Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-10T05:06:53.930Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE EVENTUAL PERIODICITY OF PIECEWISE LINEAR CHAOTIC MAPS

Part of: Topological dynamics Mathematical economics Difference and functional equations, recurrence relations

Published online by Cambridge University Press:  16 February 2017

M. ALI KHAN  and
ASHVIN V. RAJAN
M. ALI KHAN*
Affiliation:
Department of Economics, Johns Hopkins University, Baltimore, MD 21218, USA email akhan@jhu.edu
ASHVIN V. RAJAN
Affiliation:
3935 Cloverhill Road, Baltimore, MD 21218, USA email ashvinrj@aol.com
*
✉akhan@jhu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present a family of continuous piecewise linear maps of the unit interval into itself that are all chaotic in the sense of Li and Yorke [‘Period three implies chaos’, Amer. Math. Monthly82 (1975), 985–992] and for which almost every point (in the sense of Lebesgue) in the unit interval is an eventually periodic point of period $p,p\geq 3$, for a member of the family.

Keywords

Li–Yorke chaospiecewise linear functionseventual convergencealmost everywhere convergencedynamical systemsrationalisability

MSC classification

Primary: 37B20: Notions of recurrence
Secondary: 65Q10: Difference equations 91B55: Economic dynamics
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 3 , June 2017 , pp. 467 - 475
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Balibrea, F. and Jiménez-López, V., ‘The measure of scrambled sets: a survey’, Acta Univ. M. Belii Ser. Math. 7 (1999), 3–11.Google Scholar
Benhabib, J., ‘Chaotic dynamics in economics’, in: The New Palgrave Dictionary of Economics (eds. Durlauf, S. N. and Blume, L. E.) (Macmillan, London, 2008).Google Scholar
Blaya, A. and Jiménez-López, V., ‘Is trivial dynamics that trivial?’, Amer. Math. Monthly 113 (2006), 109–133.Google Scholar
Brucks, K. M. and Bruin, H., Topics from One-Dimensional Dynamics (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
Dana, R. A., Le Van, C., Mitra, T. and Nishimura, K. (eds.), Handbook of Optimal Growth, Vol. 1 (Springer, Berlin, 2006).Google Scholar
Du, B., ‘Are chaotic functions really chaotic?’, Bull. Aust. Math. Soc. 28 (1983), 53–66.CrossRefGoogle Scholar
Du, B., ‘Almost all points are eventually periodic with minimal period 3’, Bull. Inst. Math. Acad. Sin. (N.S.) 12 (1984), 405–411.Google Scholar
Du, B., ‘Topological entropy and chaos of interval maps’, Nonlinear Anal. 11 (1987), 105–114.Google Scholar
Fujio, M., ‘Optimal transition dynamics in the Leontief two-sector growth model with durable capital: the case of capital-intensive consumption goods’, Jpn. Econ. Rev. 60 (2009), 490–511.Google Scholar
Jimenez-López, V., ‘Order and chaos for a class of piecewise linear maps’, Internat. J. Bifur. Chaos 5 (1995), 1379–1394.Google Scholar
Ali Khan, M. and Mitra, T., ‘On choice of technique in the Robinson–Solow–Srinivasan model’, Int. J. Econ. Theory 1 (2005), 83–109.CrossRefGoogle Scholar
Ali Khan, M. and Mitra, T., ‘On topological chaos in the Robinson–Solow–Srinivasan model’, Econom. Lett. 88 (2005), 127–133.CrossRefGoogle Scholar
Ali Khan, M. and Piazza, A., ‘Turnpike theory: a current perspective’, in: The New Palgrave Dictionary of Economics Online (eds. Durlauf, S. N. and Blume, L. E.) (Macmillan, London, 2012).Google Scholar
Li, T. and Yorke, J., ‘Period three implies chaos’, Amer. Math. Monthly 82 (1975), 985–992.Google Scholar
Louck, J. D. and Metropolis, N., Symbolic Dynamics of Trapezoidal Maps (D. Reidel, Dordrecht, 1986).Google Scholar
Majumdar, M., Mitra, T. and Nishimura, K., Optimization and Chaos (Springer, Berlin, 2000).Google Scholar
Nathanson, M. B., ‘Piecewise linear functions with almost all points eventually periodic’, Proc. Amer. Math. Soc. 60 (1976), 75–81.Google Scholar
Nishimura, K. and Yano, M., ‘Nonlinear dynamics and chaos in optimal growth: an example’, Econometrica 63 (1995), 981–1001.Google Scholar
Velupillai, V., Computable Economics (Oxford University Press, Oxford, 2000).Google Scholar
Wilf, H. S., Generatingfunctionology (Academic Press, New York, 1994).Google Scholar