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.

References

P. Aghion, A. Banerjee, T. Piketty, Dualism and macroeconomic volatility, Q. J. Econ., 114, 1359–1397 (1999).

V. Avrutin, L. Gardini, I. Sushko, F. Tramontana, Continuous and discontinuous piecewise-smooth one-dimensional maps: invariant sets and bifurcation structures, World Sci. Ser. Nonlinear Sci. Ser. A, vol. 95, World Sci. (2019); DOI: 10.1142/8285.

V. Avrutin, M. Schanz, S. Banerjee, Multi-parametric bifurcations in a piecewise-linear discontinuous map, Nonlinearity, 19, 1875–1906 (2006); DOI:10.1088/0951-7715/19/8/007.

S. Banerjee, M. S. Karthik, G. Yuan, J. A. Yorke, Bifurcations in one-dimensional piecewise smooth maps – theory and applications in switching circuits, IEEE Trans. Circ. Syst. I, 47, 389–394 (2000).

S. Banerjee, J. A. Yorke, C. Grebogi, Robust chaos, Phys. Rev. Lett., 80, № 14, 3049–3052 (1998).

S. Banerjee, G. C. Verghese (Eds.), Nonlinear phenomena in power electronics: attractors, bifurcations, chaos, and nonlinear control, IEEE Press, New York (2001).

B. Brogliato, Nonsmooth mechanics – models, dynamics and control, Springer-Verlag, New York (1999).

A. Colombo, F. Dercole, Discontinuity induced bifurcations of nonhyperbolic cycles in nonsmooth systems, SIAM J. Imaging Sci., 3, № 1, 62–83 (2010).

W. de Melo, S. van Strien, One-dimensional dynamics, Springer-Verlag, New York (1991).

M. di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, Piecewise-smooth dynamical systems: theory and applications, applied mathematical sciences, vol. 163, Springer-Verlag, London (2007).

L. Gardini, I. Sushko, A. Naimzada, Growing through chaotic intervals, J. Econom. Theory, 143, 541–557 (2008).

B. L. Hao, Elementary symbolic dynamics and chaos in dissipative systems, World Sci., Singapore (1989).

C. Hommes, H. Nusse, Period three to period two bifurcation for piecewise linear models, J. Economics, 54, № 2, 157–169 (1991).

W. Huang, R. Day, Chaotically switching bear and bull markets: the derivation of stock price distributions from behavioral rules, Nonlinear Dynamics and Evolutionary Economics, Oxford Univ. Press, Oxford (1993), p. 169–182.

S. Ito, S. Tanaka, H. Nakada, On unimodal transformations and chaos II, Tokyo J. Math., 2, 241–259 (1979).

K. Ichimura, M. Ito, Dynamics of skew tent maps, RIMS Kokyuroku, 92–98 (1998).

K. L. Judd, On the performance of patents, Econometrica, 53, 567–586 (1985).

Y. L. Maistrenko, V. L. Maistrenko, L. O. Chua, Cycles of chaotic intervals in a time-delayed Chua's circuit, Int. J. Bifur. Chaos, 3, 1557–1572 (1993).

K. Matsuyama, Growing through cycles, Econometrica, 67, 335–347 (1999).

K. Matsuyama, Credit traps and credit cycles, Amer. Econom. Rev., 97, 503–516 (2007).

K. Matsuyama, The good, the bad, the ugly: an inquiry into the causes and nature of credit cycles, Theor. Econom., 8, 623–651 (2013).

K. Matsuyama, I. Sushko, L. Gardini, Revisiting the model of credit cycles with good and bad projects, J. Econom. Theory, 163, 525–556 (2016); DOI: 10.1016/j.jet.2016.02.010.

N. Metropolis, M. L. Stein, P. R. Stein, On finite limit sets for transformations on the unit interval, J. Combin. Theory, 15, 25–44 (1973).

M. Misiurewicz, E. Visinescu, Kneading sequences of skew tent maps, Ann. Inst. Henri Poincaré Probab. Stat., 27, 125–140 (1991).

H. E. Nusse, J. A. Yorke, Border-collision bifurcations including period two to period three for piecewise smooth systems, Physica D, 57, 39–57 (1992).

H. E. Nusse, J. A. Yorke, Border-collision bifurcation for piecewise smooth one-dimensional maps, Int. J. Bifur. and Chaos, 5, 189–207 (1995).

A. Sharkovsky, S. Kolyada, A. Sivak, V. Fedorenko, Dynamics of one-dimensional maps, Kluer Acad., Dordrecht (1997).

I. Sushko, A. Agliari, L. Gardini, Bifurcation structure of parameter plane for a family of unimodal piecewise smooth maps: border-collision bifurcation curves, Chaos, Solitons & Fractals, 29, Issue 3, 756–770 (2006).

I. Sushko, L. Gardini, Degenerate bifurcations and border collisions in piecewise smooth 1D and 2D maps, Int. J. Bifur. and Chaos, 20, 2045–2070 (2010).

I. Sushko, L. Gardini, K. Matsuyama, Superstable credit cycles and U-sequence, Chaos, Solitons & Fractals, 59, 13–27 (2014); DOI: 10.1016/j.chaos.2013.11.006.

I. Sushko, L. Gardini, K. Matsuyama, Robust chaos in a credit cycle model defined by a one-dimensional piecewise smooth map, Chaos, Solitons & Fractals, 91, 299–309 (2016); DOI:10.1016/j.chaos.2016.06.015.

F. Tramontana, L. Gardini, F. Westerhoff, Heterogeneous speculators and asset price dynamics: further results from a one-dimensional discontinuous piecewise-linear model, Comput. Economics, 38, 329–347 (2011).

Z. T. Zhusubaliyev, E. Mosekilde, Bifurcations and chaos in piecewise-smooth dynamical systems, World Sci., Singapur (2003).