×

A new method to control chaos in an economic system. (English) Zbl 1200.91195

Summary: In this paper, the method to control chaos by using phase space compression is applied to economic systems. Because of economic significance of state variable in economic dynamical systems, the values of state variables are positive due to capacity constraints and financial constraints, we can control chaos by adding upper bound or lower bound to state variables in economic dynamical systems, which is different from the chaos stabilization in engineering or physics systems. The knowledge about system dynamics and the exact variety of parameters are not needed in the application of this control method, so it is very convenient to apply this method. Two kinds of chaos in the dynamic duopoly output systems are stabilized in a neighborhood of an unstable fixed point by using the chaos controlling method. The results show that performance of the system is improved by controlling chaos. In practice, owing to capacity constraints, financial constraints and cautious responses to uncertainty in the world, the firm often restrains the output, advertisement expenses, research cost etc. to confine the range of these variables’ fluctuation. This shows that the decision maker uses this method unconsciously in practice.

MSC:

91B55 Economic dynamics
37N40 Dynamical systems in optimization and economics
37N35 Dynamical systems in control
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ott, E.; Grebogi, C.; Yorke, J., Controlling chaos, Phys. Rev. Lett., 64, 1196-1199 (1990) · Zbl 0964.37501
[2] Boccaletti, S.; Grebogi, C., The control of chaos: theory and applications, Phys. Rep., 329, 103-197 (2000)
[3] He, X. Z.; Westerhoff, F. H., Commodity markets, price limiters and speculative price dynamics, J. Econ. Dyn. Control, 29, 1577-1596 (2005) · Zbl 1198.91161
[4] Holyst, J. A.; Urbanowicz, K., Chaos control in economical model by time-delayed feedback method, Physical A, 287, 587-598 (2000)
[5] Li, C.; Liao, X.; Wong, K., Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication, Physica D, 194, 187-202 (2004) · Zbl 1059.93118
[6] Kopel, M., Improving the performance of an economic system: controlling chaos, J. Evol. Econ., 7, 269-289 (1997)
[7] Liu, X. H.; Wang, X. H., A robust output-feedback controller for a class of uncertain nonlinear systems, J. Syst. Sci. Syst. Eng., 11, 380-384 (2002)
[8] Sheng, Z. H.; Huang, T. W.; Du, J. G.; Mei, Q.; Huang, H., Study on self-adaptive proportional control method for a class of output models, Discret. Contin. Dyn. B, 11, 459-478 (2009)
[9] Song, Y. X.; Yu, X. H.; Chen, G. R., Time delayed repetitive learning control for chaotic systems, Int. J. Bifurcat. Chaos, 12, 1057-1065 (2002) · Zbl 1051.93528
[10] Stoop, R.; Wagner, C., Scaling properties of simple limiter control, Phys. Rev. Lett., 90, 154101.1-154101.4 (2003)
[11] Wagner, C.; Stoop, R., Optimized chaos control with simple limiters, Phys. Rev. E, 63, 017201.1-017201.2 (2000)
[12] Wieland, C.; Westerhoff, F. H., Exchange rate dynamics, central bank interventions and chaos control methods, J. Econ. Behav. Organ., 58, 117-132 (2005)
[13] Xiang, T.; Liao, X.; Wang, K., An improved particle swarm optimization algorithm combined with piecewise linear chaotic map, Appl. Math. Comput., 190, 1637-1645 (2007) · Zbl 1122.65363
