Sensitivity and strong sensitivity on induced dynamical systems
Publish place: Iranian Journal of Fuzzy Systems، Vol: 18، Issue: 4
Publish Year: 1400
نوع سند: مقاله ژورنالی
زبان: English
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شناسه ملی سند علمی:
JR_IJFS-18-4_006
تاریخ نمایه سازی: 30 خرداد 1400
Abstract:
Given a metric space X, we consider the family of all normal upper semicontinuous fuzzy sets on X, denoted by \mathcal{F}(X), and a discrete dynamical system (X,f). In this paper, we study when (\mathcal{F}(X), \widehat{f}) is (strongly) sensitive, where \widehat{f} is the Zadeh's extension of f and \mathcal{F}(X) is equipped with different metrics: The uniform metric, the Skorokhod metric, the sendograph metric and the endograph metric. We prove that the sensitivity in the induced dynamical system (\mathcal{K}(X),\overline{f}) is equivalent to the sensitivity in \widehat{f} :\mathcal{F}(X)\to \mathcal{F}(X) with respect to the uniform metric, the Skorokhod metric and the sendograph metric. We also show that the following conditions are equivalent:\item {\rm a)} (X,f) is strongly sensitive;\item {\rm b)} (\mathcal{F}(X), \widehat{f}) is strongly sensitive, where \mathcal{F}(X) is equipped with the uniform metric;\item {\rm c)} (\mathcal{F}(X), \widehat{f}) is strongly sensitive, where \mathcal{F}(X) is equipped with the Skorokhod metric;\item {\rm d)} (\mathcal{F}(X), \widehat{f}) is strongly sensitive, where \mathcal{F}(X) is equipped with the sendograph metric.
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Authors
D. Jardon
Academia de Matematicas, Universidad Autonoma de la Ciudad de Mexico, Calz. Ermita Iztapalapa S/N, Col. Lomas de Zaragoza ۰۹۶۲۰, Mexico City, Mexico.
I. Sanchez
Departamento de Matematicas, Universidad Autonoma Metropolitana, Av. San Rafael Atlixco ۱۸۶, Col. Vicentina, Del. Iztapalapa, C.P. ۰۹۳۴۰, Mexico City, Mexico