On distributional chaos in non-autonomous discrete systems

Abstract

This paper studies distributional chaos in non-autonomous discrete systems generated by given sequences of maps in metric spaces. In the case that the metric space is compact, it is shown that a system is Li-Yorke δ-chaotic if and only if it is distributionally δ ′ -chaotic in a sequence; and three criteria of distributional δ-chaos are established, which are caused by topologically weak mixing, asymptotic average shadowing property, and some expanding condition, respectively, where δ and δ ′ are positive constants. In a general case, a criterion of distributional chaos in a sequence induced by a Xiong chaotic set is established.

Some Kinds of Distributional Chaos for Non-Autonomous Discrete Systems

“…The related concept distributional chaotic pair as two points for which the statistical distribution of distances between the orbits does not converge, and Schweizer and Smital (1994) proved that the existence of a single distributional chaotic pair is equivalent to the positive topological entropy (and some other notions of chaos) when restricted to the compact interval case. Since then, distributional chaos has been widely concerned in dynamical system theory (see Smítal and Štefánková, 2004;Balibrea et al, 2005;Martínez-Giménez et al, 2009;Liao et al, 2009;Oprocha, 2009;Li, 2011;Dvorakova, 2011;Wu and Chen, 2013;Shao et al, 2018). Smítal and Štefánková (2004) showed that the two notions of distributional chaos used in the paper, for continuous maps of a compact metric space, are invariants of topological conjugation.…”

“…Moreover, the weighted shift operator exhibits uniformly distributional chaos and this property is preserved under iterations (in ). In 2018, Shao et al (2018) showed that three versions named However, most of the literature studies chaos in autonomous systems (X, f), but in reality, a lot of systems do not have good properties as autonomous systems. Different disturbances need the different functions to describe, suggesting that many systems in engineering practice are non-autonomous systems.…”

Some criteria of chaos in non-autonomous discrete dynamical systems

“…Then, a new concept of weak coupled-expansion for a transition matrix was introduced for NDSs, and several criteria of chaos induced by weak coupled-expansion for an irreducible transition matrix were established in [21]. Recently, Li-Yorke δ-chaos for some δ > 0 and distributional δ ′ -chaos in a sequence for some δ ′ > 0 were proved to be equivalent for system (1.1) in the case that (X, d) is a compact metric space and f n are continuous maps for all n ≥ 0 [11]. Consequently, those criteria of strong Li-Yorke chaos can be regarded as criteria of distributional chaos in a sequence for system (1.1).…”

“…has attracted considerable attention recently [1,2,7,10,11,12,14,16,17,19,21,25,26], where {f n } ∞ n=0 is a sequence of maps from X to X, with (X, d) being a metric space. Many complex systems of real-world problems in the fields of biology, physics, chemistry and engineering, are indeed non-autonomous, putting the model (1.1) into focus.…”

Inheritance of $ {\mathscr F}- $chaos and $ {\mathscr F}- $sensitivities under an iteration for non-autonomous discrete systems

“…However, many complex systems occurring in the real world problems such as physical, biological and economical problems are necessarily described by NDSs. It is more difficult to study dynamical behaviors of NDSs than those of ADSs in general [11]. Therefore, there is a strong need to study and develop the theory of NDSs, which is more involved than ADSs [8].…”