DNS Curves in a Production/Inventory Model
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Abstract
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.
Keywords
Optimal control bifurcation theory Hamilton-Hopf bifurcation intensity splittingPreview
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References
- 1.SKIBA, A. K., Optimal Growth with a Convex-Concave Production Function, Econometrica, Vol. 46, pp. 527–539, 1978.MATHCrossRefMathSciNetGoogle Scholar
- 2.BROCK, W. A., and MALLIARIS, A. G., Differential Equations, Stability, and Chaos in Dynamical Economics, North Holland, Amsterdam, Netherlands, 1989.Google Scholar
- 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).Google Scholar
- 4.STEINDL, A., and FEICHTINGER, G., Bifurcations to Periodic Solutions in a Production/Inventory Model, Journal of Nonlinear Science, 2004 (to appear).Google Scholar
- 5.VAN DER MEER, J. C., The Hamiltonian-Hopf Bifurcation, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 1160, 1986.Google Scholar
- 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.Google Scholar
- 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.MATHCrossRefMathSciNetGoogle Scholar
- 8.FEICHTINGER, G., and WIRL, F., Instabilities in Concave, Dynamic, Economic Optimization, Journal of Optimization Theory and Applications, Vol. 107, pp. 277–288, 2000.CrossRefMathSciNetGoogle Scholar
- 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.Google Scholar
- 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.MATHCrossRefMathSciNetGoogle Scholar
- 11.WAGENER, F. O. O., Skiba Points for Small Discount Rates, Journal of Opitimization Theory and Applications, Vol. 128, pp. 261–277, 2006.MATHCrossRefMathSciNetGoogle Scholar
- 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).Google Scholar
- 13.SEYDEL, R., A Continuation Algorithm with Step Control, Numerical Methods for Bifurcation Problems, Birkhäuser, Basel, Switzerland, pp. 480–494, 1984.Google Scholar
- 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.CrossRefMathSciNetGoogle Scholar
- 15.DECHERT, D. W., and BROCK, W. A., Lakegame, Mimeo, University of Houston and University of Wisconsin, 1999.Google Scholar
- 16.LEONARD, D., and LONG, N. V., Optimal Control Theory and Static Optimization in Economics, Cambridge University Press, Cambridge, UK, 1992.Google Scholar
- 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).Google Scholar
- 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.MATHCrossRefMathSciNetGoogle Scholar
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