Dynamics of the fuzzy logistic family doi:10.3934/dcdsb.2010.14.699
Juan J. Nieto - Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago de Compostela, 15782, Spain (email) Abstract: 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, attracting point, repelling point, fuzzy real number, quadratic family, logistic family.
Received: March 2009; Revised: September 2009; Published: June 2010. |
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