The logistic map and the birth of period-3 cycle
Abstract
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.
Keywords
attractor,
bifurcation diagram,
dynamical systems,
chaos,
fixed points,
logistic map,
maps,
symmetric polynomials,
tangent condition,
vector fields
510.6
About the Authors
Luis Alberto Toro
Instituto de Biotecnología y Agroindustria, Manizales
Russian Federation
Russian Federation
Carlos Ariel Cardona
Instituto de Biotecnología y Agroindustria, Manizales
Russian Federation
Russian Federation
Yu. A. Pisarenko
M.V. Lomonosov Moscow State University of Fine Chemical Technologies
Russian Federation
Russian Federation
References
1. Campos Diógenes Romero, José Fernando Isaza Delgado. Prolegómenos a los Sistemas Dinámicos. - Bogotá: Universidad Nacional de Colombia, 2002. 504 р.
2. Guckenheimer John, Philip Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. - USA: Springer Verlag, 1983. 453 р.
3. Hirsch Morris W., Stephen Smale. Differential equations, dynamical systems and linear algebra. - USA: Academic Press, 1974. 358 р.
4. Hubbard J.H., West B.H. Differential equations: A dynamical systems approach. - USA: Springer Verlag, 1990. 348 р.
5. Strogatz S.H. Nonlinear dynamics and chaos. - Pearson Books Publishing, 1994. 512 р.
6. Toro Luis Alberto. The logistic map and chaos / Applied Knowledge Paper, Course MATH 723 Chaos Theory. - Lacrosse University. MS. 2005.
7. Verhulst F. Nonlinear differential equations and dynamical systems. - Berlin: Springer Verlag, 1990. 270 p.
8. Wiggins S. Introduction to applied nonlinear dynamical systems and chaos. - New York: Springer Verlag, 1990. 843 р.
9. Boyce W.E., DiPrima R.C. Elementary differential equations and boundary value problems. - John Wiley and Sons, 1992. 605 р.
10. Wolfram S. The mathematic book. - Wolfram Media / Cambridge University Press, 1999. 470 р.
Review
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.)
Views: 198
Email this article 
