Abstract
Some economic models like the cash-in-advance model of money, the overlapping generations model and a model of credit with limited commitment may have the property that the dynamical system characterizing equilibria in the model are multi-valued going forward in time, but single-valued going backward in time. Such models or dynamical systems are said to have backward dynamics. In such instances, what does it mean for a dynamical system (set-valued) to be chaotic? Furthermore, under what conditions are such dynamical systems chaotic? In this paper, I provide a definition of chaos that is in the spirit of Li and Yorke for a dynamical system with backward dynamics. I utilize the theory of inverse limits to provide sufficient conditions for such a dynamical system to be Li-Yorke chaotic.
References
Akin, E. 1993. The General Topology of Dynamical Systems. Vol. 1 of Graduate Studies in Mathematics. Providence, RI: American Mathematical Society.Search in Google Scholar
Azariadis, C. 1993. Intertemporal Macroeconomics. Cambridge, MA: Blackwell.Search in Google Scholar
Benhabib, J., and R. Day. 1982. “A Characterization of Erratic Dynamics in the Overlapping Generations Model.” Journal of Economic Dynamics and Control 4: 37–55.10.2307/j.ctv19fvxt1.6Search in Google Scholar
Blanchard, F., E. Glasner, S. Kolyada, and A. Maass. 2002. “On Li-Yorke Pairs.” Journal Fur Die Reine Und Angewandte Mathematik 547: 51–68.10.1515/crll.2002.053Search in Google Scholar
Bowen, R. 1970. “Topological Entropy and Axiom A.” In Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, CA, 1968). Providence, RI: American Mathematical Society, pp. 23–41.Search in Google Scholar
Christiano, L. J. and S. G. Harrison. 1999. “Chaos, Sunspots and Automatic Stabilizers.” Journal of Monetary Economics 44 (1): 3–31.10.3386/w5703Search in Google Scholar
Gale, D. 1973. “Pure Exchange Equilibrium of Dynamic Economic Models.” Journal of Economic Theory 6: 12–36.10.1016/0022-0531(73)90041-0Search in Google Scholar
Grandmont, J.-M. 1985. “On Endogenous Competitive Business Cycles.” Econometrica 53: 995–1045.10.21236/ADA149289Search in Google Scholar
Gu, C., F. Mattesini, C. Monnet, and R. Wright. 2013. “Endogenous Credit Cycles.” Journal of Political Economy 121 (5): 940–965.10.3386/w17510Search in Google Scholar
Ingram, W. T. and W. S. Mahavier. 2004. “Interesting Dynamics and Inverse Limits in a Family of One-Dimensional Maps.” The American Mathematical Monthly 111 (3): 198–215.10.1080/00029890.2004.11920066Search in Google Scholar
Kennedy, J. A. and D. R. Stockman. 2008. “Chaotic Equilibria in Models With Backward Dynamics.” Journal of Economic Dynamics and Control 32: 939–955.10.1016/j.jedc.2007.04.004Search in Google Scholar
Kennedy, J. A., D. R. Stockman, and J. A. Yorke. 2007. “Inverse Limits and an Implicitly Defined Difierence Equation from Economics.” Topology and Its Applications 154: 2533–2552.10.1016/j.topol.2006.03.032Search in Google Scholar
Kennedy, J. A., D. R. Stockman, and J. A. Yorke. 2008. “The Inverse Limits Approach to Models with Chaos.” Journal of Mathematical Economics 44: 423–444.10.1016/j.jmateco.2007.11.001Search in Google Scholar
Li, T.-Y. and J. A. Yorke. 1975. “Period Three Implies Chaos.” American Mathematical Monthly 82: 985–992.10.1007/978-0-387-21830-4_6Search in Google Scholar
Lucas, R. E. and N. L. Stokey. 1987. “Money and Interest in a Cash-in-Advance Economy.” Econo-Metrica 55: 491–513.10.3386/w1618Search in Google Scholar
McGehee, R. 1992. “Attractors for Closed Relations on Compact Hausdorff Spaces.” Indiana University Mathematics Journal 41 (4): 1165–1209.10.1512/iumj.1992.41.41058Search in Google Scholar
Medio, A. 1992. Chaotic Dynamics: Theory and Applications to Economics. Cambridge: Cambridge University Press.Search in Google Scholar
Medio, A. and B. E. Raines. 2006. “Inverse Limit Spaces Arising from Problems in Economics.” Topology and Its Applications 153: 3439–3449.10.1016/j.topol.2006.03.006Search in Google Scholar
Medio, A. and B. E. Raines. 2007. “Backward Dynamics in Economics. The Inverse Limit Approach.” Journal of Economic Dynamics and Control 31: 1633–1671.10.1016/j.jedc.2006.04.010Search in Google Scholar
Michener, R. and B. Ravikumar. 1998. “Chaotic Dynamics in a Cash-in-Advance Economy.” Journal of Economic Dynamics and Control 22: 1117–1137.10.1016/S0165-1889(97)00096-1Search in Google Scholar
Misiurewicz, M. 1979. “Horseshoes for Mappings of the Interval.” Bull Acad Polon Sci Sér Sci Math 27 (2): 167–169.Search in Google Scholar
Oprocha, P. 2006. “Relations Between Distributional and Devaney Chaos.” Chaos 16 (3): 033112, 5.10.1063/1.2225513Search in Google Scholar PubMed
Robinson, C. 1995. Dynamical Systems – Stability, Sybolic Dynamics and Chaos. Boca Raton, FL: CRS Press.Search in Google Scholar
Sharkovskiĭ, A. N. 1995. “Coexistence of Cycles of a Continuous Map of the Line Into Itself.” In Proceedings of the Conference “Thirty Years after Sharkovskiĭ’s Theorem: New Perspectives” (Murcia, 1994). Vol. 5. pp. 1263–1273, translated from the Russian [Ukrain. Mat. Zh. 16 (1964), no. 1, 61–71] by J. Tolosa.Search in Google Scholar
Smítal, J. 1986. “Chaotic Functions with Zero Topological Entropy.” Transactions of the American Mathematical Society 297 (1): 269–282.10.1090/S0002-9947-1986-0849479-9Search in Google Scholar
©2016 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Steady-state priors and Bayesian variable selection in VAR forecasting
- Dating US business cycles with macro factors
- Effects of filtering data on testing asymmetry in threshold autoregressive models
- The place of gold in the cross-market dependencies
- Li-Yorke chaos in models with backward dynamics
- Hopf bifurcation in an overlapping generations resource economy with endogenous population growth rate
Articles in the same Issue
- Frontmatter
- Steady-state priors and Bayesian variable selection in VAR forecasting
- Dating US business cycles with macro factors
- Effects of filtering data on testing asymmetry in threshold autoregressive models
- The place of gold in the cross-market dependencies
- Li-Yorke chaos in models with backward dynamics
- Hopf bifurcation in an overlapping generations resource economy with endogenous population growth rate