## From bi-stability to chaotic oscillations in a macroeconomic model.(English)Zbl 1032.91093

A discrete-time economic model is considered where the savings are proportional to income and the investment demand depends on the difference between the current income and its exogenously assumed equilibrium level, through a nonlinear $$S$$-shaped increasing function. The model can be ultimately reduced to a two-dimensional discrete dynamical system in income and capital, whose time evolution is “driven” by a family of two-dimensional maps of triangular type. These particular two-dimensional maps have the peculiarity that one of their components appears to be uncoupled from the other, i.e., an independent one-dimensional map. The structure of such maps allows to understand completely the forward dynamics, i.e., the asymptotic dynamic behavior, starting from the properties of the associated one-dimensional map. The equilibrium points of the map are determined, and the influence of the main parameters on the local stability of the equilibria is studied. More important, the paper analyzes how changes in the parameters’ values modify both the asymptotic dynamics of the system and the structure of the basins of the different and often coexisting attractors in the phase-plane.
Finally, a particular “global” bifurcation is illustrated, occurring for sufficiently high values of the firms’ adjustment parameter and causing the switching from a situation of bi-stability to a regime characterized by wide chaotic oscillations of income and capital around their exogenously assumed eguilibrium levels.

### MSC:

 91B62 Economic growth models 37N40 Dynamical systems in optimization and economics 91B64 Macroeconomic theory (monetary models, models of taxation)
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### References:

 [1] Abraham R, Gardini L, Mira C. Chaos in discrete dynamical system (a visual introduction in 2 dimensions). Berlin: Springer, 1997 · Zbl 0883.58019 [2] Dana RA, Malgrange P. The dynamics of a discrete version of a growth cycle model, In: Ancot JP, editor. Analysing the structure of economic models. The Hague: Martinus Nijhoff, 1984. p. 205-22 [3] Devaney RL. An introduction to chaotic dynamical systems, 2nd ed. Reading, MA: Addison-Wesley, 1989 [4] Dieci R, Gardini L, Bischi GI. Global dynamics in a Kaldor-type business cycle model. Quaderni dell’Istituto di Matematica “E. Levi”, Università degli Studi di Parma, vol. 4. 1998 [5] Gabisch G, Lorenz HW. Business cycle theory, 2nd ed. Berlin: Springer, 1989 · Zbl 0686.90002 [6] Gardini L, Mira C. On the dynamics of triangular maps, Progetto Nazionale di Ricerca M.U.R.S.T. Dinamiche non lineari e applicazioni alle scienze economiche e sociali. Quaderno 1993;9305 [7] Kaldor, N., A model of the trade cycle, Economic journal, 50, 78-92, (1940) [8] Kolyada SF, Sharkovski AN. On topological dynamics of triangular maps of the plane, In: Proceedings of ECIT89. Singapore: World Scientific, 1991. p. 177-83 [9] Kolyada, S.F., On dynamics of triangular maps of the square, Ergodic theory and dynamical systems, 12, 749-768, (1992) · Zbl 0784.58038 [10] Lorenz, H.W., Strange attractors in a multisector business cycle model, Journal of economic behavior and organization, 8, 397-411, (1987) [11] Lorenz HW. Multiple attractors, complex basin boundaries, and transient motion in deterministic economic models. In: Feichtinger G, editor. Dynamic economic models and optimal control. Amsterdam: North-Holland, 1992. p. 411-30 [12] Mira C. Chaotic dynamics. Singapore: World Scientific, 1987 · Zbl 0641.58002 [13] Mira C, Gardini L, Barugola A, Cathala JC. Chaotic dynamics in two-dimensional noninvertible maps. Singapore: World Scientific, 1996 · Zbl 0906.58027 [14] Rodano G. Lezioni sulle teorie della crescita e sulle teorie del ciclo. Dipartimento di teoria economica e metodi quantitativi, Università di Roma “La Sapienza”, 1997 [15] Stark, J., Invariant graphs for forced systems, Physica D, 109, 163-179, (1997) · Zbl 0925.58047
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