From Poincaré to May: The Genesis of Discrete Dynamics
Keywords:
Lorenz, He ́non, May, He ́non, May, Quadratic, Map, Dynamical, Systems, Three-body Problem, Chaotic, Attractor, Cantor, Set Poincare ́, Lorenz, He ́non, May, Mapa, Cuadra ́tico, Sistemas, Dina ́micos, Problema de los tres cuerpos, Atractor, Cao ́tico, Conjunto, Cantor
Abstract
In this paper, the origin of discrete dynamics is stated from a historical point of view, as well as its main ideas: fixed and periodic points, chaotic behaviour, bifurcations. This travel will begin with Poincare ́’s work and will finish with May’s work, one of the most important scientific papers of 20th century.
This paper is based on the M.Sc. thesis ”Simple Permutations, Pasting and Reversing”([22]), written by the first author under the guidance of the second author.
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References
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O.E.Mart ́ınezCastiblanco,SimplePermutations,PastingandReversing.TrabajodeGradopresentado como requisito para optar al t ́ıtulo de Magister en Matema ́tica Aplicada, Bogota ́, Universidad Sergio Arboleda, 2020.
R. May, Simple Mathematical Models With Very Complicated Dynamics. Nature. Vol 261, Num. 459, 1976.
M. Misiurewicz, Strange Attractors For The Lozi Mappings. Annals of the New York Academy of Sciences. Vol 357 Num. 1 , 1980.
H. Poincare ́, Sur le proble`me des trois corps et les e ́quations de la dynamique, Acta mathematica, Volume 13, (1890), 1–270
J-C.Yoccoz,PolynoˆmesquadratiquesetattracteurdeHe ́nonSe ́minaireN.Bourbaki.Exp.734,1990- 1991.
P.Acosta-Huma ́nez,Genealogyofsimplepermutationswithorderapoweroftwo(Spanish).Revista Colombiana de Matema ́ticas, 2 (2008), 1–14.
P. Acosta-Huma ́nez, Pasting operation and the square of natural numbers (Spanish). Civilizar, 2 (2003), 85–97.
P.B.Acosta-Huma ́nez,Me ́todosalgebraicosensistemasdina ́micos.EdicionesUniversidaddelAtla ́nti- co, EMALCA, (2014).
P. B. Acosta-Huma ́nez. Nonautonomous Hamiltonian Systems and Morales-Ramis Theory I. The Case x ̈ = f (x, t). SIAM Journal on Applied Dynamical Systems, 8(1), 279–297.
P. B. Acosta-Huma ́nez, M. Alvarez-Ram ́ırez, D. Bla ́zquez-Sanz & J. Delgado. Non-integrability crite- rium for normal variational equations around an integrable subsystem and an example: The Wilber- force spring-pendulum. Discrete & Continuous Dynamical Systems-A, 33(3), (2013) 965–986
́
P.B. Acosta-Huma ́nez, M. Alvarez-Ram ́ırez & J. Delgado. Non-integrability of some few body problems in two degrees of freedom. Qualitative theory of dynamical systems, 8(2), (2009) 209–239.
P. Acosta-Huma ́nez, M. Alvarez-Ram ́ırez & T. J. Stuchi. Nonintegrability of the Armbruster– Guckenheimer–Kim Quartic Hamiltonian Through Morales–Ramis Theory. SIAM Journal on Applied Dynamical Systems, 17(1), (2018) 78–96.
P. Acosta-Huma ́nez, A. Chuquen, Pasting and Reversing Approach to Matrix Theory . JP Journal of Algebra, Number Theory and Applications, 38(6), (2016), 535–559
́P.Acosta-Huma ́nez,A.Chuquen,A.Rodr ́ıguez,OnPastingandReversingoperationsoversomerings, Bolet ́ın de Matema ́ticas Universidad Nacional, 17(2), (2010), 143–164.
P. Acosta-Huma ́nez, O.E. Mart ́ınez Castiblanco, Simple Permutations with Order 4n+2 by Means of Pasting and Reversing Qual. Theory Dyn. Syst. 15, 181–210 (2016). DOI: 10.1007/s12346-015-0161-0
K. Alligood, T. Sauer and J. Yorke, Chaos - An introduction to dynamical systems, Textbooks in mathematicas sciences, Springer-Verlag New York (1996).
P. Blanchard, R.L. Devaney, and G.R. Hall., Differential Equations Belmont, CA: Thomson Brooks/Cole, (2006).
R.L. Devaney, An Introduction to Chaotic Dynamical Systems Addison-Wesley Publishing Company, (1989).
M.Heno ́n,ATwo-dimensionalMappingwithaStrangeAttractor.CommunicationsinMathematical Physics. Vol 50 Num. 1 , 1976.
V. Katz, A history of Mathematics: an introduction Addison-Wesley Educational Publishers, Inc. (1998).
T-Y Li, J.A. Yorke, Period Three Implies Chaos. The American Mathematical Monthly, Vol. 82, No. 10. (Dec., 1975), pp. 985-992.
E.N. Lorenz, Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences Vol. 2 Num. 2 , 1963.
R.Lozi,UnAttracteurEtrange(?)DuTypeAttracteurdeHe ́non.J.Phys.Colloques.Vol39Num.C5 1978.
O.E.Mart ́ınezCastiblanco,Acosta-Huma ́nez,Primitivo,Unenfoquegeome ́tricodelteoremadeShar- kovskii. En Perry, Patricia (Ed.), Memorias del 20 Encuentro de Geometr ́ıa y sus Aplicaciones. (pp. 77-84). Bogota ́, Colombia: Universidad Pedago ́gica Nacional, 2011.
O.E.Mart ́ınezCastiblanco,SimplePermutations,PastingandReversing.TrabajodeGradopresentado como requisito para optar al t ́ıtulo de Magister en Matema ́tica Aplicada, Bogota ́, Universidad Sergio Arboleda, 2020.
R. May, Simple Mathematical Models With Very Complicated Dynamics. Nature. Vol 261, Num. 459, 1976.
M. Misiurewicz, Strange Attractors For The Lozi Mappings. Annals of the New York Academy of Sciences. Vol 357 Num. 1 , 1980.
H. Poincare ́, Sur le proble`me des trois corps et les e ́quations de la dynamique, Acta mathematica, Volume 13, (1890), 1–270
J-C.Yoccoz,PolynoˆmesquadratiquesetattracteurdeHe ́nonSe ́minaireN.Bourbaki.Exp.734,1990- 1991.
Published
2020-12-30
How to Cite
Martínez Castiblanco, O. E., & Acosta-Humánez, P. (2020). From Poincaré to May: The Genesis of Discrete Dynamics. Revista MATUA ISSN: 2389-7422, 7(1), 19-35. Retrieved from https://investigaciones.uniatlantico.edu.co/revistas/index.php/MATUA/article/view/2774
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