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Synchronization, intermittency and critical curves in a duopoly game. (English) Zbl 1017.91500
Summary: The phenomenon of synchronization of a two-dimensional discrete dynamical system is studied for the model of an economic duopoly game, whose time evolution is obtained by the iteration of a noninvertible map of the plane. In the case of identical players the map has a symmetry property that implies the invariance of the diagonal \(x^1=x^2\), so that synchronized dynamics is possible. The basic question is whether an attractor of the one-dimensional restriction of the map to the diagonal is also an attractor for the two-dimensional map, and in which sense. In this paper, a particular dynamic duopoly game is considered for which the local study of the transverse stability, in a neighborhood of the invariant submanifold in which synchronized dynamics takes place, is combined with a study of the global behavior of the map. When measure theoretic, but not topological, attractors are present on the invariant diagonal, intermittency phenomena are observed. The global behavior of the noninvertible map is investigated by studying of the critical manifolds of the map, by which a two-dimensional region is defined that gives an upper bound to the amplitude of intermittent trajectories. Global bifurcations of the basins of attraction are evidenced through contacts between critical curves and basin boundaries.

MSC:
91A12 Cooperative games
91B62 Economic growth models
37N40 Dynamical systems in optimization and economics
34C60 Qualitative investigation and simulation of ordinary differential equation models
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[1] R. Abraham, L. Gardini, C. Mira, Chaos in discrete dynamical systems (a visual introduction in two dimension) Springer-Verlag, 1997 · Zbl 0883.58019
[2] Alexander, J. C.; Yorke, J. A.; You, Z.; Kan, I.: Riddled basins. Int. J. Bif. chaos 2, 795-813 (1992) · Zbl 0870.58046
[3] Ashwin, P.; Buescu, J.; Stewart, I.: Bubbling of attractors and synchronization of chaotic oscillators. Phys. lett. A 193, 126-139 (1992) · Zbl 0959.37508
[4] Ashwin, P.; Buescu, J.; Stewart, I.: From attractor to chaotic saddle: a tale of transverse instability. Nonlinearity 9, 703-737 (1996) · Zbl 0887.58034
[5] G.I. Bischi, A. Naimzada, Global analysis of a dynamic duopoly game with bounded rationality, Annals of Dynamic Games, in press · Zbl 0957.91027
[6] G.I. Bischi, M. Gallegati, A. Naimzada, Symmetry-breaking bifurcations and representative firm in dynamic duopoly games, 1997, submitted · Zbl 0939.91017
[7] P. Collet, J.P. Eckmann, Iterated maps on the interval as dynamical systems, Birkhäuser, Boston, 1980 · Zbl 0458.58002
[8] R.L. Devaney, An introduction to chaotic dynamical systems, Benjamin/Cummings, Menlo Park, California, 1987 · Zbl 1226.37030
[9] Ferretti, A.; Rahman, N. K.: A study of coupled logistic maps and its applications in chemical physics. Chem. phys. 119, 275-288 (1988)
[10] Gardini, L.; Abraham, R.; Record, R.; Fournier-Prunaret, D.: A double logistic map. Int. J. Bif. chaos 4, No. 1, 145-176 (1994) · Zbl 0870.58020
[11] I. Gumowski, C. Mira, Dynamique Chaotique, Cepadues Editions, Toulose, 1980
[12] Hogg, T.; Huberman, B. A.: Generic behavior of coupled oscillators. Phys. rev. A 29, 275-281 (1984)
[13] Laj, Y. C.; Grebogi, C.; Yorke, J. A.: Riddling bifurcation in chaotic dynamical systems. Phys. rev. Lett. 77, 55-58 (1996)
[14] M. Hasler, Yu. Maistrenko, An introduction to the synchronization of chaotic systems: coupled skew tent maps, 1997, submitted
[15] Y. Maistrenko, V. Maistrenko, A. Popovich, E. Mosekilde, Transverse instability and riddled basins in a system of two coupled logistic maps, submitted · Zbl 0897.58026
[16] Milnor, J.: On the concept of attractor. Comm. math. Phys. 99, 177-195 (1985) · Zbl 0595.58028
[17] Mira, C.; Fournier-Prunaret, D.; Gardini, L.; Kawakami, H.; Cathala, J. C.: Basin bifurcations of two-dimensional noninvertible maps: fractalization of basins. Int. J. Bif. chaos 4, No. 2, 343-381 (1994) · Zbl 0818.58032
[18] C. Mira, L. Gardini, A. Barugola, J.C. Cathala, Chaotic dynamics in two-dimensional noninvertible maps, World Scientific, 1996 · Zbl 0906.58027
[19] C. Mira, Chaotic dynamics, World Scientific, 1987 · Zbl 0641.58002
[20] Ott, E.; Sommerer, J. C.: Blowout bifurcations: the occurrence of riddled basins. Phys. lett. A 188, 39-47 (1994)
[21] Pikovsky, A.; Grassberg, P.: Symmetry breaking bifurcation for coupled chaotic attractors.. J. phys. A: math gen. 24, 4587-4597 (1991) · Zbl 0766.58031
[22] Reick, C.; Mosekilde, E.: Emergence of quasiperiodicity in symmetrical coupled, identical period-doubling systems. Phys. rev. E 52, 1428-1434 (1995)
[23] A.N. Sharkovsky, Yu.L. Maistrenko, E.Yu. Romanenko, Difference equations and their applications, Kluwer Academic Publishers, 1993
[24] Venkataramani, S. C.; Hunt, B. R.; Ott, E.; Gauthier, D. J.; Bienfang, J. C.: Transition to bubbling of chaotic systems. Phys. rev. Letts. 77, 5361-5364 (1996)
[25] Venkataramani, S. C.; Hunt, B. R.; Ott, E.: Bubbling transition. Phys. rev. E 54, 1346-1360 (1996)
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