The logistic map and the birth of period-3 cycle
Аннотация
The goal of this paper is to present a proof that for the logistic map the period-3 begins at . The third-iterate map is the key for understanding the birth of the period-3 cycle. Any point in a period-3 cycle repeats every three iterates by definition. Such points satisfy the condition ,and they are therefore fixed points of the third-iterate map. This fact and the so called tangent bifurcation for the logistic map, as well as the fixed points definition, are used for finding the value. The algebraic treatment utilizes some properties of symmetric polynomials in three variables. For the purposes of this paper, the bifurcation diagram for the logistic map is also presented, as well as a program in Mathematica for its construction.
Ключевые слова
attractor,
bifurcation diagram,
dynamical systems,
chaos,
fixed points,
logistic map,
maps,
symmetric polynomials,
tangent condition,
vector fields
510.6
Об авторах
Luis Toro
Instituto de Biotecnología y Agroindustria, Manizales
Россия
Россия
Carlos Cardona
Instituto de Biotecnología y Agroindustria, Manizales
Россия
Россия
Yu. Pisarenko
M.V. Lomonosov Moscow State University of Fine Chemical Technologies
Россия
Россия
Список литературы
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7. Verhulst F. Nonlinear differential equations and dynamical systems. - Berlin: Springer Verlag, 1990. 270 p.
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10. Wolfram S. The mathematic book. - Wolfram Media / Cambridge University Press, 1999. 470 р.
Рецензия
Для цитирования:
, , . Тонкие химические технологии. 2012;7(3):71-76.
For citation:
Toro L.A., Cardona C.A., Pisarenko Yu.A. The logistic map and the birth of period-3 cycle. Fine Chemical Technologies. 2012;7(3):71-76. (In Russ.)
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