1D Piecewise Smooth Map: Exploring a Model of Investment Dynamics under Financial Frictions with Three Types of Investment Projects

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 (2013); Sec. 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 point of intersection of two curves of this kind is an organizing center, which serves as the origin of infinitely many other bifurcation curves.