American Institute of Mathematical Sciences

Advanced
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

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
[1]

Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621
[2]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297
[3]

Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017
[4]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709
[5]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692
[6]

Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979
[7]

Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381
[8]

Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271
[9]

Gernot Greschonig. Real cocycles of point-distal minimal flows. Conference Publications, 2015, 2015 (special) : 540-548. doi: 10.3934/proc.2015.0540
[10]

Tiziana Cardinali, Paola Rubbioni. Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020152
[11]

Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181
[12]

Mircea Sofonea, Cezar Avramescu, Andaluzia Matei. A fixed point result with applications in the study of viscoplastic frictionless contact problems. Communications on Pure & Applied Analysis, 2008, 7 (3) : 645-658. doi: 10.3934/cpaa.2008.7.645
[13]

Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467
[14]

Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248
[15]

Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273
[16]

Mark S. Gockenbach, Akhtar A. Khan. Identification of Lamé parameters in linear elasticity: a fixed point approach. Journal of Industrial & Management Optimization, 2005, 1 (4) : 487-497. doi: 10.3934/jimo.2005.1.487
[17]

Grzegorz Graff, Piotr Nowak-Przygodzki. Fixed point indices of iterations of $C^1$ maps in $R^3$. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 843-856. doi: 10.3934/dcds.2006.16.843
[18]

Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793
[19]

Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881
[20]

G. A. Swarup. On the cut point conjecture. Electronic Research Announcements, 1996, 2: 98-100.

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences