, normal form, intensity splitting 1. Introduction Usually, machines or production plants are constructed for a certain production intensity. In a neighbourhood of that intensity there is an efficient way of production, whereas any deviation below or above those levels leads to decreasing efficiency of production, i.e., to increasing unit production cost. Under those conditions the unit production costs show a U -shaped form. Writing these costs as a function of the production intensity v, i.e.,