1d piecewise smooth map: exploring a model of investment dynamics under financial frictions with three types of investment projects

Iryna Sushko; Laura Gardini; Kiminori Matsuyama

doi:10.3842/umzh.v75i12.7721

Iryna Sushko Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv
Laura Gardini Department of Economics, Society and Politics, University of Urbino, Italy
Kiminori Matsuyama Department of Economics, Northwestern University, USA

DOI: https://doi.org/10.3842/umzh.v75i12.7721

Keywords: 1D piecewise smooth map, border-collision bifurcation, bifurcation structure, investment dynamics under financial frictions

Abstract

UDC 517.9

We consider a 1D continuous piecewise smooth map, which depends on seven parameters. Depending on the values of parameters, it may have up to six branches. This map was proposed by Matsuyama [Theor. Econ., 8, 623–651 (2013); Section 5]. It describes the macroeconomic dynamics of investment and credit fluctuations in which three types of investment projects compete in the financial market. We introduce a partitioning of the parameter space according to different branch configurations of the map and illustrate this partitioning for a specific parameter setting. Then we present an example of the bifurcation structure in a parameter plane, which includes periodicity regions related to superstable cycles. Several bifurcation curves are obtained analytically, in particular, the border-collision bifurcation curves of fixed points. We show that the intersection point of two curves of this kind is an organizing center from which infinitely many other bifurcation curves are originated.

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