Dynamics of the fuzzy logistic family

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September 2010, 14(2): 699-717. doi: 10.3934/dcdsb.2010.14.699

Dynamics of the fuzzy logistic family

Juan J. Nieto ^1, , M. Victoria Otero-Espinar ^1, and Rosana Rodríguez-López ^1,

1.	Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago de Compostela, 15782, Spain, Spain, Spain

Received March 2009 Revised September 2009 Published June 2010

Abstract
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In this work, we study the global dynamics of the fuzzy quadratic family $F_a(x)=G_a(x,x)$, where $a \in\mathbb{R}$, $G_a(x,y)=ax(1-y)$, and $x, y \in E^1$ are elements of the set of fuzzy real numbers. We analyze the set of fixed points of $F_a$ and the behavior of each fuzzy number $x \in E^1$ under iteration by $F_a$, with $a>1$. For $0 < a \leq 1$, we study some stability properties for the fixed points of $F_a$ in $[\chi_{\{0\}}, \chi_{\{1\}}]$. We observe different types of attractors, including chaos. We show that our formulation includes and extends classical results for the real quadratic family, since the set of crisp fuzzy numbers is invariant. Finally, we present some applications and physical considerations in relation with the logistic family.

Keywords: Fixed point, periodic point, fuzzy real number, attracting point, logistic family., repelling point, quadratic family.

Mathematics Subject Classification: Primary: 03E72, 04A72, 26E50, 26E25, 74H40, 74H55; Secondary: 93C1.

Citation: Juan J. Nieto, M. Victoria Otero-Espinar, Rosana Rodríguez-López. Dynamics of the fuzzy logistic family. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 699-717. doi: 10.3934/dcdsb.2010.14.699

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