[14] Zhang, X.; Shen, K., Controlling spatiotemporal chaos via phase space compression, Phys. Rev. E, 63, 46212-46217 (2001)
[15] Zhang, X.; Shen, K., Control of turbulence in a two-dimensional coupled map lattice, Phys. Lett. A, 299, 159-165 (2002) · Zbl 0996.76037
[16] Arenas, A.; Díaz-Guilera, A.; Pérez, C. J., Self-organized criticality in evolutionary systems with local interaction, J. Econ. Dyn. Control, 26, 2115-2142 (2002) · Zbl 1100.91507
[17] Das, A.; Das, P., Chaotic analysis of the foreign exchange rates, Appl. Math. Comput., 185, 388-396 (2007) · Zbl 1120.91323
[18] Hommes, C.; Huang, H.; Wang, D., A robust rational route to randomness in a simple asset pricing model, J. Econ. Dyn. Control, 29, 1043-1072 (2005) · Zbl 1202.91110
[19] Hommes, C. H.; Nusse, H. E.; Simonovits, A., Cycles and chaos in a socialist economy, J. Econ. Dyn. Control, 19, 155-179 (1995) · Zbl 0875.90106
[20] Agiza, H. N.; Elsadany, A. A., Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Appl. Math. Comput., 149, 843-860 (2004) · Zbl 1064.91027
[21] Sonnemans, J.; Hommes, C.; Tuinstra, J., The instability of a heterogeneous cobweb economy: a strategy experiment on expectation formation, J. Econ. Behav. Organ., 54, 453-481 (2004)
[22] Du, J. G.; Huang, T. W., New results on stable region of Nash equilibrium of output game model, Appl. Math. Comput., 192, 12-19 (2007) · Zbl 1193.91036
[23] Du, J. G.; Sheng, Z. H.; Mei, Q.; Ma, G. J.; Huang, H., Study on output dynamic competition model and its global bifurcation, Int. J. Nonlinear Sci. Numer., 10, 129-136 (2009)
[24] Neugart, M., Complicated dynamics in a flow model of the labor market, J. Econ. Behav. Organ., 53, 193-213 (2004)
[25] Chen, P., Empirical and theoretical evidence of economic chaos, Syst. Dyn. Rev., 4, 81-108 (1988)
[26] Michener, R.; Ravikumar, B., Chaotic dynamics in a cash-in-advance economy, J. Econ. Dyn. Control, 22, 7, 1117-1137 (1998) · Zbl 0906.90027
[27] Onozaki, T.; Sieg, G.; Yokoo, M., Stability, chaos and multiple attractors: a single agent makes a difference, J. Econ. Dyn. Control, 27, 1917-1938 (2003) · Zbl 1178.91105
[28] Cason, T. N.; Friedman, D.; Wagener, F., The dynamics of price dispersion or Edge worth variations, J. Econ. Dyn. Control, 29, 801-822 (2005)
[29] Brock, W. A.; Hommes, C. H., Heterogeneous beliefs and routes to chaos in a simple asset pricing model, J. Econ. Dyn. Control, 22, 1235-1274 (1998) · Zbl 0913.90042
[30] Dechert, W. D.; Sprott, J. C.; Albert, D. J., On the probability of chaos in large dynamical systems: a Monte Carlo study, J. Econ. Dyn. Control, 23, 1197-1206 (1999)
[31] Ahmed, E.; El-misiery, A.; Agiza, H. N., On controlling chaos in an inflation-unemployment dynamical system, Chaos Soliton Fract., 10, 1567-1570 (1999) · Zbl 0958.91042
[32] Agiza, H. N., On the analysis of stability, bifurcation, chaos and chaos control of Kopel map, Chaos Soliton Fract., 10, 1909-1916 (2002) · Zbl 0955.37022
[33] Kass, L., Stabilizing chaos in a dynamical macroeconomic model, J. Econ. Behav. Organ., 33, 313-332 (1998)
[34] Agiza, H. N.; Hegazi, A. S.; Elsadany, A. A., Complex dynamics and synchronization of a duopoly game with bounded rationality, Math. Comput. Simulat., 58, 133-146 (2002) · Zbl 1002.91010
[35] Agiza, H. N.; Hegazi, A. S.; Elsadany, A. A., The dynamics of Bowley’s model with bounded rationality, Chaos Soliton Fract., 12, 1705-1717 (2001) · Zbl 1036.91004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.