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The logistic map and the birth of period-3 cycle

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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.

About the Authors

Luis Alberto Toro
Instituto de Biotecnología y Agroindustria, Manizales
Russian Federation

Carlos Ariel Cardona
Instituto de Biotecnología y Agroindustria, Manizales
Russian Federation

Yu. A. Pisarenko
M.V. Lomonosov Moscow State University of Fine Chemical Technologies
Russian Federation


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For citations:

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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