Abstract
Si studia il comportamento dinamico locale e globale di un modello macroeconomico che si rappresenta con un sistema nonlineare di due equazioni alle differenze (sistema dinamico discreto, o mappa). Si mostra come l'uso di un nuovo strumento di analisi, le linee critiche, consenta di studiare molte delle proprietà globali (quali i bacini di attrazione di insiemi invarianti) e delle biforcazioni globali, che si verificano in mappe del piano con inversa non unica. Si prova che il punto fisso localmente attrattivo è anche globalmente attrattivo, mentre in regimi di instabilità del punto fisso esistono curve chiuse, o altri insiemi invarianti (regolari o caotici) di attrazione, in aree assorbenti globalmente attrattive.
Summary
The local and global dynamical behaviour of a macroeconomic model, represented by a nonlinear system of two difference equations (discrete dynamical system or map), is studied. It is shown how several properties (as basins of attraction of invariant sets) and global bifurcations occurring in two-dimensional maps with a non-unique inverse can be studied by use of a new analytical tool, the critical curves. It is proven that the locally attractive fixed point is globally attractive, while closed invariant curves or other attractive invariant sets (regular or chaotic) exist in globally attractive absorbing areas, when the fixed point is repulsive.
This is a preview of subscription content, log in via an institution to check access.
Bibliografia
Andronov A. A., Leontovich E. A., Gordon I. I., Maier A. G.,Theory of Dynamical Systems on a Plane, Israel Program of Scientific Translation, Jerusalem, 1973.
Barugola A.,Détermination de la frontière d'une zone absordante relative à une récurrence du deuxième ordre, à inverse non unique, C. R. Acad. Sc. Paris, t. 290, Serie B, 1980, 257–260.
Barugola A.,Quelques propriétés des lignes critiques d'une récurrence du second ordre a inverse non unique. Détermination d'une zone absobante, R.A.I.R.O. Numerical Analysis,18(2), 1984, 137–151.
Barugola A.,Sur certains zones absorbantes et zones chaotiques d'un endomorphisme bidimensionnel, Int. J. Non-Linear Mechanics, 21(2), 1986, 165–168.
Barugola A., Cathala C., Mira C.,Annular chaotic areas, Nonlinear Analysis, T.M.&A., 10(11), 1986, 1223–1236.
Cathala J.C.,On some properties of absorptive areas in second order endomorphisms, European Conference on Iteration Theory, Batschuns, Austria, Sept. 1989.
Collet P., Eckmann J.P.,Iterated Maps on the interval as Dynamical Systems, Birkhauser, Boston, 1980.
Dana R. A., Malgrange P.,The dynamics of a discrete version of a growth cycle model, in J.P. Ancot (ed.) Analysing the Structure of Economic Models, Martinus Nijhoff, The Hague, 7, 1984, 115–142.
Delli Gatti D., Gallegati M., Gardini L.,Investment confidece, corporate debt and income fluctuations, J. of Economic Behavior and Organization,22, 1993, 161–187.
Gallegati M., Gardini L.,A nonlinear model of the business cycle with money and finance, Metroeconomica,42, 1991, 132.
Gallegati M.,Fluttuazioni del reddito e crisi finanziarie, ED. N.I.S., Roma, in corso di stampa.
Gardini L.,Some global bifurcations of two-dimensional endomorphisms by use of critical lines, Nonlinear Analysis, T.M.&A.,18, 1992, 361–399.
Gardini L.,Dinamiche complesse in una classe di modelli IS-LM, Atti XIII convegno AMASES, 1989.
Gardini L.,Comportamenti dinamici regolari e complessi, in M. Gallegati [11].
Gardini L.,Homoclinic bifurcations in n-dimensional endomorphisms, due to expanding periodic points, Nonlinear Analysis, T.M.&A. (in stampa).
Gardini L.,Absorbing areas and their bifurcations, quaderno N. 16 dell'istituto di Scienze Economiche, Università di Urbino, 1992.
Gardini L., Mira C.,A procedure to determine chaotic and absorbing areas in two-dimensional endomorphisms, Quaderno N. 9002, Gruppo Nazionale M.U.R.S.T. Modelli Nonlineari in Economia e Dinamiche Complesse, Urbino, 1990.
Guckenheimer J., Holmes P. H.,Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer, New York, 1983.
Gumowski I., Mira C.,Dynamique Chaotique, Cepadues Editions, Toulouse, 1980.
Iooss G.,Bifurcation of Maps and Applications, North Holland, Amsterdam, 1979.
Lorenz H. W.,Nonlinear Dynamical Economic and Chaotic Motion, Springer, New York, 1989.
Marotto J. R.,Snap-back repellers imply chaos in R n, J. Math. Analysis Applic.,63, 1978, 199–223.
Mira C.,Chaotic Dynamics, World Scientific, Singapore, 1987.
Mira C.,Complex dynamics in two-dimensional endomorphisms, Nonlinear Analysis, T.M.&A.,4(6), 1980, 1167–1187.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gardini, L. Insiemi invarianti globalmente attrattivi nell'interazione fra il “mercato dei beni” ed il “mercato della moneta”. Rivista di Matematica per le Scienze Economiche e Sociali 16, 41–71 (1993). https://doi.org/10.1007/BF02086762
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02086762