Bifurcation curves in discontinuous maps

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March 2010, 13(2): 249-267. doi: 10.3934/dcdsb.2010.13.249

Bifurcation curves in discontinuous maps

Gian-Italo Bischi ^1, , Laura Gardini ^1, and Fabio Tramontana ^2,

1.	University of Urbino, Department of Economics and Quantitative Methods, Via Saffi 42, 61029 Urbino, Italy, Italy
2.	Marche Polytechnic University, Department of Economics, Piazzale Martelli 8, 60121 Ancona, Italy

Received October 2008 Revised February 2009 Published December 2009

Abstract
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Several discrete-time dynamic models are ultimately expressed in the form of iterated piecewise linear functions, in one- or two- dimensional spaces. In this paper we study a one-dimensional map made up of three linear pieces which are separated by two discontinuity points, motivated by a dynamic model arising in social sciences. Starting from the bifurcation structure associated with one-dimensional maps with only one discontinuity point, we show how this is modified by the introduction of a second discontinuity point, and we give the analytic expressions of the bifurcation curves of the principal tongues (or tongues of first degree) for the family of maps considered, which depends on five parameters.

Keywords: Border-Collision Bifurcations, Discontinuous maps..

Mathematics Subject Classification: Primary: 37E05, 37G15, 39A1.

Citation: Gian-Italo Bischi, Laura Gardini, Fabio Tramontana. Bifurcation curves in discontinuous maps. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 249-267. doi: 10.3934/dcdsb.2010.13.249

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