One-dimensional and two-dimensional dynamics of cubic maps

Ilhem, Djellit; Amel, Kara

doi:https://doi.org/10.1155/DDNS/2006/15840

Discrete Dynamics in Nature and Society

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Volume 2006 | Article ID 015840 | https://doi.org/10.1155/DDNS/2006/15840

One-dimensional and two-dimensional dynamics of cubic maps

Djellit Ilhem¹and Kara Amel²

Received20 Dec 2005

Accepted24 Mar 2006

Published30 Aug 2006

Abstract

We concentrate on the dynamics of one-dimensional and two-dimensional cubic maps, it describes how complex behaviors can possibly arise as a system parameter changes. This is a large class of diffeomorphisms which provide a good starting point for understanding polynomial diffeomorphisms with constant Jacobian and equivalent to a composition of generalized Hénon maps. Due to the theoretical and practical difficulties involved in the study, computers will presumably play a role in such efforts.

References

A. Barugola and J. C. Cathala, “An extension of the notion of chaotic area in two-dimensional endomorphisms,” in Proceedings of the European Conference on Iteration Theory (ECIT '92), Batschuns, September 1992.
View at: Google Scholar
G. I. Bischi, C. Mammana, and L. Gardini, “Multistability and cyclic attractors in duopoly games,” Chaos, Solitons and Fractals, vol. 11, no. 4, pp. 543–564, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. R. Dullin and J. D. Meiss, “Generalized Hénon maps: the cubic diffeomorphisms of the plane,” Physica D. Nonlinear Phenomena, vol. 143, no. 1–4, pp. 262–289, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. Fournier-Prunaret, “Chaotic attractors in transmission system,” in Proceedings of the International Symposium on Nonlinear Theory and Its Applications (NOLTA '98), vol. 2, pp. 735–738, Crans-Montana, September 1998.
View at: Google Scholar
S. Friedland and J. Milnor, “Dynamical properties of plane polynomial automorphisms,” Ergodic Theory and Dynamical Systems, vol. 9, no. 1, pp. 67–99, 1989.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
L. Gardini, “Some global bifurcations of two-dimensional endomorphisms by use of critical lines,” Nonlinear Analysis. Theory, Methods & Applications, vol. 18, no. 4, pp. 361–399, 1992.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
L. Gardini, “Homoclinic bifurcations in $n$ -dimensional endomorphisms, due to expanding periodic points,” Nonlinear Analysis. Theory, Methods & Applications, vol. 23, no. 8, pp. 1039–1089, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
I. Gumowski and C. Mira, “Accumulations de bifurcations dans une récurrence,” Comptes Rendus de l'Académie des Sciences. Paris, Série A-B, vol. 281, no. 1, pp. Aii, A45–A48, 1975.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
I. Gumowski and C. Mira, Dynamique Chaotique, Cepadues Éditions, Toulouse, 1980.
View at: Zentralblatt MATH | MathSciNet
R. Lupini, S. Lenci, and L. Gardini, “Bifurcations and multistability in a class of two-dimensional endomorphisms,” Nonlinear Analysis. Theory, Methods & Applications, vol. 28, no. 1, pp. 61–85, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. Mira, Chaotic Dynamics, World Scientific, Singapore, 1989.
C. Mira, D. Fournier-Prunaret, L. Gardini, H. Kawakami, and J.-C. Cathala, “Basin bifurcations of two-dimensional noninvertible maps: fractalization of basins,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 4, no. 2, pp. 343–381, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. Mira, L. Gardini, A. Barugola, and J.-C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps, vol. 20 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, New Jersey, 1996.
View at: Zentralblatt MATH | MathSciNet
D. Rand, “Exotic phenomena in games and duopoly models,” Journal of Mathematical Economics, vol. 5, no. 2, pp. 173–184, 1978.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2006 Djellit Ilhem and Kara Amel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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