Chaos on Fuzzy Dynamical Systems

Abstract

Given a continuous map f:X→X on a metric space, it induces the maps f¯:K(X)→K(X), on the hyperspace of nonempty compact subspaces of X, and f^:F(X)→F(X), on the space of normal fuzzy sets, consisting of the upper semicontinuous functions u:X→[0,1] with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems (X,f), (K(X),f¯), and (F(X),f^). In particular, we considered several dynamical properties related to chaos: Devaney chaos, A-transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics).

Orbit Tracing Properties on Hyperspaces and Fuzzy Dynamical Systems

“…Jardón, Sánchez and Sanchís studied, in [10], the interplay of the topological transitivity between the systems (X, f ) and (F (X), f ). This work was extended in [11] by studying different chaotic properties, such as Devaney chaos [12], Atransitivity for a Furstenberg family A, Li-Yorke chaos [13,14] and distributional chaos [15] (see, e.g., [16] for a survey on chaotic properties), also extending results in [7] concerning Devaney chaos for linear operators on complete locally convex spaces; see also [17][18][19][20] for the study of dynamical properties of the Zadeh's extension on the space of fuzzy sets. Moreover, some applications in the computer science of fuzzy sets can be found in [21] and the references therein.…”

Kato Chaos in Linear Dynamics

“…Barragán et al [32] investigated the dynamical properties of the dynamical system (SF n m (X), SF n m ( f ). The relationships among the dynamical systems (X, f ), the hyperspace systems (K (X), f ) and fuzzy systems (F (X), f ) were studied by Martínez-Giménez et al [33].…”