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Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 586783, 23 pages
http://dx.doi.org/10.1155/2015/586783
Research Article

Design of Positive, Negative, and Alternating Sign Generalized Logistic Maps

1Engineering Mathematics and Physics Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt
2Nanoelectronics Integrated Systems Center, Nile University, Cairo 12588, Egypt
3Electronics and Electrical Communications Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt

Received 1 April 2015; Accepted 2 June 2015

Academic Editor: Agustin Martin

Copyright © 2015 Wafaa S. Sayed et al. 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.

Abstract

The discrete logistic map is one of the most famous discrete chaotic maps which has widely spread applications. This paper investigates a set of four generalized logistic maps where the conventional map is a special case. The proposed maps have extra degrees of freedom which provide different chaotic characteristics and increase the design flexibility required for many applications such as quantitative financial modeling. Based on the maximum chaotic range of the output, the proposed maps can be classified as positive logistic map, mostly positive logistic map, negative logistic map, and mostly negative logistic map. Mathematical analysis for each generalized map includes bifurcation diagrams relative to all parameters, effective range of parameters, first bifurcation point, and the maximum Lyapunov exponent (MLE). Independent, vertical, and horizontal scales of the bifurcation diagram are discussed for each generalized map as well as a new bifurcation diagram related to one of the added parameters. A systematic procedure to design two-constraint logistic map is discussed and validated through four different examples.