On stronger forms of Devaney chaos

TL;DR

This paper introduces stronger forms of Devaney chaos called $\\mathscr{F}$-Devaney chaos, explores their properties, and establishes conditions linking these forms to classical chaos, enhancing understanding of chaotic dynamics in metric spaces.

Contribution

It defines and analyzes new stronger chaos concepts, compares them with classical chaos, and provides conditions under which chaos implications hold.

Findings

$\\mathscr{F}$-sensitivity is redundant in infinite spaces without isolated points.

Examples differentiate various $\\mathscr{F}$-Devaney chaos types.

Conditions are established linking classical Devaney chaos to $\\mathscr{F}$-Devaney chaos.

Abstract

We define and study stronger forms of Devaney chaos and name it as $\mathscr{F}-$Devaney chaos, where $\mathscr{F}$ is a family of subsets of $\mathbb{N}$. Examples of maps which is $\mathscr{F}_t-$Devaney chaotic but not $\mathscr{F}_{cf}-$Devaney chaotic, $\mathscr{F}_s-$Devaney chaotic but neither $\mathscr{F}_t-$Devaney chaotic nor $\mathscr{F}_{cf}-$Devaney chaotic are discussed. Further, we show that for the maps on infinite metric space without isolated points, $\mathscr{F}-$sensitivity is a redundant condition in the definition $\mathscr{F}-$Devaney chaos. Here $\mathscr{F}=\mathscr{F}_s, \: \mathscr{F}_t, \: \mathscr{F}_{ts}$ or $\mathscr{F}_{cf}$. We also obtain conditions under which Devaney chaos implies $\mathscr{F}_s-$Devaney chaos or $\mathscr{F}_t-$Devaney chaos. Next, we define the concept of $\left(\mathscr{F}, \mathscr{G}\right)-P-$chaos and obtain conditions under which $\left(\mathscr{F}_1, \mathscr{G}_1\right)-P-$chaos implies $\mathscr{F}-$Devaney chaos for different families $\mathscr{F}_1$ and $\mathscr{G}_1$.