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Growing through chaos in the Matsuyama map via subcritical flip and bistability

Laura Gardini () and Iryna Sushko ()
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Iryna Sushko: Institute of Mathematics, NASU, and Kyiv School of Economics, Ukraine

No 1801, Working Papers from University of Urbino Carlo Bo, Department of Economics, Society & Politics - Scientific Committee - L. Stefanini & G. Travaglini

Abstract: Recent publications reconsider the growth model proposed by Matsuyama ("Growing through cycles"), also called M-map, presenting new interpretation of the model as well as new results on its dynamic behaviors. The goal of the present paper is to give the rigorous proof of some results which were remaining open, related to the dynamics of this model. We prove that in the whole parameter range of interest an attracting 2-cycle appears via border collision bifurcation, we give the explicit flip bifurcation value of the 2-cycle proving that it is always of subcritical type. This leads to bistability related to coexistence of an attracting 2-cycle with attracting 4-cyclic chaotic intervals. We give the conditions related to the sharp transition to chaos, proving that the cascade of stable cycles of even periods cannot occur. The parameter range in which repelling cycles of odd period exist is further investigated, giving an explicit boundary, as well as its relation to the non existence of cycles of period three. Length: 25 pages

Keywords: Endogenous growth models; Matsuyama map; Piecewise smooth map; Subcritical flip bifurcation; Border collision bifurcation; Skew tent map as a normal form. (search for similar items in EconPapers)
JEL-codes: O41 E32 C62 C61 D90 (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-mac
Date: 2018, Revised 2018
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http://www.econ.uniurb.it/RePEc/urb/wpaper/WP_18_01.pdf First version, 2018 (application/pdf)

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