Unbounded not diverging trajectories in maps with a vanishing denominator
Abstract
Maps with a denominator which vanishes in a subset of the phase space may generate unbounded trajectories which are not divergent, i.e. trajectories involving arbitrarily large values of the dynamic variables but which are not attracted to infinity. In this paper we propose some simple one-dimensional and two-dimensional recurrences which generate unbounded chaotic sequences, and through these examples we try to explain the basic mechanisms and bifurcations leading to the creation of unbounded sets of attraction.
References
1. R. Abraham, L. Gardini, C. Mira, Chaos in discrete dynamical systems (a visual introduction in two dimensions), Springer-Verlag, 1997.
2. B. Barucci, G.I. Bischi, R. Marimon, The stability of inflation target policies, mimeo European University Institute, Florence 1998.
3. L. Billings, J.H. Curry, On noninvertible maps of the plane: Eruptions, Chaos 6 (1996), 108-119.
4. L. Billings, J.H. Curry, E. Phipps, Lyapunov exponents, singularities, and a riddling bifurcation, Phys. Rew. Letters 6 (1997), 1018-1021.
5. G.I. Bischi, L. Gardini, Basin fractalization due to focal points in a class of triangular maps, International Journal of Bifurcation & Chaos 7 (1997), 1555-1577.
6. G.I. Bischi, A. Naimzada, Global analysis of a nonlinear model with learning, Economic Notes 26 (1997), 143-174.
7. G.I. Bischi, L. Gardini, C. Mira, Maps with denominator. Part 1: some generic properties, International Journal of Bifurcation & Chaos 9 (1999), 119-153.
8. J.H. Curry, L. Garnet, D. Sullivan, On the iteration of a rational function: computer experiments with Newton's method, Comm. Math. Phys. 91 (1983), 267-277.
9. L. Gardini, G.I. Bischi, Focal points and focal values in rational maps, Grazer Math. Ber. (to appear).
10. R. Marimon, S. Sunder, Expectations and learning under alternative monetary regimes: an experimental approach, Economic Theory 4 (1994), 131-162.
11. C. Mira, Equation de Schroeder et solutions des recurrences. Generalisation des polynomes de Tchebycheff, Proceedings of Théorie de l'Iteration et ses Applications Toulose 1982, Editions CNRS, Paris 1982, 35-43.
12. C. Mira, Some properties of two-dimensional (Z_0-Z_2) maps not defined in the whole plane, Graz. Math. Ber. (to appear).
13. C. Mira, L. Gardini, A. Barugola, J.C. Cathala, Chaotic dynamics in two-dimensional noninvertible maps, World Scientific, Singapore 1996.
2. B. Barucci, G.I. Bischi, R. Marimon, The stability of inflation target policies, mimeo European University Institute, Florence 1998.
3. L. Billings, J.H. Curry, On noninvertible maps of the plane: Eruptions, Chaos 6 (1996), 108-119.
4. L. Billings, J.H. Curry, E. Phipps, Lyapunov exponents, singularities, and a riddling bifurcation, Phys. Rew. Letters 6 (1997), 1018-1021.
5. G.I. Bischi, L. Gardini, Basin fractalization due to focal points in a class of triangular maps, International Journal of Bifurcation & Chaos 7 (1997), 1555-1577.
6. G.I. Bischi, A. Naimzada, Global analysis of a nonlinear model with learning, Economic Notes 26 (1997), 143-174.
7. G.I. Bischi, L. Gardini, C. Mira, Maps with denominator. Part 1: some generic properties, International Journal of Bifurcation & Chaos 9 (1999), 119-153.
8. J.H. Curry, L. Garnet, D. Sullivan, On the iteration of a rational function: computer experiments with Newton's method, Comm. Math. Phys. 91 (1983), 267-277.
9. L. Gardini, G.I. Bischi, Focal points and focal values in rational maps, Grazer Math. Ber. (to appear).
10. R. Marimon, S. Sunder, Expectations and learning under alternative monetary regimes: an experimental approach, Economic Theory 4 (1994), 131-162.
11. C. Mira, Equation de Schroeder et solutions des recurrences. Generalisation des polynomes de Tchebycheff, Proceedings of Théorie de l'Iteration et ses Applications Toulose 1982, Editions CNRS, Paris 1982, 35-43.
12. C. Mira, Some properties of two-dimensional (Z_0-Z_2) maps not defined in the whole plane, Graz. Math. Ber. (to appear).
13. C. Mira, L. Gardini, A. Barugola, J.C. Cathala, Chaotic dynamics in two-dimensional noninvertible maps, World Scientific, Singapore 1996.
BischiG.-I., GardiniL., & MiraC. (1999). Unbounded not diverging trajectories in maps with a vanishing denominator. Annales Mathematicae Silesianae, 13, 91-102. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14140
Gian-Italo Bischi
bischi@econ.uniurb.it
Instituto di Scienze Economiche, Universitá di Urbino, Italy Italy
Instituto di Scienze Economiche, Universitá di Urbino, Italy Italy
Laura Gardini
Instituto di Scienze Economiche, Universitá di Urbino, Italy Italy
Instituto di Scienze Economiche, Universitá di Urbino, Italy Italy
Christian Mira
GESNLA-DGE-INSA, Complexe Scientifique de Rangueil, France France
GESNLA-DGE-INSA, Complexe Scientifique de Rangueil, France France
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