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DNS curves in a production/inventory model. (English) Zbl 1181.49038
Summary: In this paper, we investigate the bifurcation behavior of an inventory/production model close to a Hamilton-Hopf bifurcation. We show numerically that two different types of DNS curves occur: If the initial states are far from the bifurcating limit cycle, the limit cycle can be approached along different trajectories with the same cost. For a subcritical bifurcation scenario, the hyperbolic equilibrium state and the hyperbolic limit cycle coexist for some parameter range. When both the long term states yield approximately the same cost, a second DNS curve separates their domains of attraction. At the intersection of these two DNS curves, a threefold Skiba point in the state space is found.

49N90 Applications of optimal control and differential games
90B05 Inventory, storage, reservoirs
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37N40 Dynamical systems in optimization and economics
90B30 Production models
91B62 Economic growth models
Full Text: DOI
[1] SKIBA, A. K., Optimal Growth with a Convex-Concave Production Function, Econometrica, Vol. 46, pp. 527–539, 1978. · Zbl 0383.90020 · doi:10.2307/1914229
[2] BROCK, W. A., and MALLIARIS, A. G., Differential Equations, Stability, and Chaos in Dynamical Economics, North Holland, Amsterdam, Netherlands, 1989. · Zbl 0693.90001
[3] FEICHTINGER, G., and HARTL, R. F., Optimal Control of Economic Processes: Application of the Maximum Principle in the Economic Sciences, de Gruyter, Berlin, Germany, 1986 (in German).
[4] STEINDL, A., and FEICHTINGER, G., Bifurcations to Periodic Solutions in a Production/Inventory Model, Journal of Nonlinear Science, 2004 (to appear). · Zbl 1167.34011
[5] VAN DER MEER, J. C., The Hamiltonian-Hopf Bifurcation, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 1160, 1986. · Zbl 0613.92017
[6] MEYER, K. R., and HALL, G. R., Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Applied Mathematical Sciences, Springer Verlag, New York, NY, Vol. 90, 1992. · Zbl 0743.70006
[7] DECHERT, D. W., and NISHIMURA, K., Complete Characterization of Optimal Growth Paths in an Aggregative Model with a Nonconcave Production Function, Journal of Economic Theory, Vol. 31, pp. 332–354, 1983. · Zbl 0531.90018 · doi:10.1016/0022-0531(83)90081-9
[8] FEICHTINGER, G., and WIRL, F., Instabilities in Concave, Dynamic, Economic Optimization, Journal of Optimization Theory and Applications, Vol. 107, pp. 277–288, 2000. · Zbl 1014.91062 · doi:10.1023/A:1026408814862
[9] DEISSENBERG, C., FEICHTINGER, G., SEMMLER, W., and WIRL F., Multiple Equilibria, History Dependence, and Global Dynamics in Intertemporal Optimization Models, Economic Complexity: Nonlinear Dynamics, Multiagents Economies, and Learning, Edited by W. A. Barnett, C. Deissenberg, and G. Feichtinger, Elsevier, Amsterdam, Holland, pp. 91–122, 2004.
[10] HAUNSCHMIED, J. L., KORT, P. M., HARTL, R. F., and FEICHTINGER, G., A DNS Curve in a Two-State Capital Accumulation Model: A Numerical Analysis, Journal of Economic Dynamics and Control, Vol. 27, pp. 701–716, 2003. · Zbl 1029.91029 · doi:10.1016/S0165-1889(01)00070-7
[11] WAGENER, F. O. O., Skiba Points for Small Discount Rates, Journal of Opitimization Theory and Applications, Vol. 128, pp. 261–277, 2006. · Zbl 1181.49028 · doi:10.1007/s10957-006-9028-5
[12] OBERLE, H. J., GRIMM, W., and BERGER, E., BNDSCO: Program for the Solution of Constrained Optimal Control Problems, User Manual M 8509, Munich University of Technology, 1985 (in German).
[13] SEYDEL, R., A Continuation Algorithm with Step Control, Numerical Methods for Bifurcation Problems, Birkhäuser, Basel, Switzerland, pp. 480–494, 1984.
[14] WAGENER, F. O. O., Skiba Points and Heteroclinic Bifurcations, with Applications to the Shallow Lake System, Journal of Economic Dynamics and Control, Vol. 27, pp. 1533–1561, 2003. · Zbl 1178.91031 · doi:10.1016/S0165-1889(02)00070-2
[15] DECHERT, D. W., and BROCK, W. A., Lakegame, Mimeo, University of Houston and University of Wisconsin, 1999.
[16] LEONARD, D., and LONG, N. V., Optimal Control Theory and Static Optimization in Economics, Cambridge University Press, Cambridge, UK, 1992.
[17] FEICHTINGER, G., KISTNER, K. P., and LUHMER, A., A Dynamic Model of Intensity Splitting, Zeitschrift für Betriebswirtschaft, Vol. 11, pp. 1242–1258, 1988 (in German).
[18] FEICHTINGER, G., and SORGER, G., Optimal Oscillations in Control Models: How Can Constant Demand Lead to Cyclical Production?, Operations Research Letters, Vol. 5, pp. 277–281, 1986. · Zbl 0615.49009 · doi:10.1016/0167-6377(86)90064-7
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