Mapa logístico

O mapa logístico ou aplicação logística é uma regra matemática que associa a um dado número um outro número através da equação:

onde é um parâmetro. Ele é um exemplo de mapa discreto, sendo comumente utilizado na introdução à teoria do caos. Foi descrito pelo biólogo Robert May[1] em 1976 como um modelo populacional para insetos, com sendo o número de indivíduos no n-ésimo intervalo de tempo, e como uma taxa de crescimento da população.

Além de funcionar como um modelo populacional, através do trabalho pioneiro de May, o estudo das dinâmicas deste mapa passaram a poder ser aplicadas em diversas áreas como biologia [2][3], ciclos econômicos [4], eletrônica [5], geração de números aleatórios [6], análises de espectro de energia [7], análise numérica [8], criptografia [9], entre outras áreas.

A explicação para a tantas áreas apresentarem aplicações para o mapa logístico é a simplicidade de sua função, polinomial de grau 2, somada à enorme variedade de dinâmicas apresentadas, principalmente aquelas associadas à dinâmicas caóticas.

Referências

  1. ↑ May, Robert M. 1976. "Simple mathematical models with very complicated dynamics." Nature 261(5560):459-467.
  2. ↑ Bruce E. Kendall. Spatial structure, enviromental heterogeneity, and population dynamics: Analysis of the coupled logistic map. Theoretical Population Biology, 54(TP981365), 1998.
  3. ↑ Sabrina Borges Lino Araujo, Ana C. Rorato, Daniela M. Perez, and Marcio R. Pie. A spatially explicit model of synchronization in fiddler crab waving displays. PLOSONE, 8:1–11, 03 2013.
  4. ↑ J. Miśkiewicz and M. Ausloos. A logistic map approach to economic cycles. (i). the best adapted companies. Physica A: Statistical Mechanics and its Applications, 336(1), 2004.
  5. ↑ Madhekar Suneel. Electronic circuit realization of the logistic map. Sadhana, 31(1):69–78, 2006.
  6. ↑ S. C. Phatak. Logistic map: A possible random number generator. Physical Review E, 51(4):3670–3678, 1993.
  7. ↑ Curado and Rego-Monteiro. Thermodynamic properties of a solid exhibiting the energy spectrum given by the logistic map. Physical review. E, Statistical physics, plas- mas, fluids, and related interdisciplinary topics, 61 6 Pt A:6255–60, 2000.
  8. ↑ Chris L. Bresten and Jae-Hun Jung. A study on the numerical convergence of the discrete logistic map. Communications in Nonlinear Science and Numerical Simulation, 14(7):3076–3088, 2009.
  9. ↑ Yuqin Luo, Jin Yu, Wenrui Lai, and Lingfeng Liu. A novel chaotic image encryption algorithm based on improved baker map and logistic map.

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